1953'^ ■?JV^ eiMEROF CALCULUS fyxmll Ultttomitg pitotg BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 LlsIij'a lh/.pl?J>.Z..., Cornell University Library 3 1924 031 252 103 olin.anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 2521 03 A PRIMER CALCULUS B^- J- ^'^ ARTHUR SI HATHAWAY PKOFESSOE OF MATHEMATICS IN THE EOSE POLYTECHNIC INSTITUTE, TEBEE HAUTE, IND. 5Ieu3 JBork THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., Ltd. 1901 All rights reserved CoPYKiGrib, 1901, '^^''^■^-t-t-^ By AETHUR S. HATHAWAY. MooEE & Lan-gen Printing Co.. TEKBE HAUTE, IND. PREFACE. This Primer has been written to meet the needs of the author, first, for a primary course in the calculus, and secondly, for an outline of topics in a more advanced course that is suitable for combined lecture and text book instruction. The author's method of development is essentially Newton's method of fluxions, as presented by Hamilton in his Elements of Quaternions, Bk. Ill, ch. II. This method is clear, logical, and scientific, and it deserves more recognition than it has received in general analysis, if for no other reason than that it is the method of the original discoverer of the calculus. Its failure to be adopted is due to want of early publication and defective notation, since it is remarkably perfect and general in principle. The subsequent discoverer, Leibintz, gained the field by publications in a desirable notation, although founded upon inferior infinitessimal principles. Lagrange attempted a modification of the infinitessimal into the idea of a principal part as determined by first terms of expansions, and made the "differential co- efficient" the primary quantity. Modern text books have returned to Newton's method of limits as applied to Lagrange's differential co-efficient; there is here offered a complete return to Newton, with the fluxion or difierential as the primary quantity. The point of view of our development is that difiier- entiation is an arithmetical process and that its resulting difierentials are numbers like other numbers, which are classifled as independent or dependent variables ac- iv PREFACE cording to the like character of the variables from which they are derived. The usefulness of a process consists in its practical applications, but nothing is gained by- attempting to show practical utility before the funda- mental principles and rules of differentiation are fairly mastered. It must be accepted at first that differentia- tion is of as much higher order of practical value than the usual processes of arithmetic as it is in advance of those processes in respect to fundamental ideas. Also, the student will have more confidence in the use of the calculus when he learns it first as a rigorous and exact arithmetical method. The following course of lessons brings the student as rapidly as it is desirable into practical applications. One object of the author has been to discourage empirical acquisition through illustrated examples worked out in full. If a student is not able to follow out careful instructions as to how to do his work, without having it done for him, he is lacking in the first elements of an engineer at least, and it is time that he began mental training in that direction (even if it is a little hard.) Attention is called to the note on page 6 which explains a general principle of notation based on da;2 = dx . dx. While unusual in the case of trigonom- etric functions, yet it is clear; i.e., there is no conflict between sin x'' = sin a; . sin a; and sin.a;2 = sin (a;*). It also removes several inconsistencies in trigonometric notation that many students do not understand. The ordinary notation may be used, however, if desired. ARTHUR S. HATHAWAY. PRIMARY COURSE Lesson Chaptee Abticles Examples 1 I 1-10 1 2 11-16 2-4 3 17-19 5-10 4 20-23 11-15 5 24 KEVIBW 17-22 6 n 25-35 1-6 7 80-38 7-12 8 39-43 9 39-43 36-41, 30-32 10 44-46 11 44-46 REVIEW 42-59 12 47-51 1-7 13 52, 59 23, 30-33, 36-39 14 60 43-56 . 15 53-55, 61 13-17 ,40-42,48,49 16 62 REVIEW 1-59 17 65 65-78 18 ni 69-75 1,2 19 76-79 3-15 20 17-21 21 80-82 22-25 p 19 p 42 p 42 p 79 PRIMARY COURSE 22 rV 109-112 1 pll4 23 24 25 26 27 28 29 V 125-130 1-7 pl24 30 131-135 31 136-139 1-10 pl34 For collected rules of differentiation and anti-differen- tiation, see pages 138, 139. EEVIBW 109-112 1 113-114 2-3 115-119 4r-5 120-122 6-10 123-124 11-13 14^17 18-19 KEVIBW CONTENTS CHAPTER I — DiFPEEENTIATION ABTICLES PAGE Preliminary Concepts and Definitions . 1-24 1 Examples 19 CHAPTER II— Principles and Rules Principles 25-35 23 Rules 36-46 29 Anti-DlfEerentiation 47-62 34 Examples 42 Reduction Formulas 63 46 Indeterminate Forms 64 49 Inverse Principle 1 65 51 Expansions in Series 66 54 Maximum, Minimum 67 56 Remainder in Maclauren's Theorem . 68 60 CHAPTER III — CoNCEETB Representation Functions Connected with a Variable Point of a Plane Curve and their Differentials . 69-79 63 State of Change of a Function . . . 80-82 74 Curvature 83-84 76 Differentiation of Directed Quantities . 85-86 78 Examples 79 Curve Tracing 87-98 87 Envelopes 99-108 96 viii CONTENTS CHAPTER IV— Integration Summation 109-H3 103 Integration 114-123 106 Potential 124 113 Examples 114 CHAPTER V Successive Difierentiatlon . . . .- 125-127 119 Partial Differentiation .... 128-130 121 Examples 124 Successive Integration .... 131-140 126 Examples 134 Rules of Differentiation 138 Rules of Integration . 139 A PRIMER OF CALCULUS CHAPTER I Differentiation 1. Variable and Constant Quantities. Letters that do not denote special numbers, as do the letters jr=:3.14159. . . and e=2.71828. . ., but which stand for undetermined numerical values in a given problem, are called variables or constants, according as their values are considered to change in that problem oraiot. Unless the contrary is stated, first letters of the alphabet will denote constants, as a, 6, c, . . . , and final letters of the alphabet will denote variables, as w, v, w, x, y, 2. 2. Independent and Dependent Variables. The independent variables are those whose values are assigned at will, each without reference to the value of any other variable. The .dependent variables are those whose values depend upon and are determined by the values of one or more of the independent variables. Thus, X, y being independent variables, then u=x ^ , v=y ^ , w=^x y are dependent variables. Every problem in which vari- ation is possible has a certain number of independent variables ; the remaining variables are dependent, and in a general sense, each is expressible in terms of the independent variables and constants, so that always as many equations connect all the variables as there are dependent variables. 3. Functions. A variable whose value depends upon and is determined without ambiguity by the values of 2 A PRIMER OF CALCULUS certain other variables is called a function of those varia- bles. Thus a; 2 is a function of x, and y^ is a function of y, and xy is a. function of x and y. In general, any expression that involves several variables, whose value is computable by means of the values of its component ■variables, is a function of those variables. Conversely, any function of given variables can have, for its repre- sentation, an expression involving the variables upon which it depends. Suppose, for example, a function were known for which no expression in terms of its variables existed ; then it would be proper to make an expression that should always stand for the value of the function corresponding to the values of the variables given in the expression. This was the case, for example, with the logarithmic and trigonometric functions when they were first considered, and the special symbols log, sin, cos, tan, sec, etc., have been introduced as charac- teristic symbols for these functions, so that sin x denotes the value of the function whose characteristic symbol is sin, corresponding to any value of x. 4. We shall often use letters as characteristic sym- bols of undetermined functions, and not as undetermined numbers, particularly the letters /, F, because each is the first letter in the word "function." Thus fx will denote an undetermined function of x that can be chosen as we please, and/Cs, y) and undetermined function of x and y. To make fx one function or another is to identify f with the process or characteristic of the function. Thus /a;=sin a; makes /^sin; fx=x^ makes /=" square of"; fx=x^ — 2a;+3, makes /=" square of, minus the double of, plus 3 ";./(«, y)=3x^-\-2xy-^^ makes f DIFFERENTIATION 3 stand for computing three limes the square of the first variable named, plus twice the product of the first and second, minus the square of the second. 5. Oeiginal Values. The original values of the variables are those which we suppose designated by the symbols of the variables, so that they are either unde- termined values, or else the numerical values that may ' be assigned to such symbols; thus x, y, x^, y^, xy, are undetermined original values, and a?=3, 2/^4, a;2=9, 2/2=16, xy^^l2 are assigned original values. The ad- vantage of leaving the original values of the variables undetermined or literal, when carrying out processes of computation with them, is that one such literal devel- opment involves the results of all possible assumptions of numerical value. 6. New Values. New values of the variables will be new symbols of the variables (in general the old symbols accented) that stand for undetermined or de- termined values that are in general different from the original values. Thus, if x, y be original values of two independent variables, then k', ^' would denote new values of those variables, and x'^, y"^, xy are the original values, and a/ 2, y'^, x!'if are the new values of the squares and the product of the independent variables. As numerical cases, taking x=Z, y=4, we could then take cc'=4, y'^7, so that 3, 4, 9, 16, 12 are original values of the independent variables and their squares and product, and 4, 7, 16, 49, 28 are new values of the same. Again instead of these new values y=4, y'=7, take new values a;'=3.1, 2/'=4.3, that are nearer the original values x=B 2/=4, and the corresponding new 4 A PRIMER OF CALCULUS values of their squares and product are 9.61, 18.49, 13.33, which are also nearer the original values than at first. When we speak of the values of the variables, without qualifying them as new values, we always mean the original values. The original values of the variables denote, in other words, the values oi the variables that are being considered. The new values are temporary values that are to be considered as approaching the orig- inal values ; and they are introduced, and made nearer and nearer the original values, for the purpose of deter- mining some questions of variation of the variables at their original values, just as, in order to determine the motion of a train at a given instant, it is practically necessary to consider its motion for a very small time thereafter, with the knowledge that greater and greater accuracy is attained the smaller this time is taken, so long as it can be accurately measured together with the corresponding distance passed over by the train. 7. Differences. The changes of value of variable quantities from original to new values are called differences of the variables. A difference is denoted by prefixing the Greek letter delta (A) to the symbol or literal value of the variable. Thus Aa;="the difierence of x"=a/ — x, A3/="the difference of y"=y' — y. A(a;2)="the difference of a;* "=a;'i'— o;2. A(3/2)="the difference of y« "=2/'?— 2/". A(»^)="the difference of xy^^=^'i/ — xy. DIFFERENTIATION 5 In the numerical cases cc=3, i/=:4, and a;'=4, 2/'=7, we have, Ax=l, A2/=3,-A(a;2)=7, A(2/2)==33, A(a;i/)=16; and for the new values a;'=3.1, 2/'=4.3, that are nearer the original values, we have the smaller differences, Aa;=.l, A2/=.3, A(a;2)— 61,A(v''')=2.49, A(a;2/)=1.33. 8. The difference of an Independent Variable is a New Independent Variable. In other words, if x be an inde- pendent variable, then Ax=:cr;' — x, may be any change of value we please. In fact the new value, a;', depends upon the original value x, and the change of value As, viz., x'=x-\-^x= original value plus the difference or change of value. The' value of As depends neither upon the value of X nor upon that of any other variable, but can be taken whatever value we please. If, however, x be what is called a real independent variable, i. e. , one limited to real values only, then Ax must also be a real independent variable. In fact, generally, the limitation of all values of a variable to real values also limits its changes of value or differences to real values. 9; The difference of a dependent variable is a new de- pendent variable, whose independent variables are the origir nal independent variables and their differences. Thus take the square of an independent variable, as a;*, then when X and Ax are assigned we have x'=x+Ax,, and x'2=x2-|-2xAx+Ax2,* so that A(x2)=x'a— x'=2xAx+Ax2, which depends, upon x, A x, as required. 6 A PRIMER OF CALCULUS Similarly ^Qcy) =x'y' — xy=(^x-\- Ax)(y-^^y) — xy =xAy-^yAx-]-Ax^y, which depends upon x, y, Ax, Ay, as required. In general, if w=f (x, y), then w'=f(pi/, y') =/ (x+Ax, y+Ay), so that Aw=f (x+Ax, y-\-Ay-)—f{x, y), which depends upon x, y. Ax, Ay. It appears also from this result, that even when the variables of a function are not independent variables, the difference of such function will depend upon its variables and their differences in exactly the same way as if the vari- ables were independent. / • 10. Peopoetional Diffeeences. Equimultiples of simultaneous differences by the same real proportional factor will be called proportional differences; they are *Such symbols as Aa;,/a;, sina;, log a;, etc., which are not separable into number factors, because one of the factor symbols is a characteristic, and not a number, are equiva- lent to single symbols of number, and exponents to such a symbol should be regarded as applying to the symbol as a whole, when no parenthesis or dot intervenes to make a separation of its parts. Thus ^^=Axhx, and not the differ^ ence of x^. The latter difference is written A(x'') or A.x^. A similar symbol is A'x=A Ax= difference of Ax, regarded as a new independent or dependent variable according as x is independent or dependent. However, by force of usage, and contrary to principles of notation, sin 'x means sin a;' and not sin (sin x) , and similarly for the squares or other powers of all trigonometric functions, except for the exponent — 1. Thus sin^'a; is not sin x~ , but conforms again to general principals of notation in which /"'a: stands for "that func- tion whose f is x," so that always ff x=x. DIFFERENTIATION 7 any proportionals to the differences, if we understand the term proportion in the sense that the ratio is a real and not an imaginary number. The literal symbol of the proportional factor will be N, so that JVAs, JVAj/, iVA(a;2), NA(y^), NA(xy) denote proportional differ- ences of the variables x, y,x^, y^, xy. E. g., let x=3, y=4c, h.x=l, ^■y=^^,- JV=4, then iVAa;, JVAj/, iVA(a;2), NA(y^), NA{xy) =4, 12, 28, 132, 64. For the new values x', y',^3.1, 4.3, corresponding to Aa;=.l Ai/=.3, which are nearer the original values than before, and the larger factor iV=49, we find, N\x, NAy, iVA(x-2), A'A(2/2), iVA(x2/) =4.9, 14.7, -29.89, 122.01, 65.17. Note that although the differences have all been de- creased in value from their first values, yet the corres- ponding increase in the proportional factor has left the proportional differences about the same as before. 11. DiFFEEENTiATiON. Differentiation is the process of finding limits of proportional differences of variable quantities, as the differences tend toward zero and the proportional factor tends towards infinity. Such limits are called differentials of the variables. A differential is denoted by prefixing the letter d as characteristic of differentiation to the literal value of the variable. Thus, for independent variables, x, y, d x=differential of a;=lim JVAa:==lim N{x' — x), dy==differential of y^lim. iVAi/=lim N(y' — y). The hypotheses of this differentiation of independent 8 A PRIMER OF CALCULUS variables, x, y, are firstly, that the differences approach zero (or the new values approach the old) while the proportional factor correspondingly approaches infinity, so that the proportional differences approach limits, and secondly, that these limits are designated by dx, dy. Turning to dependent variables, we have similarly d(c(;2)=lim iVA(a;2)=lim NQii^ —a;") =lim. [(aZ+ce) . iV(a;'— a;)] =2xdx ; * d(3/2)=lim iVA(2/2)=lim N{y"^ —y^) =lim [(2/'+ yWiy'—y)] =2ydy; d(xy)=]im. NA(a;2/)=Um N(x^y' — xy) =lim [2/N (x' — x) -\-xN(y' — y)] =ydx-\-xdy. As further exercises show similarly that d(x^')=3x^dx, d- = — — J, d^x=^j^, d(x^y^)=2xyHx+Zx'^yHy. 12. To understand differentiation, and the exact signification of the resulting differentials as variable numbers, some points in the process of differentiation must be discussed more fully, and in particular they must be illustrated by numerical values. 