Chapter 20 of 47 · 3978 words · ~20 min read

Part 20

When such a coordinate as u is connected with x and y by a functional relation of the form [f](x, y, u) = 0 the curves u = const. are a family of curves, and this family may be such that no two curves of the family have a common point. When this is not the case the points in which a curve [f](x, y, u) = 0 is intersected by a curve [f](x, y, u + [Delta]u) = 0 tend to limiting positions as [Delta]u is diminished indefinitely. The locus of these limiting positions is the "envelope" of the family, and in general it touches all the curves of the family. It is easy to see that, if u, v are the parameters of two families of curves which have envelopes, the Jacobian ð(x, y) / ð(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ð(u, v) / ð(x, y) vanishes, a curve of the family u touches a curve of the family v.

If three variables x, y, z are connected by a functional relation [f](x, y, z) = 0, one of them, z say, may be regarded as an _implicit function_ of the other two, and the partial differential coefficients of z with respect to x and y can be formed by the rule of the total differential. We have

ðz ð[f] / ð[f] ðz ð[f] / ð[f] -- = - ---- / ----, -- = - ---- / ----; ðx ðx / ðz ðy ðy / ðz

and there is no difficulty in proceeding to express the higher differential coefficients. There arises the problem of expressing the

## partial differential coefficients of x with respect to y and z in

terms of those of z with respect to x and y. The problem is known as that of "changing the dependent variable." It is solved by applying the rule of the total differential. Similar considerations are applicable to all cases in which n variables are connected by fewer than n equations.

Extension of Taylor's theorem.

45. Taylor's theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the form

[f](a + h, b + k) = [f](a, b) + d[f](a, b) + (1/2!) d²[f](a, b) + ...

1 1 + -------- d^(n-1) [f](a, b) + -- d^n [f](a+[Theta]h, b + [theta]k), (n - 1)! n!

and

The last expression is the remainder after n terms, and in it [Theta] denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor's theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see MAXIMA and MINIMA).

Plane curves.

46. In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.

(i.) If [psi] denotes the angle which the radius vector drawn from the origin makes with the tangent to a curve at a point whose polar coordinates are r, [Theta] and if p denotes the perpendicular from the origin to the tangent, then

cos [psi] = dr/ds, sin [psi] = r d[Theta]/ds = p/r,

where ds denotes the element of arc. The curve may be determined by an equation connecting p with r.

(ii.) The locus of the foot of the perpendicular let fall from the origin upon the tangent to a curve at a point is called the _pedal_ of the curve with respect to the origin. The angle [psi] for the pedal is the same as the angle [psi] for the curve. Hence the (p, r) equation of the pedal can be deduced. If the pedal is regarded as the primary curve, the curve of which it is the pedal is the "negative pedal" of the primary. We may have pedals of pedals and so on, also negative pedals of negative pedals and so on. Negative pedals are usually determined as envelopes.

(iii.) If [phi] denotes the angle which the tangent at any point makes with a fixed line, we have

r² = p² + (dp/d[phi])².

(iv.) The "average curvature" of the arc [Delta]s of a curve between two points is measured by the quotient

| [Delta][phi] | | ------------ | | [Delta]s |

where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and [phi] is the angle which the tangent makes with a fixed line, so that [Delta][phi] is the angle between the tangents (or normals) at the points. As one of the points moves up to coincidence with the other this average curvature tends to a limit which is the "curvature" of the curve at the point. It is denoted by

| d[phi] | | ------ | | ds |

Sometimes the upright lines are omitted and a rule of signs is given:--Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. The differential d[phi] is often called the "angle of contingence." In the 14th century the size of the angle between a curve and its tangent seems to have been seriously debated, and the name "angle of contingence" was then given to the supposed angle.

(v.) The curvature of a curve at a point is the same as that of a certain circle which touches the curve at the point, and the "radius of curvature" [rho] is the radius of this circle. We have 1/[rho] = |d[phi]/ds|. The centre of the circle is called the "centre of curvature"; it is the limiting position of the point of intersection of the normal at the point and the normal at a neighbouring point, when the second point moves up to coincidence with the first. If a circle is described to intersect the curve at the point P and at two other points, and one of these two points is moved up to coincidence with P, the circle touches the curve at the point P and meets it in another point; the centre of the circle is then on the normal. As the third point now moves up to coincidence with P, the centre of the circle moves to the centre of curvature. The circle is then said to "osculate" the curve, or to have "contact of the second order" with it at P.