13. The differentials of independent variables are new independent variables. In illustration, to make rfa:=5, we may take successively •Observe that since a/ approaches x, therefore xZ+x ap- proaches 2x, and also that N increases as a/ approaches x, so that JV(a/ — x) approaches the limit, dx. The limit of a product being the product of the limits of its factors, we therefore find that lim [(x'+x).N(x'—x)]=2xdx. DIFFERENTIATION 9 Aa;=l, .1, .01, .001, .... limit=0 J\r=4, 49, 499, 4999, .... limit=oo ; then iVAa;=4, 4.9, 4.99; 4.999, .... limit=5. Thus, in this way, we determine da^lim N\x=5. If instead of the above series of values of N we should take another in which every proportional factor becomes double its preceding value so that we have successively, iV=8, 98, 998, 9998, .... then with the same series of values of Aa; as before wc should find, iVAx=8, 9.8, 9.98, 9.998, .... limit=10, which gives another value da;,=lim NAx,=10, which is also double the preceding value of dx. In general dx can be determined any value we please with- out regard to the value of x or of any other variable, since the value of Ax may be assigned at will, and its series of values approaching zero, assigned likewise as we please, in respect to value and law of contin- uation, so. that whatever series of values approaching infinity may have Ijeen already assigned to N, we can make the proportional difference NAx take a series of successive values that will approach any limit we please. Take, for example, Ax= -—, which gives, as N in- creases indefinitely, a corresponding value of Aa; that is 10 A PRIMER OF CALCULUS approaching zero as required; then the series of values of N Ax will be N^=a, a, a, . . . . whose limit approached is a. Or take Ax=aN/(^N^-\-5)=a/(N -{--—) which approaches zero; then JV"A2;=ai^2 /(iV«+5)=a/'(l+ A), which approaches a. There are, in fact, an innumerable number of different ways of making each independent difference approach zero, and the common proportional factor approach in- finity, so that the proportionals of those differences shall each approach any assigned Value we please. If, how- ever, we are considering a real independent variable, x, then since iV and Aa; are real; therefore NAx is real and must approach a real value. In words, the differential of a real independent variable is a new real independent variable. 14. Understand that in the differential process such as Aa;^l, .1, 01, .001, . . . . , a,ndN=4, 499, 49, 4999, in which the limit of NAx is sought, we do not consider Ax as ever actually zero, or N as actually in- finity, so that we are not trying to find a value of "infinity times zero." In fact, a little common sense will show that since neither zero nor infinity are any actual values, therefore "infinity times zero" is a phrase that is in itself meaningless. Nor can this DIFFERENTIATION 11 phrase be given a definite meaning in accordance with the usual acceptance of zero as denoting the nominal limit of a value that becomes smaller and smaller without limit of smallness, and of infinity as the nom- inal (but not-existing) limit of a value that becomes larger and larger without limit of largeness. Since one factor of a product can become smaller and smaller, and the other factor larger and larger, so that the product shall approach any value we please, it foUows that even these limit ideas of zero and infinity cannot give determinate significance to infinity times zero. Should the student see cause, from these facts, to ob- ject to the independent differentials as too indeter- minate in value for mathematical consideration, then the same objection would be equally vaUd against any independent varia1?les, and against the whole idea of variation of value, which must be founded on the initial idea of certain indeterminate values, which can receive or change value at will, and of other related values which depend upon these undetermined or inde- pendent values. 15. The differential of a dependent variable is a new dependent variable that is dependent upon and determined by (i. e., a fu/nction of) the independent variables and their differentials. This result is a matter of definition and as a test of differentiability. For the only way in which the limit of the dependent proportional differ- ence might be changed in value, without changing the values of the independent variables and their differentials, would be to take different series of values of the inde- pendent proportional difierences, but not so as to change 12 A PRIMER OF CALCULUS their assigned limits ; and when such variations of ap- proach, alone cause variations in the hmit of the de- pendent proportional difference, we may consider that there is no definite limit or differential^ and that the de- pendent variable is therefore non-differentiable. The test of differentiability is therefore the determination of the dependent differential solely in terms of the inde- pendent variables and their differentials. 16. For example, x^ is differentiable, because d(x^)= lim iVA(x2)=lim (z'+s). N(x' — x)^2x(lx, an expres- sion that is definitely obtained in whatever way we suppose a/ to take an indefinitely continued series of values that approach the hmit x at the same time that N takes a corresponding series of larger and larger values so that Nipc" — X) approaches the limit dx. Similarly xy is differentiable, because we find invariably lim NA.(x y)=liai Q/N^Jc-\-xNAy')=y dx-\-x dy, in whatever manner A^, Ay approach zero and iV^ ap- proaches infinity so that we have lim NAx=dx, lim NAy^dy. It will be found that all ' ' continuous' ' variables,* for which expressions known to the student exist, are differentiable, except that in some expres- sions, for certain values of the variables involved, it *" Continuous " means varying by small amounts when the variables change by small amounts, the dependent change approaching zero when the independent changes do so. The expression /a;=a;4- integer part of x, is not continu- ous at integral values of x; viz. when x' increases towards 2,fx' increases toward 3, but/ 2=4, so that /a/ — fx does not tend to vanish as x' approaches x=2. DIFFERENTIATION 13 may happen that the value of the differential is ambig- uous. This will be shown in the differential expres- sion itself, so that it need not be regarded as affecting the general differentiable charg,cter of the variable in question. 17. The differential of a fuTwtion of one or more varia- bles is the same fwnction of its variables and their differen- tials, whether the variables are all independent or one or more of them are dependent. This is a consequence of the definition of differentia- bility which makes a function, w=f{x, y) that is differ- entiable have a differential, dw= lim NMo= lim N [/ (jc-j-Ajc, y-\-/S,y') — -/ (x, yj] , that is a definite expression in terms of x, y, and dx= lim NAx, dy=liin NAy; say d'w=f'(x, y, dx, dy). If such a result holds when x, y are independent vari- ables, so that we have arbitrary methods of making lim NAx=dXj lim NAy^dy, in which dx, dy, are arbitrarily selected values, then it must all the more be true when we have only certain dependent methods of making lim NAjc=dx, lim NAy=dy, where dx, dy are dependent values. In other words, the dependent methods of approach, and the dependent limits, are included among the arbitrary methods of approach and the arbitrary limits. Thus d(i/2)= lim (y'-\-y)NAy=2ydy, whether y is independent, so that lim NAy=dy is also independent, or whether y is dependent, so that lim NA=dy is also dependent. In the latter case there 14 A PRIMER OF CALCULUS remains the finding of dy in proper terms, from the value of y in terms of the independent variables, before the differentiation of y^ can be considered as com- pleted. Similarly d(m/)=.ydx-\-xdy, whether x, y are independent or dependent variables ; and if We have actually y=x^, so that dy=2xdx, then xy=x^ and d(x^)=x^dx-{-x .2xdx==Bx^dx. Although this result is obtained indirectly yet it must verify directly. Thus, N^(x»)=N(af^—x»)=(x'^^afx-\-x^')N(x'—x) whose limit [as a/ approaches x and N increases so that N(3f — x) approaches da;] is easily seen to be 3a;'da;. Again, in d(x y)=y dx-\-x dy, we can put y=x^ so that d y^=3x^ d X, and making these substitutions for y and dy, we find d ( a; * ) ^=a; * dai-l-a;. 3a; * da;=4a; ' dx. Let the student verify this result directly, and also go over the differentiation of the product xy which gives the value d(xy)=ydx-\-xdy, and try to find how any supposed dependence of y upon x could do more than make dy correspondingly dependent upon x and dx (assuming, of course, that the given dependence of y upon X makes it a differentiable function.) 18. The differential of a given function is therefore seen to be a fixed rule for differentiating that function, even when its variables, instead of being simple inde- pendent variables, are any complex functions of other variables. Thus from d(u^)==2 udu, we have it equally true, by replacing u by x^-{-y^, that DIFFERENTIATION 15 d(x2+2/'')''=2(a;«+2/2)d(x2+2/2), which finally reduces to since we will find that It is on this account that rules of differentiation become important and of wide application, whether expressed in terms of one set of letters or another, since it will be indifferent what letters are employed to denote the variables. In fact the more important rules are best memorized in words. 19. Thus, d(xy')=y dx-\-x dy, is in words: The differential of a product of variables equals the sum of the products consisting severally of the differential of each factor into the remaining- factor. This rule extends, also, to a product of any number of factors, e. g. , dixyz)^^zdx-\-xzdy-\-xydz; etc. To prove this, let the product xyz change to x' y z, then to t! y' z, then to x' y' 2'. This is a succession of partial changes of value due to first changing x alone, then y alone, then z alone, and the sum of these partial changes equals the total change. Thus, a/y'2' — xyz=(x'yz — xyz')-\-(oiy'z — x'yz)-\-(x''i/z' — a;y«) or A(a; 2/ z)=y z^x-\-x'zAy-\-x!y'Az. 16 A PRIMER OF CALCULUS Multiplying this equation by N and remembering that d=lim NA, and we find, lima/=a;, lim3/'=y, etc., d(x y z)=y z dx-\-x z dy-\-x y dz. 20. When the vahws of the variables of a function are assigned, then the value the differential of the function varies proportionally with the values of the differentials of its variables. For, let x,y,—, be the assigned values of the variables, and w the corresponding value of the function; then Aw will be assigned when Aa;, Ay,... are assigned, and limi\'Aw is determined when lim JVAa;, limiVAj/,... are determined. If the latter limits be made x^, y^,— and the former consequently becomes Wj, then to make the latter change proportionally to faj, ky^,--, we have only to take new multipliers each k times as large as before, with the same values of Aa;, Ay... as before, since lim kNAx=k lim iVAx=i=fc x^ ,lim kNAy=:k lim NAy=ky^ , etc. But in this method of approach each Am; remains the same as before, and the limit of the new propor- tional difference is lim k N\ w=k lim iVA w=k w ^ . In other words, if Wj, x^, y^,-- be corresponding values of dw, dx, dy,--, and we change dx, dy,— proportionally to new values kx^, ky^,— then dw changes in the same proportion to the new value kw^ . DIFFERENTIATION 17 21. Since proportional factors must be real num- bers, it follows that the proportional factor k of the preceding proposition must be real and not imaginary. An important consequence of that proposition is that: In the differential of a function of one real variable, the differ- ential of the variable appears only as a factor of the result, or, d foc;=f' X dx where /'a; is a function of x called the differential co- efficient of fx as to X, and also, the derivitive oi fx as tox. In fact x being assigned, if any two values of dx are in the ratio k : 1, (where k must be real because x and therefore dx are real variables) then the corresponding values of dfx are in the same ratio by Art. 20; thus the quotient dfx/dx does not change value when dx changes value; and thence this quotient depends on the value of x alone, so that it is some function, fx, of x. 2i. The theorem of Art. 21 does not hold for all functions, when the variable is not limited to real values. Thus if z=x-\-yV — 1 be an imaginary variable whose real components are cc, y, then mod 2= ^(x8 +2/2) is a function of z, whose differential will be, as the student may verify by the work in full, d mod z=(x dx-\-y dy) /mod z. If this differential contain dz=dx-\-J — \.dy as a factor only, so as to be of the form /'z dx -\- J — If'z dy what- ever values dx, dy may have, then 18 A PRIMER'OF CALCULUS fz=x/modi z=y/(mod zj — 1), or x= — yj — 1 which is impossible, remembering that x, y are any real values. On the contrary d.z2=2zdz, d.28=32«d«, etc. 23. Analytical Functions. Any dififerentiable function of one variable, whose differential contains the differential of its variable only as a factor, is called an analytical function. Any function of a real variable is (art 21) an analytical function.; but for an imaginary variable z, mod z is not an analytical function of z, while z^, z^, etc., are such. 24. Derivation. Derivation is the process of differ- entiation followed by division by the differential of a variable. The result of derivation is the derivative of tfie function as to the variable, and must be a function of the variable alone if derivation is possible. In other words, derivation is a process that is applicable only to analytical functions of one variable. Derivation can have a definition of its own not depending upon differ- entiation, viz., it is the process of finding the limit of the quotient of the difference of the function by the difference of the variable as the differences approach zero, provided there is a definite limit depending on the value of the variable alone, and not at all upon the manner of approach of its difference to zero. This fol- lows from ■dfx T. NAfx .. A/a; 4- =lim ■ = hm -7=^. dx N^x Ax DIFFERENTIATION 19 Examples. I. 1. If a;=3, 2/=4, and we take successively Aa;=l, .7, .07, .007, and so on smaller and smaller, ^y=2, 1.3, .13, .013, and so on smaller and smaller, then find the corresponding series of values of A(x2), A(2/2), A(;Ky). 2. If in Ex. 1, we also take successively N^l, 9, 99, 999, and so on, larger and larger show that we thus determine dK=7, dy=13, d(cB2)=.42, d(i/2)=104, dixy)^67, and verify the last three from their literal values d(x^)=2xdx, d(y^)=2ydy, d(xy)^ydx-\-xdy. 3. If in Ex. 2 we double each value of N in its series of values, show by full numerical computation, that the values of dx, dy and also those of the dependent dififer- entials are doubled. 4. Show that, if a;=3, 2/=4, then however we make Ax, Ay approach zero and N approach infinity so that NAx, NAy, approach 7, 13, respectively, we shall have NA(x^), NA(y^), NA{xy) approaching the limits 42, 104, 67, respectively. [iVA(a;2)=iV[(3+Ax) «— 9]=6.A'Ax+Aa;.iVAx, etc.] 6. Prove that d(z^)=2zdz, d(y^)=3y^dy, d(y»)= Qy^dy. Also prove the last equation from the preced- ing ones, by putting z=y^. 20 A PRmER OF CALCULUS 6. Prove that ,1 dx ^x= x-^ 2dy yZ > "i- 4dx [so COl verify the last equatioUby' taking ad. y=x^ in the 7. Prove that d^zi=dz/2^z, dJ(x^+y^)=:(xdx+ydy)/J(x^+y^) Also verify the last equation by putting z:^x^-\-y'' in the first. Also verify the first from (^z)2=z. 8. Prove that d^(a^-\-s^)=sds/ J(a^+s^). 9. Prove that d.yi=^-^y^dy. [,^.yi=y'i-^i=(y'*-i/*)/(y'i+y'iyi-\-yi), etc., or let w=y', then w^=y*, whence 3w^dw:=4y^dy, etc.] 10. Prove that d(x^—Sx^+6x—4:')=^(x^—2x+2)dx, and that d.(a;»— 3a;2+6a^-4)*= 12(a;— 3a;2 +6x—4y(x^—2x+2)dx. 11. Prove that d(ax-\-by)==adx-\-bdy. Thence show that the characteristic d of difierentiation is distributive over a sum and commutative with a constant factor, just as if it were a number multiplier. 12. Prove that d. -^ J — . State this as a rule X x^, for difierentiating fractions. 13. Prove that d. ^= ^(2xdy—3ydx'). DIFFERENTIATION 21 14. Prove that cZ^!±5^=0, also that d [(x+ 5) 2 — x^ — lOx] =0. 15. If m, n be any given positive integers, prove that d.x "= — x° dx, d.x "^ = x"^ dec. n n State this result as a rule for differentiating powers to fixed fractional exponents. m [Leti/=x "' then y^=x^, y'^^=x'^^ and y'^ — y°-^^ a;'™ — X™, which may be written, (x'm-i-|-x'™-2x+. . . +x'x™-^+x'"-i) (x'— x) . Multipling this by N and proceeding to the limits, lim a:'=«, lim 2/'=2/, lim iV(x' — x)=(ix, lim JV (y' — y)=dy we find n^"~id2/=mx"'~idx, and divided by 2/"^ ^=x'", this 7 , 7 m c?x ^ , \s n ay / y^=m, ax / X, ordy= — 2/ — ^''^■J 16. Prove that d.x^yi=^x'^y^dx-\-^xiyidy. 17. Show that the successive derivatives of a;5_7a;4_|.4a;3_9x2-f2x— 7, are 5x*— 28x3+12x2— 18x+2. 20x8— 84x2+24x— 18, 60x2—168x4-24. 120x— 168, 120, 0. 22 A PRIMER OF CALCULUS 18. Show that the successive derivatives of (l-|-a;)* are4(l-|-a;)3, 12(l-\^y', 24(l+x), 24, 0. 19. Take various algebraic expressions and differ- entiate them by the full process, and also by rules, i. e., by examples already worked. 20. Expand (l-|-a;)* by derivation. [We know that (l-[^y=A-{-Bx-\-Cxi-\-Dx'-\-Ex*, for all values of x, where A, B, C, D, E, stand for some unknown numerical coefl&cients. Deriving this equa- tion we iind other identities, 4(l+a;)s=B+2C7a;+3Da;a+4£'a;3 12(l-fa;)2=2C7+6Z)j;+12£i«;2 24Cl+a;)=6Z)4-24£x 24=24^; Taking x=o in these equations, since they are iden- tities and so true for all values of s, we find A=l, B=4, 0=Q, D=i, E^l, and, (l+a;) *=l-}-42;+62;2 +4x'>-\-x*.] 21. Expand (l-|-a;)" by derivation. [(l+.)« = l+..+!^ ,. + ^fo-l),C^-2) ,a+.. where 2!=2.1, 3!=3.2.1=6, 41=4.3.2.1=24, etc.] 22. Expand a;» — Zx'^-\-2x — 1 in ascending powers of X — 4, by derivation. [a;3_3;c2+2a^l=23+26(x— 4)-f 9(a>— 4)24-(a^ 4) s] . PRINCIPLES AND RULES 23 CHAPTER II Principles and Kules 25. Principle 1. If two variables are always equal, or if ihey always differ by a constant, then their differentials are always equal. For, let X, y be original values of two variables and a;', y' any new values as near as we please to the original values, then the conditions are, if the variables are always equal, that y=x and y'=^^, or if the second always exceeds the first by a constant c, that y=x-\-c, j/'^aZ-f-c. In either case y' — y=x' — x, and therefore iV(2/' — y)=N(x' — x), and as the new values are made to approach the old, while N increases so that either member approaches a limit, the other member must approach the same limit, i. e., dy=dx. 26. The proof of the above principle shows under what circumstances the differentiation of equals gives equals, viz., the equation must remain true when the variables change from their original values by any cor- responding amounts, however small. This principle is therefore not applicable to such an equation as x^ — 3a;-|-2=0, which is true for certain values of X (x=l or 2), but which does not remain true, when x changes from those values. The equations to which the principle applies are of three classes, first, absolute identities, such as (x-\-y)^^=x^-\-2xy-\-y^ -j-second, limited 24 A PRIMER OF CALCULUS identities which are equal only for certain ranges of value of the variables, such as 1/(1 — x)^l-\-x-\-x^ -\-x^-\-, etc., which is true only when x is smaller than 1; and thirdly, equations that practically define one of the variables in terms of the others, such as x^-\-y^^a^ which makes y^=J(a^ — a;^). 27. An alternative form of Principle 1 is that : The differential of a constant quantity is identically zero. For a constant can be made a function of any vari- ables we please, as 2=a;+2 — x, l=x/x, etc. ; and as such a function, its change of value is zero; likewise any proportional change of value is zero, and hence the limit of such proportional change, or the required dif- ferential, is zero. 28. Inverse Peinciple 1. IJ the differentials of two variables are always equal, then the variables are either always equal or always differ by a constant quantity. Two proofs of this will be given later, one geometric, and one algebraic. An alternative of this inverse prin- ciple is: (a). If the differential of a quantity is identically zero, then that quantity is a constant. 29. Peinciple 2. The characteristic, d, of differentia^ tion, is distributive over a sum, and commutative with a con- stant factw.. In symbols, d(x-\-y')=dx-\-dy, d.ax^=adx. The proof will be left as an exercise. It is one of the PRINCIPLES AND EULES 25 first results the student would naturally notice in the practice of differentiation, and he would probably state it in some such form as, the differential of a sum is the sum of the differentials of its terms, and, the differential of the product of a constant and a variable is the constant into the differential of the variable. It is, however, important to consider it in the above form as a symbolic law of the characteristic d. The second part is really a conse- quence of the first, viz., d.2x=d(x-\-x)^dx-\-dx=2 dx, etc. 30. By the partial differentiation as to x, of a function of two or more variables x, y, etc., we mean differen- tiation as if X were the only variable, and the others were constants. The characteristics of partial differ- encing and differentiation as to x will be Aj;, d^, and as usual da;=lim iVAj;. Thus Aa;(x52/*)=^'^2/' — x^y^, dx(x'^y^)=2xy^dx. Similarly, Ay(x'^y^)=.x^y'^—x^y^, dy(xHj^)=ZxHj^dy. 31. A partial differential as to x is sirriply a special value of the complete differential corresponding to any value of ,dx', and the values dy==o, dz=o, etc., since these are the values of dy, dz, that result by making y, z, constants. Thus if d/ (a;, y, z)=f'(x, dx, y, dy, z, dz), then dxf(.x, y, z)=f{x, dx, y, o, z, d), etc. 33. Peinciple 3. The complete differential of a fuiic- 26 A PRIMER OF CALCULUS tion of severed variables equals the sum of its partial differ- entials as to each variable. In symbols, df(x, y, z) ^da,f(x, y, z)+dy f(x, y, z)-\-d^f(x, y, z). For let the complete change be made first by chang- ing X alone, then y alone, then z alone, giving the successive partial changes from fQc,y,z) to f(oc',y,z) to f(.^,y',z) to f(3f,y',z') which are denoted by ^xfix,y,zX Ayf{x',y,z), A^f(x',y',z). The com- plete change of value of the function is easily seen to be the sum of these successive partial changes of value, i. e., ^/(a;,2/,z)=Ax/(a;,i/,z)+Aj,/(a;',j/,«)+A^/(a/,jf',3). Multiplying by N, we find an analagous result for the proportional differences, which is precisely the principle for differentials we wish to prove, except that the original values of the variables in the proportional differences are in the second, x' instead of x, and in the third, a:', y' instead of x, y. However, a;', y', become ' X, y in the limit, and the proportional differences become the differentials, so that if the general differ- ential is a continuous function of its variables, xf, y' will be replaced by x, y in the differentials. For ex- ample in Nh.yf(x!,y,z) in which a;', z are treated as con- stants, when y' is very nearly y, this proportional dif- ference is by definition, very nearly dyf(p^,y,z'), and this will be very nearly dyf(x, y,z) when a/ is very near X, if the latter differential is a continuous function, i.e., if dfix,y,z) is a continuous function (of which PRINCIPLES AND RULES 27 dyf(K,y,z) is a special value obtained by making dx=o, dz=o.') Thus df (^, y, «)=4/ (a;, y, z) -\-dyf (x, y, z)-\-d, / {x, y, z), assuming, as is always the case in the calculus, that the functions considered are continuous, so that limits are found by substituting the limits of the variables. 33. That the limit of fvf as a/ approaches x is not always the same as fx, may be seen from the example fx=x-\~ integer part of x; taking a;=2 and cc' less than and approaching 2, fx' approaches, 3, but /x=4. This cannot occur when fx is continuous, since then by definition, fx'—fx approaches zero when x' approaches X, -and therefore fx' approaches^/a;. 34. When x, y, z, are real variables, or, more gener- ally, when w^f(x, y, z) is an analytical function of its- variables whether they are real or imaginary, then by Article 21 dxW/dx is independent of dx, and therefore a function of x, y, z alone. This quotient is called the partiai derivative of w as to x, and is denoted by 9w/3x, the script d being notice of partial differentation, while the denominator shows the variable of differentiation. In this notation, we have. (a). dw=g^dx+^dy+-^dz =/i («, y, 2) d^+fi (^, y, z)dy+f^ (x, y, z)dz. 35. The use of Principle 3 greatly simplifies the dif- ferentiation of many complicated expressions. In the first place it reduces differentiation to the consideration 28 A PRIMER OF CALCULUS of one variaj)le at a time, and secondly, an expression involving one variable only may be made a function of several variables by replacing selected component parts of the expression by new letters for the time being; thence differentiating as to the several variables and adding, we find the differential of the whole. Thus d.x^=d.xy, where y=x, =dx.xy-\-dy.xy =y dx-\-x dy = xdx-\-xdx = 2 xdx ; d.x^^d.x.x.x=x^dx-\-x^dx-\- 1 or ^^ < 1. 53. The following differentials are logarithmic, but their bases are not readily discovered; and it is best to include them among standard forms, da; dx dx 1 whose integrals may be verified as. ' dx dx '\ .X — a x-\-ai 38 A PRIMER OF CALCULUS 1 , X — a 54. On the contrary the differentials (similar to the above), dx dx dx ^(a^ — x^y x^(x^—a^y a;2+a2' are anti-trigonometric forms, their integrals being, . _^x 1 ^x 1 X sin 1—, -sec~i— , — tan— 1-. a a a a a d% X The anti-vers form, —777; — - = d. vers— ^ - is also J{2ax-^x^) a the anti-sin form d. sin— ^ . In fact the two in- a tegrals differ by the constant 5r/2. 55. Other logarithmic differentials are, seca;da;, esc x da;, tan a; if x, cot a; da; = cos a; dx/ sin a;. By multiplying and dividing the first three by sec X -\- tan x, esc x — cot x, sec x, respectively, the integrals are seen to be, log (seca;-|-tanx), log(csca; — cota;), log sec x, log sin x. 56. The differential — ■. t-t j— , reduces to a sm x-|-o cos x-}-c PRINCIPLES AND RULES 39 the form ^ _^_ ^ , according to the values of a, b, c, by the transformation 2=tan^a;, and therefore dz=^sec^x^ dx. To make the transformation, first put sina;=2sin Jxcos Jx, cosx=cos-Jx2 — sinjx^, 6=6 (cos^x^ -|-sin Jx*) and multiply both numerator and denominator by sec ^x^. 57. The integral i ^ , where n is a positive /c' dz , by Z — j— c successive applications of the formula, This formula may be verified by differentiation. /f X '-j^ dx, Jn X ■where /mX, /„x are entire functions of x, of degrees m, w, respectively, with real coefficients. If m is not less than n, we divide the numerator by the denominator to a remainder of less degree than n. The entire part of the quotient is integrated term by term under Inverse Rule 1. We have therefore only to consider a proper fraction of the above form, i.e., one in which m<^n. By the theory of equations, the denominator /,iX factors into real irreducible or prime factors, of the forms X — a, or (x — 6)2-|-c2, each occurring to certain 40 A PRIMER OF CALCULVS powers. By the theory of resolution into partial frac- tions, the given proper fraction fm^/fn^ will reduce to a sum of fractions each involving a power of one prime factor only in its denominator, and a numerator one degree less than the prime factor of the denominator, pro- vided, all fractions of this form be included in the sum whose denominators are divisors of fnX. The integration of any such partial fractions comes under preceding methods. The factoring of the denominator, and the resolution of the fraction into the sum of its partial fractions, are algebraic problems. 59. The Exponential Differential, d.ca''= clog a. a'^dx. This differential consists of a power with constant base, multiplied, to a constant, by the differ- ential of the exponent. It is anti-differentiated by dividing it by the product of the differential of the exponent and the natural logarithm of the base. 60. The Trigonometric Differentials, csinxdx, c cos X dx, c sec x * dx, c esc x^ dx, c sec x tan x dx, c CSC a; cot a; dx. These consist of certain trigonometric functions of a variable, multiplied, to a constant, by the differential of the variable. The integration consists in replacing these functions in each differential by the corresponding function of the same variable from which it is derived and dividing the result by the differential of the variable, and also by — 1 if the resulting function is a ' 'co' ' function. Thus, the integrals are respectively, — ccosx, csinx, ctanx, — ccotx, csecx, — ccscx. 61. Integration By Parts. Principle 3 can be reversed in integration. Its particular application is to PRINCIPLES AND RULES. 41 a differential udv, where m is a chosen variable factor and dv a, known differential. From d{uv)=ud v-\--vdu, we find I udv=uv — I ■moJm. In words, to integrate by parts, integrate as if a chosen variable factor were a constant for the first term, and complete the integration by subtracting the complete integral of the differential of the first term as if the assumed constant alone varied. The success of this method, whenit is applicable, depends upon the choice of the variable factor, which must be such that the new integral to be found is easier of solution than the given one. E.g. Cx^ e''dx=x^e'' — [2Cxe''dx =x^ e^— 2x6^-1-2 Ce^dx ^x^e^ — 2xe^4-2e^. This is by taking e^dx^=d.e^ as the known differen- tial each time, and the remaining factor as a constant in the partial integration. But if we take a'" as the assumed constant, we find Cx^ e^dx==^e»^ — i ( x^ e^ dx. and the new integral is more difiicult than the old. 62. In the following examples that give a differen- tial, followed by one or more of its integrals, it is required : first to verify the differentiation by the prin- ciples and rules of preceding articles, as an exercise in 42 A PRIMES OF CALCULUS differentiation; secondly, to obtain the integral from the differential by the inverse rules or methods of the preceding forms, as an exercise in such methods and rules. The more important integrations may be taken as fundamental forms in any subsequent examples of integration. Examples. DIPPEEENTIAIiS. INTEQBALS. 1. i8x^—9x^ + Gx—7)dx; 2a;*— 3a;8+3a;2— 7a;+8 2. iB^x + j — 9x^+\)dx; 2a;'^+12Vx— 3x8— ar-2 3. (ix—l)dx; 2a;2— a;+3, (4a;— 1)V8. 4. (4— 3a;)2da;; 3+16a;— 12a;2+3a;3, —(4—3x^/9 5. V(4a+9a;) da;/ 2(4a + 9x)^/27 6. sds/V(a*+s2); ^(a^+gS). 7. x'^'^{a-{-b3i^ydx; (a+6a;")"+V6M(A + l) 8. a;™(a+6a;«)'^ j(m+l)a+(m+nA+7i+l)6a;" \dx; a;ro + i(a+&a;»V'+^ 9. a;-^("+i)-i (a -|- 6 a;" ) '' da; ; _a^»(ft+i)(a_|_6a;")''+Van (A + 1) 10. x^dx/'(a^-\-cx^)^; a;8/(a2+ca;2)"^3a2 11. a;^da;/(a2+ca;2)~2"; a;'H-V(a24-ca;2)~ra2(A4-l) ft A+2 12. a;(a2+ca;2pda;; (a2+ca;2)^~/c (A + 2) 13. V(a''' -\-cx^)dx^^'^ — — — - . X dx, for integration. by parts; J,V(a^+<^^) + ^J T^a-^^^ PRINCIPLES AND RULES 43 14. J(a^—x^)dx; \xJ{a^—x^)-\-%r-&vci-^- a2. 16. (a2 — x^pdx^^ j-^^.x^dx, for integration ~(5a^—2x^) 7 (a2— x2)+ ^sin-i ^ 15. ^(.a^+x^)dx;^^ — V(2ax— :B2)+a-vers-i- 1 q da . 7 (2aa: — g^) ' xj(2ax — a;2y ox 20. V(2ax— x2)rfx; ^=^ V(2aa;— x2)+^vers-i^ 21 '^^ Jja^—x^) 1 log 92 t^a: J{x^—a^) . 1 _ifc^ a:37(x2— a2)' 20^x2 "T" 2a3 ®^° a 2^- x(a4-6x») ' -^^"S^"''' +*)'^^°SM=^ 44 A PRIMER OF CALCULUS 25. -J^ dx; log 2o. —. — ^^TT<"; log r-j 27- §^, d^^ -+i log '-^ - V3 tan-i ^3 „„ da; 1 ■a;2+sV2+l . 1 , _, 0:^2 2^- ^*+i' 472^°g x^^V2+l + 2V2^"^ li-» ^y- (a;a+2)2'^'^' 41og(..+2)-^^tan-^^-j±=^^ 30. dcc/ai.log a;; _ log' a; = log log a; 31. loga;da;/a;y loga;2/2^1oga;. logx/2 32. 2a;da;/a;2; log.x^ 33. , ; •( , (a function not tabulated) loga; J log a; 34. J^^dx; V(a+a;) (6+a;) + (a— &) log ( Va+a;+ V6+x) [Put 6+a;=2/'') &=22/c?2/] 35. V^. &;V(a-a;)(6+:«) + (a+6) sin-y^ 36. (f'+er^ydx; i(,e^='—(r-^+4x[) 37- J^^^^^' 21og(e- + l)-x 38. a^b^dx; a^ 6^ /log a 6 PRINCIPLES AND RULES 45 39. ^^; ^e^^^e<-\-log{e'> — l) C -1- 40. x^e'^dxy e*(a;2— 2x+2) (int. by parts) 41. x^logxdx; -J-x^CSloga; — 1) 42. sm~^xdx; Ksin— ix + «y(l — ^^) 43. sin 2 s. da;; — J cos 2 x 44. CQSX^dx; Ja;+Jsin2x 45. cosx^dx; sinx — ^sinx^ 46. (tan x + cot x) ^ dx; tan x — cot x 47. (tan2x — l)2dx; ^tan2x-[-^og cos 2x 48. e™^sin«xc?x; e™"'(msinnx — ncosnx}/(m,^-\-n^) [Integrate twice by parts with e™* as constant] 49. e™-'^cosnxdxy 6™^(nsinnx+mcos«.x)/(m2-|-n2) 50. tanx'dx; tanx — x 51. tan x^ dx ^sec x^ tan x cZx — tan x cfx ; J tan x^ -|- log cos X tan x"dx=- — — | tan x"-^ ^^ teg f. . . +(— l)"+i(tanx— x) 53, tanx^^cix, w a positive integer; tan x2"-i tanx2"-3 2n— 1 2W— 3 54. tanx^"— 1 dx, n a positive integer; tan x^"— ^ tanx^"— * , , ^ ^^ ianx^ , , -2^^=12 -2^;i=^+- + (-l)n-2- + logcosx) 65. secx* dx:=secx2 (l+tanx2)dx; tanx+Jtanx' 46 A PRIMER OF CALCULUS 56. cosa;*da;=J(l-{-cos2a;)2 dx; ■gij (12 a; + 8 sin 2a! + sin 42;) 57. cos a;* sin x* da; =|- (1 + cos 2 x) (sin 2a;) " dx = ^ (1 — cos 4a;) c?x+^ sin 23;" cos 2a; dx; X sin 4a; , 1 . „ „ 16 6r- + 48-'^''2a;« 58. cosa;* sina;sda; = cosa;* (1 — cos a; 2) sin a; da;; — ■J-cosa;5 -|.icoss'' ^^• 5-4tos2a; = J tan - 1 (3 tan a;) 60. If sinh a; = ^ (e* — e-^ ) cosh a;=^ (e* +^~*) then cosh x^ — sinha;2=zl d sinh a; = cosh x dx dcosha;=sinhxda; dsinh-ia;=da;/^ (a;2+ft2) dcosh-ia;=da;/ J {x^ — a") 63. Reduction Formulas Let v=:^(a^-\-cx^'); then we have the differential rules, d.v^ = Gn ■u"~2a;cZa; , C a; V" » f a; 'I "-1 da; These rules may be used to integrate x™ v" dx by parts in six different ways, so that the new integral shall be I x^' v'^' dx where m', n' are one or both two units smaller than m, n; and repeated applications of such integrations wiU thereiore reduce the given integral REDUCTION FORMULAS. 