(vi.) The following are formulae for the radius of curvature:--

1 | { /dy\² }-3/2 d²y | ----- = | { 1 + ( -- ) } --- |, [rho] | { \dx/ } dx² |

| dr | | d²p | [rho] = | r -- | = | p + ------- |. | dp | | d[phi]² |

(vii.) The points at which the curvature vanishes are "points of inflection." If P is a point of inflection and Q a neighbouring point, then, as Q moves up to coincidence with P, the distance from P to the point of intersection of the normals at P and Q becomes greater than any distance that can be assigned. The equation which gives the abscissae of the points in which a straight line meets the curve being expressed in the form [f](x) = 0, the function [f](x) has a factor (x - x0)³, where x0 is the abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x - x0) occurs (n + 1) times in [f](x), the curve is said to have "contact of the nth order" with the line. There is an obvious modification when the line is parallel to the axis of y.

(viii.) The locus of the centres of curvature, or envelope of the normals, of a curve is called the "evolute." A curve which has a given curve as evolute is called an "involute" of the given curve. All the involutes are "parallel" curves, that is to say, they are such that one is derived from another by marking off a constant distance along the normal. The involutes are "orthogonal trajectories" of the tangents to the common evolute.

(ix.) The equation of an algebraic curve of the nth degree can be expressed in the form u0 + u1 + u2 + ... + u_n = 0, where u0 is a constant, and u_r is a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, u0 vanishes, and u1 = 0 represents the tangent at the origin. If u1 also vanishes, the origin is a double point and u2 = o represents the tangents at the origin. If u2 has distinct factors, or is of the form a(y - p1x)(y - p2x), the value of y on either branch of the curve can be expressed (for points sufficiently near the origin) in a power series, which is either

p1x + ½ q1x² + ..., or p2x + ½ q2X² + ...,

where q1, ... and q2, ... are determined without ambiguity. If p1 and p2 are real the two branches have radii of curvature [rho]1, [rho]2 determined by the formulae

1 | | 1 | | ------ = |(1 + p1²)^{-3/2} q1 |, ------ = |(1 + p2²)^{-3/2} q2 |. [rho]1 | | [rho]2 | |

When p1 and p2 are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is an _isolated point_. If u2 is a square, a(y - px)², the origin is a _cusp_, and in general there is not a series for y in integral powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to analytical geometry and the theory of algebraic functions than to differential calculus.

(x.) When the equation of a curve is given in the form u0 + u1 + ... + u_(n-1) + u_n = 0 where the notation is the same as that in (ix.), the factors of u_n determine the directions of the _asymptotes_. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If u_n = L1 L2 ... L_n, where L1, ... are linear in x, y, we may resolve u_(n-1)/u_n into partial fractions according to the formula

u_(n-1) A1 A2 A_n ------- = -- + -- + ... + ---, u{n} L1 L2 L_n

and then L1 + A1 = 0, L2 + A2 = 0, ... are the equations of the asymptotes. When a real factor of u_n is repeated we may have two parallel asymptotes or we may have a "parabolic asymptote." Sometimes the parallel asymptotes coincide, as in the curve x²(x² + y² - a²) = a^4, where x = 0 is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions.

Integral calculus.

47. The formal definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of classes of "integrable" functions, extensions of the notion of integration to functions which become infinite or indeterminate, and to cases in which the limits of integration become infinite, the definitions of multiple integrals, and the possibility of defining functions by means of definite integrals--all these matters have been considered in FUNCTION. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.

Methods of integration.

48. The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.

From the equation d(uv) = udv + vdu we deduce the equation _ _ / dv / du | u -- dx = uv - | v -- dx, _/ dx _/ dx

or, as it may be written _ _ _ _ / / / du / / \ | uw dx = u | w dx - | -- ( | w dx ) dx. _/ _/ _/ dx \ _/ /

This is the rule of "integration by parts."

As an example we have _ _ / e^(ax) / e^(ax) / x 1 \ | xe^(ax) dx = x ------ - | ------ dx = ( --- - -- ) e^(ax). _/ a _/ a \ a a² /

When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express [f](x) in terms of z. Let [phi](z) denote the function of z into which [f](x) is transformed. Then from the equation

dx dx = -- dz dz

we deduce the equation _ _ / / dx | [f](x) dx = | [phi](z) -- dz. _/ _/ dz

As an example, in the integral _ / | [root](1 - x²) dx _/

put x = sin z; the integral becomes

_ _ / / | cos z · cos zdz = | ½(1 + cos 2z)dz = ½(z + ½ sin 2z) = ½(z + sin z cos z). _/ _/

Integration in terms of elementary functions.