47 eventually to dependence upon standard forms, either algebraic, logarithmic or anti-trigonometric, when m, n are any integers positive or negative. These formulas are, (a) iv'^.x' ■^dx 'm-fl (b) ja;™-i. OT" dx _ j ^m+Xyn — en j a;™ + 2v"-2 dx |a;'»-^^"+2 — (m— 1) Cx'^-^v'^+^ dx c{n —.^ + ^dx -_ J a;m+i ^n _(_ a2 m ja^^'Bn-a j;;^; m+w+l (. J (d) Joj-^+s.-^dx = — i— rr I — a;*+^ r"+2_|_(-^ i „_|_3) jajm ^rH-2 (f^ /x '^ — ^ -.x«"H-»-i da; = — — ; T-^rrl a;™-i?7'H-2 — a^Om — 1) \x™-^v^dx c(m-{-n-\-l) (. , J (f) /«'^'^«- 1^ = —-. — —rr \ a;™+i v'^^—c (m+n+3) Cx">+^ •^ dx a^ (m+l) ( ^ ' ' J 48 A PRIMER OF CALCULUS In these formulas, a* may be changed to — a* throughout, and c is usually 1 or — 1. Similarly, letu:=smx, 'y=cosx, so that d . w" ^ nu^—hi dx, M"— 1 dx J r ■" V" f""-^ rfa; M»H-1 and then (a') iv'^^.u'^vdx = — T-j I w^+i ^"-1+ (n— 1) rM™+2 'yn-2 cfa; | (bO j M™-i.'u"wc7x = — j— - j M'«'+l|;"-l+(?l— 1) jw'W'yn-S cfa; I == — pj I — M™+i ?)«+i-|-(m-|-n+2) Tm™ i;"+2 cfs I (eO \-r-:,.if'+^^udx. REDUCTION FORMULAS. 49 = ■ ■ I — M'"-! ■W+l + rm— 1) rM»n-2 -y" dx | m-\-n (. ^ J J m+1 61. sinx8(ix==tanx5. cosx^ sinaidx; — ^tan x^ cos x^-j- f | tan x^ cos x^ sin x dx, — ^tanx^ cosx' — -i^tanx3cosx*4-|' I tanxcosxsinxdx, — -^sincc^ cos a; — -/^sinxs cosx — -j^j- sin x cos x + jw-- 62. cos X* sin X 2 dx; cosx sin X ,„, „ „ „ is , ^ 4g (3+2cosx3— 8cosx*)+ ^ 63. -^ — -7 J ; r (tt-- — J + TT-- — ■ — 4 sin x) smx^cosx-^ cosx^ Ssmx^ ' Ssmx "^ ^ -\-^ log (sec x-{- tan x) 64. Indeterminate Forms fx Rule. To evaluate '^ for a given value of x that makes fx=o, Fx=o, differentiate both numerator and denomina- tor, before substituting the given value of x; and similarly for values of x that make /x^oo, Fx=^ao . Before mnkin g such differentiations, any factor of f{x')/Fx that is not zero or infinity for the given value of x may he replaced by Us value for such value of x. 50 A PRIMER OF CALCULUS For let/rt=o, Fa:=o, then : fa ,. fx' ,. fa/ — fa ,. Afa d/a ta x'-aP^ xi=aFy! — Fa AFa dFa If/a = oo, Fa=^ao , then l//o^o, l/Fa = o, and fa ,. 11 ,. dFx , dfx == 't;- == lun -rr / -F-= hm ^^-— / -±— Fa Fx fx Fx^ ix^ = h^ dFa/dJa; i.e., h=dfa/dFa. Finally, any factor whose limit is finite, can, by the principle that the limit of a product equals the product of the limits of its factors, be at once replaced by its limit, and the limit of the remaining factor may be found by itself. Exponential indeterminate forms must be evaluated through their logarithms. E.g. y=(l-\--Y when a; = 00 whose form is 1°°, must be evaluated from logy=a;log(l + - )^log (1+z) /z where z =-=o. This is 0/0, and is therefore when z=o, dlog(l+z) /dz = ]n:^ = l; log2/=i, y^e. 64. Evaluate the following functions for the given values of x : ^,. a;8— z2— a;+l . ,j. INDETERMINATE FORMS 51 (d) logcB/Cx— 1), a;=l (1) (e) (e^+e-^)/'a;, x=o (2) ^.. xsina; — a;^ „, (^) 27 osx + x^-2 ' ^^^ ^-2^ (g) log sin x/ cos a;, a;=7r/2 (0) (h) sin~^a;/sma;, x=o (1) (i) taij x/log cos K, a;=ff/2 (oo) (j) e*(cosa; — l)/a;log(l+a;), a;=o ( — \) [Note that the factor e* can be replaced by e^ = l before differentiating numerator and denominator] . (k) secx — tanx, x=7r/2 (0) (1) (i+xM^;x=o (1) * i_ (m) (1+x) ==% x=o (00,0) (n) (l-^-mx)", a;=o (e™) (o) (logx)^, x=o (1) (p) x'°« ''+'), xr^oo (e) 65. Applications of Inverse Principle 1 65. If xdx-\-ydy=o and y=a when x=o, then x2+2/2 = a2 ! A PRIMER OF CALCULUS 66. If ——-4-^^=0 and y^=b when x:=o, 67. If -r+-r^o, and y=a when a;=o, then ck' + 2/3 = a' 68. If -f^^ — and v=o when x^o, then v^=4aa; da; 2/ ' ^ j ./ 69. If —= — =c, and r=a when S=o, then r=a e" 70. If (l+2;)8=l+3a;+3a;2+a;8, then hy integration, (l+x)*=l+4x+6a;2+42;3+x* 71. From---^ = l+a;2+a;*+a;6+... , (x24. Therefore y^5-\-S2 — 16 = 21 is the maximum value of y. Also y can be as much less than zero as we please by taking x large enough.] 79. 2/=4+(a;— 3)5— (s— 3)S. [-r = 7T7 t;^; which is zero for a;=17/'5, and "- ax 3(x — 3)i ' dy discontinuous for x=3. At a;=17/5, -p changes from dy -}- to — and 2/ is a minimum; at x=B, -p changes from — to + and y is a maximum. Also y can be as great as we please by taking z enough less than zero, and y can be as much less than zero as we please by taking x great enough, so that both the maximum and the minimum values of y, are only with reference to ad jacent values.J 80. y=a sin x-\-b coss. dy [-p =&sina; — a cos a;; tanx==a/b, y='^J(a^-\-b^). -J- -J- = — (a cosx-\-h smx) = — y. Therefore, when y is positive, -^ is decreasing, hy (b) with -p in place of y, and remembering that -5=- is zero for the given value of X, therefore it changes from -|- to — or MAXIMUM AND MINIMUM VALUE^*'^^ 59 y=^a^-\-b^ is a maximum value. When yisnega-^ dy '^'^ '' tive -~ is increasing, and being zero, it is therefore ^ changing from — to +, and y^ — V(a^+&^) is a minimum value. These are true greatest and leasts /j, values of y, since y cannot increase or decrease in- definitely.] 81. 2/=(z-fl)2 (x— 3)S. [min., x= — 1; max., x=2; min., a;=3]. 82. w=x^ -{-y'', -where lx-\-my-\-n=o. [_w =n^ / (Z^ -f-™^ )) ^ minimum] . 83. Find the largest rectanglar area that can be en- closed by a boundary of 200 feet. [2500 sq. feet.] 84. Find the largest rectangle that can be cut out of a circular sheet 6 feet in diarneter. [18 sq. feet.] 85. Find the altitude of the maximum rectangle that can be cut from an isoscles triangle, one side being part of the base. \_\ altitude triangle.] 86. Find the altitude of the maximum right cone that can be inscribed in a sphere of radius a. [Let a+a:= altitude, 2/= radius of base =J(a^ — x^); ' a; ==^ a for required maximum.] 87. Find the basin of largest volume, round or square, that can be made with a given number of square feet of tin. [Width = double the height]. 88. A Norman window consists of a rectangle sur- mounted by a semicircle. Given the perimeter of th§ 60 A PRIMER OF CALCULUS frame, what dimensions give the window that will admit most light. [Height = width]. 89. How far must one stand from the base of a column to obtain the largest angle of vision of a statue on the top. [The distance is a mean proportional between the entire height of column and statue, and the height of the column] . 68. Remainder in Maclaurin's Theorem. Lemma. If Fz, and its derivative F'z, be real and con- tinuous functions of the real variable z from z=a to z^:b, and if Fa=Fb, then will F'z be zero for same value of z between a and b. For, when z changes continuously from a to 6, Fz must in the beginning either increase from the value Fa, or decrease from that value, and since it returns to the same value (_Fb=:Fa) in the end (and does so by continuous change of value), therefore there must be an intermediate yalue of z at which Fz changes from increasing to decreasing or from decreasing to increas- ing. Let z=3f be such intermediate value of z ; then by Art. 67, Fx" is a maximum or a minimum value of Fz, and therefore F'af is either zero or discontinuous, and since the possibility of discontinuity is excluded by supposition, therefore J"a;' =o. Theorem. The Remainder in MaclaiHrin's theorem after n terms is /("V. — r, where i! is some value between o and x. n\ REMAINDER IN MACLAVRIN'S THEOREM 61^ a:" For, let such remainder be R—r, so that R is that n\ fanction of x which is given by the equation (a) Sx= The conditions to which fz and its derivitives /'z, f'z, .../f")2, must comply are that they are all real and continuous functions of the real variable 2, from 2=0 to z=x ; and we can then therefore make up from these functions, the following function Fz which, with its derivative F'z, are also real and continuous from z^o to z=x: Fz=fz+ (x-z)f'z+^^^f"z+ ... ^ (n— 1)! •' ^ n\ F'z^fz-rz+{x-z)f"z-^{x-z)f'z+^^^^f"z-... («,— 1)! since preceding terms all cancel. But Fz=fx when z^o, ii). consequence of the value of R given in (a); and Fz=fx when z=x, in conse- quence of the vanishing of every power of a; — z. Hence, by the lemma, F'z= for some value of z, between 62 A PRIMER OF CALCULUS and X, say z=z3f. Substituting this value of z in Pz and dividing out factors not zero, we find R=f^^h'. 90. Showthat the error of log (l-\-x)==x ^ + • • • =F — is between K»+V(m+1) and ^c»+V(«+l) (l+a;)"+S so that when x is positive and not greater than 1, log(l+x)^a; ^-|-... How many terms of log2=l — \-\-\ — J+--- must be taken to compute log 2 to an error between .0001 and .00005? 91. Show that the errors of j;3 /rSn— 1 sin a;=a; — -5-7 + gj-r-- -t- (-2„_)!» a;2 _ a:'''"-^ cosa;=l 2!" + ^(2n— 2)!' are ±sinex.a;2»/(2w) !, ±cos5x.x^-i/(2n— 1) ! where d is some number between and 1. Show that Cx^l" fx'l"— 1 these errors are smaller than I — /nl, x\ — yn !, respectively; and that therefore, however large x may be, n can be taken large enough so that the above approximatons to sinx, cosx are as accurate as we please. How many terms of these series must be taken to compute cosl, sinl to errors certainly smaller than .0000001? 92. ltyx=x" -\-aj^x'>^'--\-... -{-ar^ix-^ttn , where « is a positive integer, and ttj, a,, ... a^, are real numbers, show that between two real roots of /x=o, lies at least one real root of/'x=;o. CONCRETE REPRESENTATION 63 ^ CHAPTER III Concrete Representation 69. Algebraic quantities are represented by concrete quantities such as length, area, volume, etc. Negative numbers are represented only by the assignment of opposite characters of measurement, and then a negative measurement of one character means the corresponding positive measurement of the opposite character. E. g. , — 2 units to the right == 2 units to the left, — 3 units up = 3 units down, — 4 radians counter clockwise = 4 radians clockwise, etc. Imaginary numbers can be represented by directed lengths in a plane in accordance with the principle that J — 1 denotes change of direction through a counter-clockwise right angle, as ^ — 1 units to the right =1 unit up. This is applicable when the concrete quantities are such as forces acting at a point, but not for ordinary lengths, or areas or volumes. 70. The differential of a variable quantity must be a quantity of the same kind. In fact, the change of value, the proportional to this change of value, and consequently its limit the differential, must be the same kind of quantities as the given variable. In other words, the differential of a length is a length, of an area, an area, of a force, a force, etc. Concrete representation of variable numbers will therefore give corresponding representations of their differentials, and the determina- tion of the differentials from the variables is important not only for its applications to concrete problems, but 64 A PRIMER OF CALCULUS also because it gives concrete ideas of differentiation that illustrate this algebraic process and its principles. 71. Let OX, OY (Fig. 1) be horizontal and vertical axes of reference in the plane of the paper. A variable point P in this plane is determined by two variables X, y called its co-ordinates, which are respectively the measures of the distances of Pto the right, and up, from the axes. Negative measures in these directions mean positive measures in the opposite directions. The first co-ordinate is called the abscissa of P, and is OL=x units to the right (or briefly Oi=a;); the second co- ordinate is called the ordinate of P, and is LP=y units up (or briefly LP^y). If y be a definite real function of X, this means that each value of x gives one and only one value of y, or that P is represented on each vertical line by one and only one point; iiy=fx be a continu- ous function of x, then the locus of the point P is a continuous curve, crossing each vertical line not more then once. As an example of discontinuity find the locus of P from x = l to x=3 when 3/=: x -|- integer part of X. Conversely, any continuous curve drawn from left to right, and crossing each vertical line once only, would, if we understand that P always lies on this curve, make y a definite function of x. In Figure 1 the curve drawn is actually a circle of center C, and vertical radius AC. The upper half of this circle cor- responds to a different function of x from the lower half. With a certain unit of length,^ we have 0B=8, BC=d, AC=5. For P=(x, y), any point on this circle, we find from the right triangle on CP as CONCRETE REPRESENTATION 65 hypothenuse, with sides parallel to the axes, that (y_9)2 + (a;_8)2z=25, i.e., 2/=9±V25 — ( x— 8)^ are the two functions in question. 73. The curve y=fx is smooth when it has a definite tangent PT at each point P, and when the direction of this tangent changes continuously for continuous varia- tion of P. The tangent at P is defined as the limiting position of the indefinitely produced chord PP' as P' ap- proaches coincidence vnth P. This condition of smooth- ness is in fact the condition that fx is differentiable and that such differential, fx dx, is a continuous function. Continuity and smoothness are implied conditions on all curves. There may be exceptional or singular points, in this respect, but the continuous changes of value of the independent variable that are considered in general statements must not include such singular points. 73. In a given curve there are other functions of the abscissa x, of P, besides the ordinate y. Thus, let A be an assigned initial position of P on the curve, and let the tangent line PT and the normal line OP (perpen- dicular to the tangent) meet OX in M, N, and also meet a perpendicular to OP through in M', N'. Then: the arc of the curve is the arc AP=s; the ordinate area is the area ABLP:^^u (described by the ordinate) ; the slope angle, is the angle XMP=4> radians ; the slope is tan <^ ; the tangent and normal (lengths) are, MP and PN; the subtangent and subnormal are, ML and LN; 66 A PRIMER OF CALCULUS the polar radius and angle are OP==r and <.XOP=(t radians ; the polar radius area is the area OAP:=v; the polar slope angle and slope, are < 0PM ^ip radians, and tan tj/ the polar tangent and normal, are M'P and PN' the polar subtangent and subnormal, are M'O and OiV'. 71. The co-ordinates r, are poZar co-ordinates of P; the unit of measure for the angle is a. counter-clockwise radian; the unit of measure for r is the unit to the right turned through the angle 0, so that it is in the direction 6; r is therefore positive or negative according as the direction 6 is towards or from P. We generally suppose d taken so that r is positive. The unit of measure on M'N' is in the direction 6 — ^. The units on tangent and normal are in the direction <^ and <^+^; the direction 4> can be taken as the direction of increase of s. The area described, by the ordinate y is divided into positive and negative parts determined by the product of the sign of the value of y and its positive or negative direction of motion along OX. The area described by the radius vector r is also divided into positive and negative parts according to its positive (counter-clock- wise) or negative direction of turning about 0, whether '/■ is positive or negative. Conventions of sign are for definiteness of general statements, i. e., with such con- ventions, general theorems can be made holding for any position of P on its given locus, that must otherwise be CONCRETE REPRESENTATION 67 separated into several distinct theorems depending upon the position of P. In other words, results that are obtained from a congtruction in which all the quantities are positive will hold for any possible construction when proper conventions of sign are used to interpret the quantities, which would not thus generally hold when magnitude only is concerned. This is because con- ventions of sign make continuously varying quantities change from positive to negative when their magnitudes are to change from additive to siibtractive, as the posi- tion of P changes continuously. 75. To construct the differentials of abscissa x, or- dinate y, and arc s, of a given curve, for assigned values ofx, dx. Let P he the point on the curve whose abscissa is x; take PR=dx units to the right; draw the tangent at P, and draw RS parallel, to OY to meet this tangent in S; then is RS=dy units up, and PS=ds units in the direction of increase of s. For, let P' be the point on the given curve whose abscissa is the new value x'; let the new ordinate y'=L'F meet the Une PR = dx at Q; lay off on PR the length PR'=N.PQ; draw R'S' parallel to OF to meet the chord PP' in S'. Then by similar triangles, R'S'=N.QP', and since PQ^^x, QP'=Ay, there- fore, PR'^N.^x, R'S'=N.^y. The differential pro- cess \\mK\x=dx, or lim PR' =PR, consists in making Q, and therefore P', approach coin- cidence with P, while making N correspondingly 68 A PRIMER OF CALCULUS increase so that the point R' approaches coincidence with R. Two constructions are shown in the figure, the first is lettered as described and N= 3, the second is unlettered, Q is nearer to P than in the first con- struction, R' nearer to R, and JN=7. We are to imagine a series of such constructions, unlimited in number, in -which Q is taken nearer and nearer to P, with the object of determining the limit of S' knowing that the limit of R' is R. It is easily seen that S' approaches S, for, in the first place, R'S' is by con- struction always parallel to Y, and therefore its limit- ing position is a line RS parallel to OY, and secondly> since P' approaches P, and S' lies by construction on the chord PP' (produced), therefore S' must approach coincidence with a point on the tangent at P, which is by definition the limiting position of the chord PP' produced. Hence dy =lim NAy =lim R'S'^RS, where RS is a line parallel to OY, and meeting the tangent PT in S. Next produce the chords from P to each point of the arc PP'=^s, in the ratio N:l, and let arc PS' be the curve in which such extended chords terminate. The arc PS' is then similar to arc PP' by construction, and its length is N. arc PP'=N. A?/ both arcs have also the same tangent PT, since the tangent is determined by the limiting position of the same chords produced, in either case. Thus when P' is so near to P that the arc PP' is always between its chord and tangent and of one direction of bending throughout, the similar arc PS' must lie between its chord PS' and tangent PS, and be of one direction of bending throughout. Hence as P' CONCRETE REPRESENTATION 69 approaches P, the arc PS' must approach point to point coincidence throughout with the straight line PS, since PS' does so; i.e., ds=limN^s=limarcPS'=PS. Observe that the two triangular figures PQF, PR'S', each with an arc side, are similar figures, with P as center of symmetry, and N as ratio of similitude. Since PR'S' approaches coincidence with the triangle PRS, it FIG. I. I Y /IS" appears that the difference figure PQP' approaches similarity with the differential triangle PRS, as its sides indefinitely diminish. The difference figure reduces to the point P; we have left, however, in the triangle PRS what might be called its ultimate form. (a) The limiting ratio of any arc to its chord as the arc is taken smaller and smaller, is unity. 70 A PRIMER OF CALCULUS For, PP': arc PP'= PS': arc PS', whose Umit is PS:PS= 1 (b) The slope of the curve y^fx at the point (x, y) is dy/dx^f'x. For, the slope is tan (l>^RS/ PR=dy / dx. (e) Inverse Principle 1. If dw=o identically, then w=constant. [Or, if du=dv identically, then d(u — v)=o identi- cally, and u — v=^constant.'\ For the slope of y=fx being dy/dx, then if dy=o for a particular value of x, at such point (x, y) S coin- cides with R, or the tangent is parallel to OX; if dy^o for every value of x, then the tangent is parallel to OX at every point, and the curve y=^fx must be a straight line parallel to OX, so that y=^fx remains constant as X changes. This proves the principle for real functions of one real variable. More generally, if w be a real function of real independent variables x, y, then (^10, = -A- dx-\--^ dy, can be identically zero only when -;r-^o,-^=o, identically, since dx, dy, are arbitrary values. Thus w=f(x,y) is constant when either x or y changes alone, whatever value the other variable may have, and it must then be constant when both vary, since A/(a;, y) = ^Jix, y) + Aj,/(x', y)=o -{-o—o. Similarly for real functions of any number of real variables, and this includes imaginary variables, regarded as depending upon their real components. If w is imaginary, it is w^w^-\-w^^ — 1, where lOj, Wj CONCRETE REPRESENTATION 71 are real, and dw=dWj^-\-dw^. J — l=o only when ■dWj^=o, dw^=o, so that Wj, Wj, and therefore w, will he constant, if dw=o, identically. 76. To construct the differentials of the polar radius and angle, r, 6: Draw from P=(r, 0), the length PS=ds on the tangent ■at P, (Art. 75); draw SR.^ perpendicular to the polar radius OP at jRj ; with center draw the arc PjSj equal to R^ S in ■length; then is PR^=dr, R,S=rdd, y^ds where <^ is the slope angle (in radians) and s is the arc length. (a) When the curvature is zero at every point the curve is a straight line. [Jjf d=conitant']. (b) The curvature of a circle is the same at every point and equal to the reciprocal of its radius. Let be the center of a circle of radius a, A its lowest point, and P ainy other point (Fig. 1); then , and ds=ad4>, or d4>/ds=l/a. 84. The circle of curvature at a point P on a given curve, is the tangent circle at that point with the same direction and magnitude of curvature as the curve. Its radius is therefore i2=ds/(/<^, audits center C=(x,y), is distance R from Pin the direction ^-{-7:/2. HE is negative this means that the center is actually in the opposite direction, since ds/d will be positive or negative, according as <^ increases or decreases as s in- creases, i.e., according as the curve bends towards the direction -\-Tt/2 or <^ — n/2. Thus equating pro- jections oi 00 and the broken line OPC upon the axes, we find (a) x=x4-i2cos(<#>-fjr/2)=x — Rsm^x — dy/d<^. (b) ^^=y-\-RsiVi{<^-\-r. /2)^x-\-Rcos4>=x-\-dx / d^. In rectangular co-ordinates, ds=V(dx2+d2/2) = V(l+^2)dx, [p=d3//dx=tan<^]. d^=dp/(l+p^); R=(l+p^)i/(dp/dx-). 78 A PRIMER OF CALCULUS In polar co-ordinates, [jj=idrXdd^=r cot ^] ; ^z=^ + e, [triangle OMP, Fig. 1]; d^^dil>+d0=\(r^ + 2q^)de—rdq]/(_r'i+q^); i2=(r2+g2)t/(r2+2g'2— rdg/de). Differentiation of Directed Quantities 85. The differential of a directed quantity OP (Fig, 2) that varies definitely with the time, and whose values add by the parallelogram law, is a directed quantity PS, whose com- ponent PR^ along OP is the differential of the magnitude of OP, and the perpendicular component R^S is the product of the magnitude of P and its differential change of direction (in radians). According to the parallelogram law, (true for veloci- ties, forces, etc.) OP+PP'=OP', so that ^.OP=PP', JVA . OP=NPP'=PS', and when F approaches P, d. OP^ iim PS' = PS, a tangent to s=arc^P, of .length ds. Also, the components of PS along and perpendicular to OP, are PRj^=dr, R^S=^rdO. 86. The velocity of P is the time derivative of the displacement OP. Its magnitude is s, and its direction is 4>- The acceleration of P is the time derivative of the velocity. Its components are therefore s in the direc- tion , (tangential) and sd

Xds=o, or the path is a straight line. No force acting means then uniform motion in a straight line. If s^o and d^/ds=constant, then the particle is moving uni- formly in a circle, and there is an acceleration towards the center at every point, of constant magni- tude, (speed) 2 /radius. , In a particle constrained to move in a circle, this is the acceleration of the tension along the radius. It appears that the normal component of the force on a particle is the deflecting component, and the tangential component is the speed accelerator. In straight line motion, the normal component is zero, since the cur- vature is, zero. Examples III. 1. In the parabola ay^x^ find P, and construct PRS of Fig. 1, in the cases: x=2a, dx=^a; x=a, dx=2a; x^o, dx=3a; x^ — a, dx= — Za; x=—2a, dx=a. With a given value of x, what change is made in PRS by changing dx? , 2. Construct points (P) and tangents (PS) of the semi-cubical parabola ay'^ =x^ for x=o, a, 4a. Why is there no point corresponding to a negative value of x? 80 A PRIMER OF CALCULUS 3. Find one arc of the semi-cubical parabola from x=o to any value of x. 4. Find P and construct PR^S of Fig. 2 for the equi- angular spiral r=ae^, when 6=o, dO=a; 6=^~/Q, de=2a; (9=-/'3, (W=a; 0^-/2, 66==— a. Show that the radius always meets the curve at an angle of 45°. 5. Draw the points and tangents at 6=o, t^/A, 5r/'2, i-/A, -, for the curves r=2acos6, r=a cos25. 6. Find the curve in which r^a when 6=o, and whose polar radius meets the curve at a constant angle ijj=ta,n~^c. [rdd/dr=c; r=ae^/'']. 7. Find the length of the arc of the equi-angular spiral of Ex. 6, from 6=o. lds=J(l + c^)dr; s=J(l+c^) (r— a)]. 8. A point P moves so that its distance (x) from a fixed directrix OYis in a constant ratio (e:l) to its dis- tance (r) from a fixed focus F; show that a tangent to the locus between the directrix and point of contact subtends a right angle at the focus. \r=ex, dr=edx; take PS=ds, where S is on the directrix; then dx= — x, and dr= — ex= — r^PF, so that (Art. 76) F=R^, the foot of the perpendicular from S on ZP.] 9. A point P moves so that the sum of its distances DIFFERENTIA TION OF DIRECTED Q UANTITIES 81 (r, r') from the fixed focii F, F', is a constant (2a). Show that the tangent at P bisects the angle between one focal radius and the other produced. r-|-r' = 2a, dr-\-dr'^o; therefore take PR^ on FP produced for dr, whence an equal length PR\ on PF' is dr', and the perpendiculars to r, r' at i2j, R'^ must meet in a point S of the tangent at P (Art. 76). 10. If the point P moves so that the difference of the focal radii r, r' of Ex. 9 is a constant, show that the tangent bisects the angle between the focal radii. 11. If the focal radii of P as to fixed focii- F, F' satisfy the condition r-\-2r'^^a, find a construction for the tangent at P; similarly if r — 2r'=3a. 12. Find the ordinate areas, and the polar radius areas from 0, in the curves ay=x'^, ay^=x^, a^y^x^. 13. Find the ordinate area of the circle x'^-\-y^=a^ from x^o, and of the ellipse x^ /a^ -\-y'^ /b^ =1. 3C Ans. ^xj{a^ — a;^ )-)-|a2sin-i-, and & /a times the same. The areas of circle a, and ellipse (a, 6) are found by putting a;=a and multiplying by 4, giving -a^, Tzah. 14. Find the polar radius area of the circle r^a from 0=0. 15. Find the polar radius area from 6=o of the curves of Exs. 1-6. 17. Find the volume of a hemisphere of radius a, 82 A PRIMER OF CALCULUS between its base and a parallel section at distance xj also the convex surface. [ The radius of the section at distance x is y=^ (^a^ — x^), and if V be the required volume i^V=^-y\^x where T^y\ is the area of some section between the distances x and a;-|-Ax, so that \miy^:=y. Thus dV=-y^dx=T:{a'^—x-^)dx, V=r.x(a'^ — ^x^). If S be the convex surface, s its arc section by a diametric plane perpendicular to the base, then dS=2-y ds=^2-adx since s is an arc of the circle y=J (a^ — x'^) from a;=o; and S=2ffax]. 18. Find the moment of the spherical segment of Ex. 17 as to its base, and its center of volume. [The moment of volume as to a plane is, volume times distance from plane to center of volume — distances on opposite sides of the plane having opposite signs. The moment of a volume equals the sum of the moments of its parts. These suffice to .determine moment and center. Thus, momy=momF-(-mom AF, and hence A.momF=momAF^Xi AF where x^ is some dis- tance (to the unknown center of volume) between x, x', so that limXj^x, and d.momV=xdV=TaoTndV, considered as concentrated at the distance x to which it pertains. Hence dmom V=7c(a^x — x^)dx, inom.V=~(2a^x^ — x*), and distance of center of volume = (mom F) / V=-j-. k— 5 : . Take x= o to 4 3a^ — x^ make F=vol. hemisphere, etc.] DIFFEREN TIA TION OF DIRECTED Q UANTITIES 83 19. Find the volume of a right circular cylinder of radius x and altitude c and its moment of inertia about its axis. [AF^2-Xj A.t; f, where Xj is between x and x'. dV=2r,cxdx, V^^ircx^. The moment of inertia of a volume as to an axis is, the volume into the square of its radius of gyration as to the axis. Such radius is between the longest and shortest radius to volume. Also the moment of inertia of a volume is the sum of the moments of inertia of its parts. Thus A . mom-iner. V= mom-iner. A V=xl A F, d. mom-iner. V^x^ d V^ mom-iner. d V, considered as concentrated at distance X, Moment inertia F==irca;*/2; radius of gyration =x/V2]. 20. Find the volume of a cone of altitude x whose base area is a^ when x^l; also find its moment as to a plane through its vertex perpendicular to its altitude, and the distance of its center of volume from the plane. 21. Find the moment of inertia, radius of gyration about its axis, and the convex surface, of a cone of revolution, of altitude x and semi-vertical angle /?. [7rx5tani34/10,a;tan,'3V. 3, TX^sec/Stan/S]. 22. Show that when a; ^4 the quantity Jx is chang- ing one-fourth as fast as x, and that for small values of h, 2-\-lh is the principal part of ^(4+A). 23. A man walks 3 feet per second towards a tower 80 feet high. If he should continue to approach the top as at 60 feet from the base, in what time would he reach the top? 84 A PRIMER OF CALCULUS [s2=6400+2:^, dx= — Zdt; dt is required when z=60 and ds= — 100, and is 55.5 4--- seconds]. 24. Two pien starting together walk in paths at right angles, each 3 feet per second; show that one leaves the other 3^2 feet per second. 25. A vessel is anchored in 18 feet of water, and the cable passes through a sheave in the how 6 feet above water. If the cable is hauled in 18 feet per minute, what is the speed and acceleration of the vessel when 30 feet of cable are out ? [If I = cable out at start, a = horizontal distance to anchor, and s,x= same after t minutes, then (Z— s)2 = (a— 2;)2+242; ds=18dt, and when ^-s=30, we have a;=30, x=32]. 26. A particle P moves in a plane curve about a fixed point 0; find its radial and radial normal com- ponents of velocity and acceleration. Take an initial axis OX, and let {r,6l stand for a directed quantity whose magnitude is r in the direction radians fi-om OX, this symbol in particular standing for OP so that e=_ _ a J^y — J(a±yy—a^ ) which gives n ^ — ^ X a catenary. Using the expansion of e" we have approx- imately y=x^/2a, a parobola. ] Curve Tracing- 87. To trace the locus of F(x, y) =o, take a series of values of x, and for each value x^a find from the 88 .1 PRIMER OF CALCULUS equation the corresponding values y=b,b'... , and plot the points j( a, 6), (a,h'},... The points so plotted on each verticfo line x=a are the points where the several branches of the locus cross that line. When the vertical lines are close enough the form and continuation of each branch will be shown by its dotted construction. This is the primitive method; an improvement consists in drawing a short dash at each plotted point in the direction of the tangent, when fewer points are neces- sary. The tangent is drawn from its slope dy/dx which is — F^{x,y) / F^{x,y) where the numerator and denominator are the partial derivatives of F(x, y) as to X, y, respectively; viz., since F{x,y') remains zero as x, y change continuously, therefore dF{x,y)=o, or by Art. 34 (a), F^{x,y')dx-\-F^(x,y)dy^o. A further im- provement is to obtain an accurate idea of the general form of the locus from a systematic study of the equa- tion, when it will be necessary to plot the locus with care only at a few critical points. The methods of such study will be considered in detail. 88. Examine the equation for symmetry as to axes and origin. The test of symmetry is that the substitution, for the co-ordinates (x, y), of the co-ordinates of the symmetric point, in the equation of the locus, must leave the equation unaltered. E.g., the locus of ^4-^ = 1 is symmetric as to the a;-axis because changing (x,y), into (x, — y), the symmetric point as to the a; axis, leaves the equation unaltered. Similarly, this locus is sym- CURVE TRACING 89 metric as to the y-axis because changing (x,y) into ( — x,y) leaves the equation unaltered; and it is sym- metric as to origin, because changing (x,y) into ( — X, — 2/) leaves the equation unaltered. In general, the substitution, for the co-ordinates x, y, of the co- ordinates of the symmetrical point, in any equation, gives the equation of the symmetric locus ; e.g., the locii of 2/2 = 8x+12 and y^=—8x-\-12 found by replacing (x,y) in the first by ( — x,y) are symmetric locii as to the y-axis since if any point (a, 6) satisfies the first, then ( — a, 6) satisfies the second. 89. Examine the equation for limits of real value of X and y. If y is imaginary when x lies between a and 6 then no part of the locus lies between the vertical lines a;=a and x=6; for although the imaginary value oiy is in such case an algebraic solution of the equation, and (x, 2/)) is a point of the locus in an algebraic sense, yet no point in the plane of representation corresponds to it. E.g.. in x^ /a'^-\-y'^ /h^=l, where a,b are real numbers, if a;'>a'' then y is imaginary, and if y^>b^ then x is imaginary; hence the locus lies be- tween the vertical lines x=±a, and the horizontal lines2/=±&. 91. Directions to and at infinity. The direction whose slope is the limiting value of y/a;=tan^ as the point (x, 2/) approaches infinity on a distant branch is the direction to infinity of that branch. The direction at infinity of the branch is the direction given by the limit of the slope dy/dx=taji at the distant point (x,y) on the branch. This is identical with the direction to 90 A PRIMER OF CALCULUS infinity; for ]imy/'x=]iindy/dx by the theory of indeterminate forms, when both x and y approach infinity, and when ]imy/'x=o or oo in consequence of 2/ or a; approaching a finite limit then also limdy/'dx=o or oo. E.g., if for a finite value a; we have y^oo , then xf being near to x, 3/' will be finite, and Ay^y' — 2/=°°) so that dy=ao when dx is infinite, and dy/dx=oo=yXx. It appears that a distant branch with a limiting direction is very nearly a straight line of slope Iimrf2//da;=lim2//a;. A spiral winding indefinitely around the origin and extending indefinitely outward is an example of a distinct branch with no direction to or at infinity. 92. Asymptotic Line. When all the points of a distinct branch approach more and more nearly coinci- dence with the distant points of a given straight line, such line is the asymptote of the branch. To have an asymptote, it is evident that the branch must have a direction to infinity; also, the tangent to the branch at a point approaching infinity must have this asymptote for its Umiting position, so that the asymptote is a line tangent to the branch at infinity. If (x, y) be a point on the branch approaching infinity, then excepting vertical branches for which \\va.y/x=vi , we shall have \miy/x=m, the slope of the asymptote, and the ordinate of the asymptote that passes through the point (a;, y) on the branch wiU be mx-\-b, where h is 2/-intercept of the asymptote. The condition that the branch approaches coincidence with the asymptotic line is that the difierence of their ordinates to the same CURVE TRACING 91 abscissa, or (mx-{-b) — y, opproaches zero when x approaches infinity. Thus 6=lim (y — mx). Denoting y — mx by q, we have then to put y^mx-^q in the equation of the locus and in the resulting equation for q in terms of x find the limiting value of 5 as a; approaches infinity, under the condition that q is to be finite. If the equation between q and x is algebraic, we divide it by the highest power of x and find the limiting equation by putting cc^co. The value of q from this equation is the required ^/-intercept of the asymptote. 83. For example find the directions to infinity and the asymptotes of xy^ — x^ — 2ay^-\-a^=o. Divide this by a;* and put a;=oo ; if y be supposed finite we obtain 0=0, which finds no finite value of y when x = oo ; if y/x be supposed finite and equal to m, we find m'^ — 1 = 0, which gives two directions to infinity of slope 1 and — 1 respectively. Divide the equation by 3/' and put y^cc ;{{ x be supposed finite, we find X — 2a=o. This is therefore a vertical direction to infinity whose asymptotic line is x=^2a. To find the asymptotes of slope m^l or — 1, put y=mx-\-q in the given equation, and it becomes, since m^^l, 2 (mq — a)x^-\-(q^ — 2amq)x — 2aq^-\-a^=o; dividing this by x^, we see that if q remains finite as x increases indefinitely, we must have in the limit) mq — a=o or q = a/m=ma. Thus y^=m (^x-\- a) ^±{x-\- a) is the equation of the asymptote of slope m=±l. (a) Show that, tTie terms of highest degree in the equation of a locus, when equated to zero, give the equation of the lines X, 92 A PRIMER OF CALCULUS to infinity through the origin; viz., in the above example, xyi — a;3=o, are such lines, etc. 94. Examine the eqvMtion for regions of rising and failing branches (from left to right') and resulting crests and hollows. In other words, note from the equation where y in- creases, where y decreases, and where it is at a maximum or minimum value. If the equation does not show this readily, it is determined by the values of (x, y) that make dy/dx positive in the first case, negative in the second case, and where dy/dx is chafaging sign in the other cases. 95. Examine the equation for regions of concavity v/pward, concavity downward, and consequent points of inflection, In other words, note where the tangent turns counter- clockwise as its point of contact advances to the right (which is shown by the slope p=d2//da; increasing or by dp/dx positive) where the tangent turns clockwise (which is shown by the slope p=dy/dx decreasing or by dp/dx negative) and where the tangent is changing direction of turning (which is shown by dp/dx chang- ing sign.) 96. Examine the equation for multiple points, and trace the locus in ihe neighborhood of a multiple point. A multiple point is a point where two or more branches of the locus intersect; it is a double or triple point, efc., according to the number of branches. CVEVE TRACING 93 At such a point p=idy/dx= — F^(,x,y) /F^(x,y'), (Art. 87) must be correspondingly multiple valued, which can only be (excluding discontinuity) when this fraction is o/o for the point (x, 3/). Thus a multiple point must be a solution of the simultaneous equations F(x,y)=o, F^(x,y)=o, F^{x,y)^o. For such a point we have by the theory of indeterminate forms, p= — dF^(x,y)/dF^{x,y') which becomes, after re- placing dy/dx by p, and x, y by their values at the multiple point, a quadratic for p. If this quadratic is determinate the point is a double point; and the branches intersect, or touch, or are imaginary, accord- ing as p has two different or equal real values or two imaginary values. In the latter case the multiple point is an isolated point of the locus with no real point next to it. In general, if (x^, y^) is the double point, we put x=x^ -\-h, y^y-i -\-k in the equation of the locus, and trace the locus for small negative and positive values of h. This is, in effect, transforming the axes to parallel axes through the multiple point, with h, k as co-ordinates to the new axes. It is easily shown that the terms of lowest degree in the resulting equation for (A, k) when put equal to zero, will be the equation of the tangents at the multiple point, which is now made the origin; and this equation determines at once by its degree, the order of the multiple point. If m be the slope of one of these tangents, we trace the branch to this tangent by putting k=mh-{-q in the equation of the locus, aud determine from the resulting equation in q, h, the principle part of q for small positive and negative values of h, and thence the position of the 94 A PRIMER OF CALCULUS corresponding point (h, k) above or below the tangent, according as q is positive or negative. 97. Trace the locii of the following equations, taking convenient lengths for the constants. 2ay=x^ — 2ax-{-db; iia^y=x0—3ax^+3(a^±b3)x+a»c (b>o) a^y^=a'^x*—x^. 2/2=x3/(2a — a;), the ctssoid; if OP intersect the ver- tical Ijne a;=2a in Q, and the circle on the abscissa 2a as diameter in R, show that OR=PQ. ay^=x»; ay^ = (x—ay(x—b) (a >,<,=&). xi-\-yi=a^, the hypocycloid described by the rolling of a circle of radius a/4 inside a circle of radits 0.4:=o, the tracing point being on the circumference of the rolling circle and on the axes when in contact with the fixed circle. y=8a^ / (x^ -\-4a^), the witch; draw a circle with' vertical diameter 0A=2a; draw a line from to meet the circle in R and the tangent at A in Q, when the abscissa of Q and ordinate of R are the (x, y) of a corresponding point P of the witch. j;3 _|_ 2/3 = Sax^ / x^ -\-y^ =Baxy ; !c*+3/* = o3y x* — 2a^y^ay^. (x" +2/2)2 ^a* (a;? — y2)^ ti^g lemniscate, if r, / be radii to P from focii F, F' on the x-axis such that CURVE TRACING 95 F0=^0F'=a/J2, then r/=a2/2; with polar radius r from 0, r^ ^a^ cos 26. a - -- X y^jr{e^-{-e ")=acosh -, the catenary. 98. The same methods may be employed in tracing the locii of polar equations. In looking for symmetry the symmetric of (r, 6) has several forms that must be tried separately: e.g., the symmetric points of (r,6) as to OX are (r, — 0), ( — r/w — 6), etc. A direction to infinity is a value of that makes r^oo , and the cor- responding asymptote is found from the limit of the polar subtangent OM'=r'^dd/dr, whose direction of measurement is 6 — r./2. A direction 6 that makes r=o is the direction of a tangent at the origin, since that is the limiting direction of the chord OP as P approaches 0. The locus recedes or approaches the origin as 6 increases if r^ is increasing or decreasing, and it is concave towards or from the origin according as i&-a.\p=rdd / dr increases or decreases with 6. The following equations are given to trace : e r=a6, the spiral of Archimedes; r^ae", the equi- angular spiral. r=2asin^, r=2acos^, r=a sec26, r:^a/9. r=asec-jr ; r=o sin 26, r==a cos 26; r=a sin ^ . r^=a^cos2d; r=^a(l-^coBO), the cardioid. r=a-\-h CSC 0; r^a (sec26-f- tan 2ff); r=a^ CSC 0^-\-b^ sec^^. r=acos^+&sin^; r^a cos2i? + ^sin2^. 96 A PRIMER OF CALCULUS Envelopes 99. Let three variables x, y, t be always connected by a given equation F(x, y, €) =0/ then to a given point (z, y) corresponds one or more numbers (the numbers of P) which are the solutions of the given equation for t in terms of the given values of (.x,3/). We assume every point of the plane to be so numbered by this equation. The equation F(x,y,t')=o may be the equation of any locus we please in the plane by select- ing for t a proper functional value t=f(,x,y). In other words, consider any given locus in the plane, and select from point to point of that locus one of the numbers of each point so that this number varies continuously with the position of the point; then the continuous assemblage of such numbers form a definite function i=/(a;, y). It is obvious, on account of the multiplicity of the numbers of each point, that it may be possible to find different functional values of t such that for each, F(^x, y,i)=o shall be the equation of the same locus. Since dF(x,y, i)=o on such curve, we have (using the notation of Art. 34 for partial derivations as to the first, second, and third variables) (a) F^ (x,y, t)dx-{-F^ (x,y,t)dy-\-F^{x,y,t)dt=o. This is an equation for the slope dy/dx at the point (x, y) on the given curve, remembering that dt is of the form Ldx-\-Mdy depending upon the given locus. 100. The n-curve. The locus of all points having the same number n is the w-curve. Its equation is F{x,y,n)=o. Thus the o-curve is F(x,y,o)=o, the ENVELOPES 97 i-curve is F{x,y,T)^o, etc. The slope of the n-curve is given hj t^n, dt=ova. 99a and is (a) f 1 (x, y, n)6jXi-\-F^ (x, y, n) dy=o. 101. The self -intersections. The points (x,y) that simultaneously satisfy F(x,y,n')^o F(x,y,n)^o may be called the n'. n points, because they are each points of number n' and n; thej'^ are the inters ections of the m'-locus with the n-locus. The limiting positions of these intersections as n' is taken nearer and nearer and to its limit n are the n.n points, or self-intersections of then-locus. To find these n.n points we must replace the n'-locus by another that always intersects the w-locus in the n' . n points and only those, and that does not reduce to the n-locus itself when we put n'=n. This locus is given by [^F(x, y,n') — F{x,y,n)'\/{n' — n)=o. since any point (x. y) that satisfies this equation and F(x,y,n)=o, will also satisfy Jf (x,y,n')=o and so be an n'. n point, and conversely every n'. n point satisfies the above equation. Taking n as the original value and n' as the new value of t, so that the equation is, A tF(x,y,t) / At=:o, we see that its limit is F^{x,y,n)=o. Hence (a) The self-intersections of an n-locuk are its intersections with the locus of the partial derivative of its equation as to its number (regarded as the original value of the variable t); i.e., the n.n points are the values of (x,y) that simul- taneously satisfy F(x, y, n) = o, F^ (x, y,n)^o. 102. By solving the preceding simultaneous equa- 98 A PRIMER OF CALCULUS tions for (x, y) we find the co-ordinates of seif-intersec- tion of the n-locus each in terms of the number n of that locus; then by giving n all values, we find the assemblage or locus of all n . n points for all values of n, or the locus of self intexsections. By eliminating n between the above simultaneous equations (by solving one for n and substituting its value in the other) we evidently obtain the equation of the locus of self intersections in terms of (x,y) alone. We may select the variable function t^f(x, y), so that F(x, y,t):=ois the equation of the locus of self-intersections, viz., f(x,y) is a solu- tion of F^ (x, J/, i) =0 for t in terms of x, y. The slope of the locus of self-intersections is then given by 99(a) which reduces since Fg(x,y,t)=o to (a) F^ (x, y, t) dx-\-F^ (x, y, t) dy=o. 103. The multiple points of an n-locus are points of self-intersection of that locus. For at a multiple point (s!,2/) of the n-locus, F(x,y,n)=o, we have also Fi(x,y,n)=o, Fc^{x,y,7i)^o (to vcieike dy/dx=o/o'). Thus, substituting t=n in 99(a) which is true for all values of (x, y) of number t, even when dt is not zero, we find that F^(x,y,n)=-o; i.e., by 101a, the multiple point (a;, y) on the n-locus is a point of self-intersection. We divide the locus of self-intersections into that of ordinary points and that of multiple points of the n-locus. 104. The hens of ordinary self-intersections is met tangentiaily at each point by the Vrlocus on which that point is a sdf-intersecton. For the slope of the locus of self- intersections at such point (x, y) whose number is n, is ENVELOPES 99 found by making t=n in 102a, and since the differ- ential co-efficients are definite non-zero values (because the point is an ordinary point on the w-locus) therefore this slope is the same as the slope of the n-locus, given by 100a; i.e., the two locii meet tangentially. This result does not hold on the multiple point locus, since then Fj^(x,y,n)=o F^{x,y,n)=o; and in general dy/dx=o/o signifies that the value of dy/dx at such limiting point depends upon the manner of approach of (x, 2/) to their limiting values, and is otherwise absolutely indeterminate. Now on the n-curve, we are to find the limit of — F^ (x, y, n) /F^ (x, y, n) as (x, y) approaches its limit on the locus F(x,y, 71)^=^0; and on the multiple point locus we are to find the limit of — F^ (x, y,t')/F^ (x, y, t) &sx,y,t approach their limiting valued wherein f is a variable approahing n and con- ditioned by F^(x,y,i)=o. These are certainly differ- ent methods ot approach and give in general different limiting values for dy/dx. E.g., on the w-curve {y — n)*:=(a; — a)* the point a;=a, y=n is a multiple point whose locus, as n varies through all values, is the vertical line x^a. This is the only locus of self- intersections, as may be shown by eliminating n be- tween this equation and the ■n-derivative, y — n=oJ and its slope is dy/dx=cc at every point. On the contrary, the n-curve has two branches y — n=±(x — a)i that meet to form the multiple point at a;=a, y^n, and on these branches dy/dx=±'^(x — a)* whose limit when X approaches a is zero, i.e., the slope of every n-curve is zero at its multiple point, and it therefore meets the multiple point locus everywhere at right 100 A PRIMER OF CALCULUS angles — quite the reverse of tangential meeting. In general any definite motion of a curve with a multiple point, as a lemniscate which is a figure 8, generates a system of n-curves, in which n may be taken as the time at which the generating curve is an «-curve; and such motion can be so determined that the locus of the multiple point shall meet the «-locus at any angles we please, constant or varying with the locus. 106. Jf a given locus in met tangentially at every point by an n-locus through that point, then it is a locus of self- intersections. For, take the equation of the given locus as F(x,y,t')^o where the variation of* with (x,y) is determined by the condition that <=ri at the point of tangency (a;, y") of an n-locus. The condition of tangential meeting at (x, y) is then that the slope given by 99a when t=n\& identical with the slope given by 100a. Thus, making t^n and subtracting, remembering that i is a variable so that dt=o for any continuous series of values of (x, y) on the given locus is inadmissible, we find F^(x,y,n)=^o i.e., any point {^,y) of the given locus is a self-intersection. The complete locus that satisfies the above condition of tangency or envelop- ment by the n-curves will be called the envelope of the system of «-curves. This envelope will not in general include the multiple point locus and will be simply the locus of ordinary self-intersections. 106. Let F(x,y,t')=o be the equation connecting the volume x, pressure y, and temperature t of a unit mass of gas; then the n-curves of this equation are the so-called isothermal lines of the gas of temperature n. ENVELOPES 101 For the so-called perfect gas xy^at, and the isothermals are hyperbolas. An intersection of two isothermals of different temperatures implies an unstable condition of the gas, and is in general impossible. 107. Let F(x,y,z):=o be the equation of a surface in •which XOYib a horizontal plane, and z is the height at the point x,y; then the n-curve F(x,y,n)=o is the contour line, on the plane XOY, of points on the surface of altitude n. In contour maps, we have also no inter- section of contour lines of different altitudes, because to each point (x, y) corresponds only one altitude number. 108. Find the envelope of the following systems of curves, m, p, q, t, etc., denoting variable parameters. For straight line systems draw also a sufficient number of lines of each system to show the envelope graphically. (a) y^mx-\-a/m,; y^=4ax (b) (2/— »nx)2=a2m2±62; ai^/a^ ±2/2/62 = 1 (c) X cos t-\-y sin t — a=o; x^-\-y^=a^ (d) If points Q, E^ move uniformly along straight lines OA, OB, show that QR envelopes a parabola. [s/J+i//(ai+6)=l; (aa;+2/)2+26(aa;— 2/)+62=o] (e) Find the envelope of a line of constant length (a) moving with its ends in the axes. [x/p-\-y/q=l, p^-{-q^=a^, treat p, q as functions oft, then on the envelope xdp/'p^-{-ydq/'q'^=o, pdp-{-qdq=o, which gives x/p^=yXq^ by eliminating dp, dq. To elimin- nate p, q denote for the moment the common value of the members by r; then substituting in preceding equation gives r=l /a 2, etc. Ans. a;S+2/S=a*] 102 A PRIMER OF CALCULUS ( f ) Particles are started from the origin with equal speeds in varying directions S, in a vertical plane; find the envelope of their paths. [x=^atcos0, i/=a«sinS — gt^/2, and the path is y—xta,ne—gx^/2a^cos0'i. Ans. y=a^ y2g—gx^ X2a] (g) Find the envelope of the variable ellipse 3;2/p2_|_2^2/g2_.l Qf constant area 7ia^ (M='^^)! also the one of fixed director circle (p^-\-q^=a^) ; also the one in which p-\-q=a. Ans. 4x^y^=a^; (x±y)^=a^; a^-\-yi=ai. (h) Find the envelope of the normal to the parabola 2/2=4aa;. [y^m,{x — 2a) — am^; 27ay^=4:(x — 2a)8] (i) Show that the self-intersection of the normal of a given curve at z, y is the center of curvature of the curve at (x, y). [y — y= — ^(x — x)Xp where x is the variable para- meter, y a function of x given by the equation of the curve, and p=dy/dx. Ans. x=a; — pO^-\-p^)/(x,Ax), then lim S^/(a;, Ax) =lim S^ <^ (x, Ax). For the difference of these integrals is the limit of the difference of the corresponding sums, which is limS^[/(x,Aa;)— ^(x. Ax)], by Art. 112a; and this integral is zero by the preceding lemma, since by hypothesis limiV[/(x, Ax) — ^(x,Ax)]=o identically. While the two integrals are identical for any same putn of integration from a to u, i. e., for the same method of approach to continuous variation from a to u, yet each may change value with change of path of integration. INTEGRATION 109 118. Notation. Since the value of the integral depends only upon the corresponding differential and the path of integration, whether the difference of the sum has one or another of the many values whose proportionals approach the given differential, therefore the integral is named most definitely as ' ' the integral oj the differential over the given path." Since we can write lim%lf(x,^x)=\im-S,lN-^.limNf(x,Ax) by multi- plying by N~^N=1 at each stage of the approach, we therefore find limS-iV-i= 1 say, as the characteris- J a tic of integration of the differential from a to u; and if lim Nf(x',^x) = lim N^ (x, Ax) =...=' a; da;, the identir cal integrals limS"/(a;, Ax)=limS'„'^{!«,^»:)^-" , are <^'xdx:= "integral from a to w of a ^xdx." 119. Theorem 3. The. integral of the differential of a fandion is independent of the path of variation between given limits, and is equal to the total change of value of the function between the limiis. /U lU dx = \ ^x. a \a For since limiVA<^x:=d<^a;, therefore by the notation established by Th. 2, and by Th. 1, I da;=lim S"A^a;=lim(<^w — ^a)==<^M — 0a. From this result follows d <^x = "a function whose differential is" a I 110 A PBIMER OF CALCULUS du=d~^. d<\>u, and thai vanishes when u=a] i.e., Since d!=lim i\'A therefore, formally, d~i=lim A— ij\-i =lim2iV-i= J , from S=A-i. This result shows that the formal relations of d, j , are consistent, i. e., in ac- cord with established facts. 120. Inverse Prin. 1. If dx — ijix a, constant for variations of a;. For trom the given identity and Th. 2 d^x= \ d\j/x a J a i. e., by Th. 3, 4>u — a=fu — xl/a, or u — ^m=<^o— ■i/'O, a constant for variations of x=m. 121. The preceding results extend to variations, sums and integrals, in any number of variables. Thus a variation of (x,y) will be a series of sumultaneous changes of (x, y) from given initial values (a, 6) to any final values (u,v); such variation determins a series of values of (Az, Ay) and a series of values of any function f(%, Ax, y,Ay). There is no change in any of the pre- ceding theorems and proofs except the slight changes consequent upon the introduction of the additional letters required for the values, changes of values and limits of the additional variables. The differentials in real variables x, y, etc., are of the forms /' X . dx, /i (x, y) dx -\-f^ (x, y) dy, etc. INTEGRATION 111 The differential f'xdx where f'x is a continuous function of x is always a perfect differential dfx. E. g., iif'x is real, then draw the curve y=f'x, when the ordi- nate area, fx, of that curve from a;=a to any final value ot X, is a function of x whose differential has been shown to be dfx=ydx=f' xdx, (Art. 77). The differential in more than one real variable is not, however, in general perfect, since this requires that the differential co-efficients be partial derivatives of a given function of the variables. The differentials in two real variables x,y include differentials in an imaginary variable z=x-\-yJ — 1. When the differentials are imperfect then their integrals between given limits (a,h), (u,v) depend upon the manner of continuous variation, which is called the path of integration. When the dif- ferentials are perfect their integrals are independent of the paths of integration and functions of the final values of the va,riables (the initial values being given con- stants). A path of integration is determined when the corresponding values of the variables are determined in terms of one real variable, since this reduces the difier- ' erential to a perfect differential in that one variable. 133. As illustrations, take (x, y) as the co-ordinates of a point P in the plane XOY, then the path of inte- gration is shown by a path of P from its initial to its final position. In this case the imperfect differential y dx is the differential area described by the ordinate y, and I y dx along a given path is the total, continuously described, ordinate area of the path. For, P, P' being 112 PRIMER OF CALCULUS two successive positions (x, y) (a;', if) of the point on the path between its initial and final positions, then the ordinate area PLL'P'=y^Ax, where j/j is some ordi- nate between y= LP and 'i/=L'P', is the typical difference f(x, Ax, y, Ay) whose sum is the total ordinate area. Thus S^'^yj As denotes the total ordinate area, described by any n successive changes along the given path. As we take n larger and larger, such area is described more and more nearly continuously, i.e., by sums of smaller and smaller differences, so that y dx represents the result of con- tinuous summation pf ordinate areas described by con- tinuous motion along the path. If along' the given <^' zdx a which can be determined when a function <^ can be found such that dx:^tj>'xdx viz., it will be Iu {j>x^(x,y,z) as to ^,y,z, and — ^ (x, y, z) is called the potential of the field on the particle P. The existence of such a function was dis- 114 PRIMER OF CALCULUS covered by Lagrange; several years later Greene pointed out that it represented potential energy, or energy of position in the field with reference to the initial posi- tion; viz., — (x,y,z)=c is the equation of the surface of potential c, and c is the amount of work that will be done by the field in moving the particle along any path from this surface to the zero potential surface, vis., jdw = jd (x, 2/, 2) = I ix, y,z)=c. Examples. 1. Find the sums from x=o to a;=4 of x^^x, (a;2-|-Ax) Ax, A.a;3/3, for uniform variations of 2, 4, and n changes. Also represent in each case the terms of the sum by rectangles in the plane XOY, ofordinates y=x^, 2/=a;2-|-A2;, y^A(^x^}XAx respectively, and base Ax. (Representation by ordinate areas). 2. Show that the limit of each sum in Ex. 1 for r!=oo is 64/3; also verify that the proportionals of the dififerenences approach the same diiferential x^dx. Show that in the ordinate area representation, the common limit of these sums is the ordinate area of the parabola y=x^ from x=o to a;=4. 3. Find lim 2*a;Aa; by uniform variation; also geomet- rically by its ordinate area representation; also by the fundamental theorem of integration (Art. 119). 4. Find by direct integration the functions whose dififerentials are u du, u^ du, and that vanish when w. vanishes. Ans. m2/2, m3/3. (See Art. 114). INTEGRATION 115 xdx== -r ; also by Th. 3 and d. x^ /'2=^xdx. 6. Verify by ordinate area that | ^(a^ — x^)dx= 7ra2 /4; also by Th. 3 and Ex. 14 p. 43. 8. If the ordinates of two curves to any same abscissa are in a constant ratio h/a, then their ordinate areas between the same bounding ordinates are in the ratio h/a. 9. If parallel chords between two curves vary as their distances from a fixed point, then the area of a segment between two chords as bases is that of the rectangle on the altitude and the half sum of the bases. [A chord at distance « is ex, and the area between /w cxdx^= a (u — a) (cu-{-ca)/2.'] 10. If the areas of parallel sections of a tubular surface vary as the squares of their distances from a fixed point, find the volume between two parallel sections in terms of the bases &i, 63, middle section b^, and altitude h. Ans. A(6i+462+63)/6. 11. Find the volume of a cone or pyramid in terms of its base and altitude, and also the distance of its center of volume from the base. Ans. bh/3, A/4. 116 PRIMER OF CALCULUS 12. Find the volume of a hemisphere of radius a, and the distance of its center of volume from its base. 27ra3 '=£ -(a2— x2)dx^ 3 r^ Ba M^ Jo 13. Find the moment of inertia of a right circular cylinder of altitude c and radius a, about its axis. j: X* . 27!Cxdx=: ■ 2 14. A wedge is made from a right circular cylinder of radius a and altitude h, by plane sections through a diameter of one base and the tangents to the other base that are parallel to such diameter; find the volume of the wedge. [A plane perpendicular to the diameter at distance x from the axis cuts each side piece that is taken off to make the wedge in a triangle whose area is to ah/2 as a^ — x^ is to a^, by similar triangles; thus V=i:a^h—i f""— (a2— a;2)d2;=a2A(-— f).] 15. Find the volume common to two circular cylin- ders of common upper base and tangent lower bases. 2 p - (a2 — a;2) dx=^ a^ h. 16. The axes of two right circular cylinders intersect at right angles; find the ijicluded volume, and the surface. 17. A sphere of radius o is charged with ijca^X units of electricity (^ per unit area); find its potential at a point C whose distance is c from the center. INTEGRATION 117 Draw the diameter 00, and let CP=r, ^OOP=0 where P is any point of a small circle of the sphere abont 00 as axis. The charge of the zone between the circle P=(r, 6) and the circle P'^r', 8' (whose altitude is A. r cos C) is 2Tza^A. rcosd and its potential' at is 2?: aA A (r cos (?) /rj where r^ is an average disiance between r and /. Since r^ -^c^ — 2r c cos 0=a^, there- fore r dr^cd.r cos Oj and the differential potential is 27C a^d(r cos 0) y r =27c aXdr/c. When c^-a then r changes from c — a to c-\-a giving the total potential ATza^X/c. When c/^)) but not =log log a; + 2 log ^/x+JJx, since it is not true that log (a; + j/) ^ log « + log 2/> J(x + y)=Jx-\-Jy and log Jx = J log x. Conversely, the n values Aa;, A^a;, ... A" a; determine the variation of x, viz., SUCCESSl VE DIFFERENTIA TION 121 127. In the case of an independent variable x, the n successive differences Ax, A^x, ... A"x are assignable at will, and can each be made to approach zero in such a way that for any proportional factor iV that approaches infinity, the proportional differences JVAx, N^A^x, ... iV" A"x shall approach any assigned limits dkc, d^x, . . . d" X. At the same time, the successive proportional differences NAw, N^ A'^w ... N^A^w, of a successively differentiable dependent variable w, must approach limits dio, d^w, ... d"w, that depend only upon the values of the independent variables and their successive differentials. E.g., A2x2/ = X2 3/3 — 2Xj2/i+X2/ = (x+2Ax+A2x) (2/-\-2Ay-\-A^y)—2(x-\-Ax) (y^Ay^^xy = xA^y-{-2AxAy + yA^x-\-2AxA^y+2A^xAy-{-A^xA^y so that d'^ .xy^= lim N^ A^ .xy^=xd'^y -\-2dxdy -\-yd^x. Partial Differentiation 128. Differentiation under the suppositions that certain variable quantities are constants is called partial differentiation. When the suppositions affect only in- dependent variables and not all of those, then partial differentiation of equals give equals. Thus in (x -|- y) 2 = x^ -f- 2xy -\-y'^, we may consider either x or y to vary alone, or both to vary together so that xy is constant, and the differentials of each member are equal under any of these suppositions. But in 122 PRIMER OF CALCULUS ^^+2/^=4, where both variables must change in order to maintain the equality, partial differentiation as to a; or 2/ does not give an equation that follows from the given one. Prin. 4. Tvm successive partial differentiations of any function are commutative in order of operation. In symbols d, d^w = d^ d^ w, where d^ affects certain variables x, ... of w, and d^ affects certain variables y, ... of w. By Prin. 3 d^w=(^dy-\-...')w=dyW-\-... and d^^d^w=^dx-\- ...)d^w = dxdyW-\-.,. Similaily d^w=(_dx-\- ...)io = dxW -\-... and d^d^w=:(dy-}-...)diW^dydxW-\-.,. It therefore only remains to prove the principle for partial differentiations as to any two variables x, y, and this ib done at once by dx dyf{x, y) = dx lim iV[/(a;, y') —f(x, yj] (definition) =lim N \_dxfix, y') — dxf(x, y)] (Prin 2) = dy dx f(x, y) (definition) By dividing this result by dx dy, we find, 9 9w_ 9 9w ^^^ 9x'9y~9y9x 129. It is not possible to mark a differential symbol so as to show all possiole suppositions under which the differentiation is taken, and /orm is often used instead of marks for the purpose. It follows that changes of form that are legitimate when the differentials are suf- ficiently marked to indicate their significance, are not PARTIAL DIFFERENTIATION 123 legitimate when the form is changed so as to lose its assigned significance as a mark of a certain kind of differentiation. For example, the early practice was, with a function w of x, y, to use dw/dx, dw/dy as forms for partial differentiation as to x, y, so that in this notation dw = -r- dx -\- -j—dy. To cancel dx, dy here gives the incorrect result dw = 2dw; but if we mark the differentiations by subscripts, then dx, dy, may be cancelled, giving d'w = dx'w-\-'dyW, a correct result. 130. Again, if y be a function of x, whose successive derivatives are y', j/", ... so that dy = y'dx, d^/ = y"dx, . . . then d^y = y" dx^ -\-y'd^x, d^y = y"' dx» + V dxd^x-\- y' d% etc. If we suppose dx to be constant in successive differen- tiations then d'^y^=y"dx^, d^y = 'if'dx^,... and the latter forms are understood as indicating partial differen- tiatiation with dx assumed constant; so that the quotients d^y/dx^=y", d^y/dx^=y"' become abbreviated forms for the successive derivatives of y as to x. The full forms for these derivitives are d dy d d dy dx dx' dx dx dx' in which d is unrestricted. Since every differehtiation, in these successive derivations, is performed upon a function of x alone, it follows that it is independent of differentials and may be performed under the supposi- tion that any differential we please is constant; in particular, da; = constant reduces them by Prin. 2 to 124 PRIMER OF CALCULUS the above abbreviated forms. If x is not the inde- pendent variable but is a function of the independent variable z, say x =fz, then we may transform deriva- tives as to X into derivatives as to z by substituting dx=f'zcbi in the complete forms, but not in the abbrevi- ated ones. Thus, 3-:^=7rT- jt-t', but d'y = y"dx^ ' dx dx fzdz f zdz' " " does not become d'^y^if'f'z'^ dz^ because the accepted significance of the latter form is that dz = constant whereas for the first, it is that dx^ constant, so that the two symbols d^y are not symbols of the same quantity. The true equality resulting from dx=f'zdz is here, (d^y)dxconsta.nt = y' f'z^ dz^. „r , , d^y d dy d^ydx — d^xdy . Wehavealways ^ = gj- 5^= "^"d^i '- m which, as pointed out above, the differentiations of the last form are unrestricted, and may be made as to any differential a constant. In particular to make y the in- dependent variable, we let dy = constant, and so find d^y _ dy« d^x dx^ ~ dx»' dy^ ' Examples IV 1. 2/ = a;2e*; d^y/dx'^ =e''{x^ -\-^-\-^) 2. y = x^; d^ y/dx^ = 5\ 3. y = x^\ogx; d*y/dx* = b/x 4. y = logsinxj d^y/'dx^= — csca;^ 5. y^e-™*(acos«a; + 6sinma;); PARTIAL DIFFERENTIA TION 125 6. y = e-'^ (a-\-bx); 7. y = a e— ™^ + & e~"^; dt^ij 3.11 8. Solve the equation ^-|^ -{■A-j--\-By = o where A, B are constants. Try y = e"^, whence c"^ -\- Ac -\- B = o io determine c. Show that the sum of two solutions, each multiplied by an arbitrary constant, is a solution, and thence, if c= — m, — n, derive the solution of Ex. 7. This solu-, tion is general, because it involves two arbitrary con- stants a, b, such as would be obtained by two successive anti-differentiations. If m = n show that besides e— ™'» also x«— ™^ is a solution, so that the general solution is thatofEx. 5. Ifc = — m-\-nJ — 1, — m — n^ — 1, then replace the Iwo solutions e" by their sum multi- plied by 1/2 and their difference multiplied by 1/2^ — 1, and so obtain the general solution of Ex. 5. d'^v dfii 9. l{y=f^x be a solution of -j-y + A ~-\-By=fx find the general solution. [Let y=zf^x-\-z be the general solution; whence z is found from -^ + ^ ^ + -B 3 = O.J 10. Solve .^ + 2 ^ + 26 2/ = 154 cos 4a; 4- 8 sin Ax. dx^ ' dx ' " Ans. 2/ = 9 cos 4x + 8 sin 4x -{- er" (a cos 5a; + 6 sin 5a;) . 126 PRIMER OF CALCULUS 11. Showthat^u.= -^«+n^^,^ + . ■4 ,4^1" . (ix '" dx') uv. 12. Showthat-j^e<^/a;=e°^(a + T-)"/'a;. [Verify for n = l and thence for n = 2, 3, ...J 13. -j^e«*a;2=e»^[a''a;2+2na"-ia;-|-7i(7i^l)a''-2] 14. Verify the following transformations of inde- pendent variable, and solve the equations. d«y „ dj/2 n d«8 ^ d^x , ^dx , . d'^y , 2x dy , y ^ d^y ^+1+^5^ + 0+^^ =oto^f +2/=o,a;=tan.. Successive Integration 131. We consider certain "multiple" integrals in two or more variables, where the integration is partial with respect to each variable in turn, and as if the remaining variables were constants, while the limits of variation of each variable are constants, or at most functions of the variables that are assumed as con- stants. The integral is written so that the order of successive integration is from right to left, so that each integration reduces the multiple integral to one of next lower multipUcity. Thus „ j ,/(^, y) dxdy^j^[j J {X, y) dy J dx. SUCCESSIVE INTEGRATION 127 is a double integral. The bracketed integral is first evaluated, with a; as a constant, and 6, v are at most functions of x, so that the bracketed integral is simply a cjix term in a single integral. Similarly =J " j^ I J ^-P (a;, y, 2) dz I dx dy is a triple integral. The bracketed integral is first evaluated, with x, y as constants, and c, w are at most functions of x, y, so that the bracketed integral is simply an/(x, y) term in a double integral. Similarly for multiple integrals in four or more variables. 132. The single, double, and triple integrals have geometric representation as the limits of sums of dif- ferences over a line, surface, and volume respectively. We illustrate by examples worked out in detail, show- ing also in the first illustration the sums considered, of which the integrals are limits for continuous variation. The student should make the drawings as described. 133. lind the moment of inertia of a rectangular paral- klopiped of edges a, b, c, about the edge c as axis. Let OA^a, OB = b, be the two edges in the plane of the diagrarn. Confine attention to the rectangle AB, knowing that a length c of the volume is above every point P. Take OA, OB as axes of x, y. Lay off OL = x, OL' = x', LP=y, L'P'=y', where P, /" are neighboring points within the rectangle AB. Then cAx^y is the difference volume whose base is the 128 PRIMER OF CALCULUS rectangle PP' and (x^ -\- y^) c^r ^y is its moment of inertia as to the axis OC, where (Kj, y^) is some point of the rectangle PP'. Here, x^-{-y^ is a function, (x, Ax, y, Ay), that reduces to x^ -\-y^ when Ax==o, Ay = 0. If we assume x, Aa; to be constants and give y any variation from o to 6, then the sum of the differ- ence volumes PP', is S* cAxAy, ==bcAx, the difference volume whose base is the rectangle LU Q' Q of altitude 6 and base Ax; and the sum of the moments of inertia of the difference volumes PP' is ^^^(r^ -\-y'^) cAxAy= the moment of inertia of the difference volume LQ'. Next give x any variation from o to a, and then the sum of the difference volume LQ, is ^gbcAx^abc = the entire volume; and the sum of their moments of inertia is the entire moment of inertia, S° Sg (x^ + y^)cA xA y. The results hold for any vari- ation, first of y from o to 6 with x, Ax constant, and then of a; from o to a; and in particular for continuous variations, in which the sums become integrals of the differentials corresponding to the vanishing differ- ences. Thus c dx dy is the differential volume P, and (x^ -^y^)cdxdy is its moment of inertia. Also, cdr.dy^= be dx is the differential valume LQ, while /b b^ (x^ -\-y^) cdxdy^=bc (^x^ -{-'-^^dx is its moment of inertia; and finally, ( I cdxdy=\ bcdx=abc, J oj Jo SUCCESSIVE INTEGRATION 129 is the total volume, and j C (x^-^y^^cdxdy^ J J be (a;2 -(- — . ) (ix = a 6 c — ^ — is the total moment of inertia. 134. Find the moment of inertia of a right cylinder about its axis, 00 ^c, where the base is an elliptic quadrant of radii OA = a, OB=:b. As before cdxdy, c (x^ -{-y^) dxdy are the differ- ential volume at P and its moment of inertia; and these integrals from y = oioy = yy = LK= b J (a^ — x^) / a = the ordinate of the point K on the ellipse ^B whose abscissa is a; = OL, are J: j '^c dxdy=:cy^ dx, c (a;2 +2/2) dxdy = c (x^ y^ +~) dx. These are therefore respectively the differential vol- ume LK and its moment of inertia. Finally volume = j cy^dx = '—j- [Ex. 14, p. 43.] c (x^ 2/i + -~) dx ■K abc a^ -f- 6^ ^~i 4 [Ex. 16, p. 43.] 135. Find the moment of inertia of the preceding elliptic cylinder about the axis OA. Show the axis 00 and the cyhnder OOAB in pers- pective on the plane of the diagram. Draw a plane 130 PRIMER OF CALCULUS through the ordinate LK perpendicular to OA, and therefore intersecting the cylinder in elements KN, LM; draw RQ parallel to LM and intersecting LK in R, MN in Q; and on RQ take any point P=x,y,z= iOL, LR, RP). The differential volume P is dx dy dz, and its moment of inertia as to OA is (y^ + z^) da; dy dz. Integrating from z = o to z^c we find the differential volume RQ and its moment of inertia, dxdydz = cdx dy, £ r. (y2 -)-s2)da;d3/cfc!^(j/2c-|-— ) dx dy. Integrating again from y^o to y=y^=:LK=^b'J'a^ — x^/a we find the differential volume KLMN and its moment of inertia, cy^dx, cy^^^—dx These are results that can be found directly from geometry and Art. 134. Integrating then from x = o to x^a, we find the total volume and its moment of inertia, iTzabc, i7ra6c(62/4 + cV3.) 136i Find the volume and moment of inertia about OC of the octant of an ellipsoid of radii OA = a, OB = 6, 00=0. The figure is shown in plane diagram by projections of quadrants of the ellipses AB, BO, CA; the section by a plane perpendicular to OA through the ordinate LKoi arc AB is an elliptic quadrant KM from AB to OA; SUCCESSIVE INTEGRATION 131 draw in this plane RQ parallel to LM from LK to arc X'^ .y2 ^2 KM, and we have since ~ + ^ + ^ = 1 foi^ ^.ny point (x,y,z) on the ellipsoid, LK^=h J (a^ — x2)/a, RQ = cJ (_ 1 — — — =— ). Take on i? Q any point a P=(x,y,z) = (OL,LR,RP). The volumes and moments of inertia about 00 are, I dx dy dz Jo and the same integration ot Qx^ -\- y^) dx dy dz. The results of the first two partial integrations may be found geometrically from Art. 134, applied to the elliptic cylinder of length dx on the base LKM, and are -. LK . LM dx, -T LK . LM dx. 7 or fsmce 4 '4 4 ■- LM = c 7 ( a* — a;2 ) / a], -^ ( a^ — x^ ) dx, — 7^ — i (a' — a;2-)2 (i^j;^ and the integrals of these from a;= to x = a give finally -o— > —o— — c ^• 137. If m denote the volume (or mass of a homo- geneous volume) then the rectangular parallelopiped, elliptic cylinder, and ellipsoid, whose semi-axes of symmetry are OA^a, OB = b, 00 ^c, and of which the volumes of the preceding articles are octants, have the following moments of inertia 132 PRIMER OF CALCULUS (j2 _l_j!)2 Parallelopiped : m — ^ — about OQ ; etc. a2 -1-62 Cylinder : m j about OC ; m (jj- -\- -^) about OA, etc. Ellipsoid : m — ^ — about OC ; etc. These are easily remembered, and are useful, especially in connection with the theorem of Art. 139. 138. CerUer of gravity. The differential element of mass at (x, y,z) called a particle P, is u dxdy dz where u is in general a function of x, y, 2 denoting the demHiy at that point. If (x, y, z) be the center of mass (center of gravity) then computing moments as to the planes yz, zx xy directly, and by the sum of the moments of the particles, we find m X = j I luxdxdydz m y = I I luydxdydz m z = I I iuzdxdydz 139. The Tnoment of inertia of a given mass about a given axis is equal to the moment of inertia of the mn,ss about a paralled axes through the center of mass plus the product of the mass and the square of the distance oetween the axes. Let OZ be the axis through the center of mass, and let OA^a, be the distance between the axes. We have I I luxdxdydz = m's. = o by hypothesis; and S UCCESSIVE INTEGRA TION 133 therefore mA2 = | | j [(a; — o)^+ 2/"] m t^a; dy dz = I I I (a;2 -j- 2/3) w da; dj/ & + a^ j j j « cf a; dy <^2 ^ 140. A cylinder with elements perpendicular to the a;, 2/ plane intersects a surface z =f (?c, y) ; required the surface area intercepted by the cylinder. If A^S be the portion of the surface intercepted by the cylinder on the rectangle Ax Ay as base, the cor- responding differential element of surface d^S is inter- cepted on the tangent plane to -the surface at P^(x, y,z) by the cylinder on da; dy as base; so that if y be the angle between this tangent plane and the plane x y, i. e. , the angle between the axis of z and the normal to the surface at P, we have d^ S cos y = dxdy so that d^ iS^^ da; dt/ sec ;-, and S= j j secydxdy, over the base of the given cylinder on the x y plane. To find sec ^ in terms of x^ y observe that if m = o be the equation of a surface, then for variations of P on that surface w^o ov -^:r- dx -\- -^^ dy -1 — =- dz = o. This 3x ' dy " ' 3z shows that the line PN whose components on the axes are-iy-, -w~, -w- is perpendicular to the tangent line PS = ds, whose components are dx, dy, dz* Thus PN is a normal to the surface at P since it * If r, r' be two lines that make an angle A with each other and angles a,b,c, a', V, cf, with the axes, then eqating the projection of r upon r' to the sum of the projections of the components of r upon r', we have r cos A = r cos a . cos a' + r cos 6 . cos 6' + r cos c . cos cf or rr' cos A = IV + mm! + 1171' in terms ot the components of r,r'\ if this is zero then ^ is a right angle. 134 PRIMER OF CALCULUS is perpendicular to every tangent line PS, and PNcosr = 9w/3z,=:l when m; = z — }{x,y). Also PiV2 = 7=f- + -^j- + Tp- =l+x- +7r- where 2 =/(«', 3/). Examples. 1. Find the center of gravity of an arc of a circle of radius a and length I. [Take the center as origin and axis of y parallel to the chord, whose length is fc=2asin— ^ {I /2a). Then l=a~^. xds\ I ds^ I ady since y varies from — Jfc to \k when s varies from o to I. /'I I /•! /'ft . y=j yds] I ds= I — adxll^o. ^ '«' ^ ft I 2. Find the center of gravity of a straight wire whose density varies uniformly from end to end. Let a, h, be the densities at the ends, and I be the length of the wire; then the density at distance x from the first end is M=a+(6 — d)x/l; and x=j uxdxl I udx=^l(a-\-2b)/(a-{-b') 3. Find the center of gravity of the first quadrant arc of the hypocycloid xi-\-y^=ai. 4. Find the center of gravity of the first quadrant area of the circle x^ -\-y^ =a*. SUCCESSIVE INTEGRATION 135 x= 1 1 xdxdyl I t da;rf2/ 5. Find the center of gravity of the first quadrant area of the hypocycloid x^-\-'yi=^ai. 6. Find the center of gravity of the area of the cardioid r=2a(l — cos 5). 7. Find the center of gravity of the parabloic area 2/2=:4aa; from x=o. 8. Find the center of gravity of a hemisphere whose density varies as the distance from the center; find also its radius of gyration about its axis of symmetry. 9. Find the center of gravity and the radius of gyration of the volume generated by the revolution of ^2=4aa; about the axis of 2 (from a:=o). 10. Find the radius of gyration of a sphere about a tangent line as axis. (Arts. 137, 239). 11. Compare and verify when necessary the following formulas for straight and rotary motion of a rigid body; dm = mass of a differential particle; v = its linear velocity; r == its distance from the axis in rotary motion. The integration extends to every particle of ' the body, and for this integration v = constant in linear motion, and v/r=a/=constant in rotary motion; v and w are, however, variable with the time. 136 A PRIMER OF CALCULUS STRAIGHT MOTION m^inertia (mass) = I dm D= velocity mv = momentum ROTARY MOTION I=moment of inertia = I r^dm w=v/r=angular vel. Ize/=mo. of momentum = ivdm == I rvdr. -- = a cceler ation dt dw dt = angular acceleration dv J. m ^- = force at 'dv rd^ dm ^i;3=kinetic energy = I ^v^ dm I -r-=moment of forces dt S'Tt^"^ ■J I zy 8= kinetic energy = I ^v^ dm In finding the moment of the forces in rotary motion dv the tangential acceleration of dm, viz, -j-, is the only effective component, since the normal component v^ /r passes through the axis. (See Art. 86). In straight motion there is no normal component of acceleration since the curvature of a straight line is zero (or its radius of curvature infinite). 12. Find the surfaces cut from one another by a right circular cylinder of radius a and a sphere of radius 2a, whose center is on the cylindrical surface. 13. Find the volume enclosed in Ex. 12. SUCCESSIVE INTEGRATION 137 14. Find the volume enclosed by the surface 15. Show that if an arc of length s be revolved about an axis in its plane, the surface described is the product of s into the length of path of the center of gravity of s. State and prove a similar theorem for revolution of a plane area about an axis in its plane. 16. Find the volume and surface of the anchor ring, generated by revolving a circle of radius " 6, about an axis in its plane at distance a from the center. 17. In polar co-ordinates (r, 6, <^) of a point P with reference to an initial line OA and initial plane OAB, we have r=OP, 6 = = diedral angle between OAP and OAB. Show that the differential element of volume is . . dV==dr. rdO . r sin dd. =r^ sin ddrdddtj). 138 A PRIMER OF CALCULUS Rules of Differentiation I. d.u^^lu'^^du. (b) d.u^ /'v'^=u'^^(lvdu — mudv)/v^'*'^. 11. d.av—oylogady; d.ev=e!ydy. (a) d . v^ =yvV-^ dw + zty log w . dy. III. dlogoV=d»/t'logay d\ogv=dv/v. (a) dlog(a;+Vx2+ca2)=da;/V(a;2+ca2). a; ada; (b) dlog a+Vo2^_ca;2 j;V(a*+ca:*)' , . „ a+cc 2ada; rV 'dsva.v=cosv.dv; dcosv= — smv.dv. dtanw=sec2)* di;; d' ) a °^ (x + V x^^+cs^"- III J a; da; 1 - = - sec~ XfJ{x^ — '*^) ^^ ** II d-i . a* dx=:a^ /log a; d— i . e^ da; = e? . IV d^-^ .smvdv = — cosv; d— ^ . cos « d« = sin -y. ' III d— 1 . tanv . di)^ — log cosb; ' d"-i . cot V dv= log sin v. Ill d—i . see vdv=^ log (sec ■« + tan v) ■ d— 1 . CSC vdv= log (esc 'v — cot v) . .IV d— 1 secB* di7 = tan'y; d—''- esc v^ dv ^ — cotv. IV ' , d~^ see ■e-tan vdv = secv; , d"^i CSC V cot V dv = — cse v. d— 1 . sin «^ dv =i (2t) — sin 2v) ; dr-^ cos D 2 du = J (2v 4" sin 2*) . Inv. Prin. 3. d-i . da;/(x, ?/) =/ (a;, y) — d-i dj,/ (a;, y) .