49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called "elementary" functions. The elementary functions are rational algebraic functions, implicit algebraic functions, exponentials and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form [f](x, y), where [f] denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of e^x; (v.) all rational integral functions of the variables x, e^(ax), e^(bx), ... sin mx, cos mx, sin nx, cos nx, ... in which a, b, ... and m, n, ... are any constants. The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any simple root of the denominator there is a logarithmic term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. The operation of resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Hermite, _Cours d'analyse_, Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, possibly after preliminary integrations by parts. In the case of rational functions of x and a radical of the form [root](ax² + bx + c) the radical can be reduced by a linear substitution to one of the forms [root](a² - x²), [root](x² - a²), [root](x² + a²). The substitutions x = a sin [theta], x = a sec [theta], x = a tan [theta] are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin [theta] and cos [theta], and it can be reduced to a rational function of t by the substitution tan ½[theta] = t. There are many other substitutions by which such integrals can be determined. Sometimes we may have information as to the functional character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax^4 + bx³ + cx² + dx + e)^-½ the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see FUNCTION).

Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. Hardy's tract, _The Integration of Functions of a Single Variable_ (Cambridge, 1905), and to the memoirs there quoted. A few results are added here _ / (i.) | (x² + a) - ½ dx = log {x + (x² + a)^½ }. _/ _ / dx (ii.) | ----------------------------- _/ (x - p) [root](ax² + 2bx + c)

can be evaluated by the substitution x - p = 1/z, and _ / dx | --------------------------------- _/ (x - p)^{n} [root](ax² + 2bx + c)

can be deduced by differentiating (n - 1) times with respect to p. _ / (Hx + K)dx (iii.) | ------------------------------------------------------ _/ ([alpha]x² + 2[beta]x + [gamma]) [root](ax² + 2bx + c)

can be reduced by the substitution y² = (ax² + 2bx + c)/([alpha]x² + 2[beta]x + [gamma]) to the form _ _ / dy / dy A | ---------------------- + B | ---------------------- _/ [root]([lambda]1 - y²) _/ [root](y² - [lambda]2)

where A and B are constants, and [lambda]1 and [lambda]2 are the two values of [lambda] for which (a - [lambda][alpha])x² + 2(b - [lambda][beta])x + c - [lambda][gamma] is a perfect square (see A. G. Greenhill, _A Chapter in the Integral Calculus_, London, 1888).

(iv.) [f]x^m (ax^n + b)^p dx, in which m, n, p are rational, can be reduced, by putting ax^n = bt, to depend upon [f]t^q (1 + t)^p dt. If p is an integer and q a fraction r/s, we put t = u^s. If q is an integer and p = r/s we put 1 + t = u^s. If p + q is an integer and p = r/s we put 1 + t = tu^s. These integrals, called "binomial integrals," were investigated by Newton (_De quadratura curvarum_). _ _ / dx x / dx (v.) | ----- = log tan ---, (vi.) | ----- = log (tan x + sec x). _/ sin x 2 _/ cos x

(vii.) [f] e^(ax) sin (bx + [alpha]) dx = (a² + b²)^-1 e^(ax){a sin (bx + [alpha]) - b cos (bx + [alpha])}.

(viii.) [f] sin^m x cos^n x dx can be reduced by differentiating a function of the form sin^p x cos^q x;

d sin x 1 q sin² x 1 - q q e.g. -- ------- = ----------- + ----------- = ----------- + -----------. dx cos^q x cos^(q-1) x cos^(q+1) x cos^(q-1) x cos^(q+1) x

Hence _ _ / dx sin x n - 2 / dx | ------- = ------------------- + ----- | -----------. _/ cos^n x (n - 1) cos^(n-1) x n - 1 _/ cos^(n-2) x _ _ / ½[pi] / ½[pi] (ix.) | sin^(2n) x dx = | cos^(2n) x dx = _/ 0 _/ 0

1·3 ... (2n - 1) [pi] ---------------- · ----, (n an integer). 2·4 ... 2n 2 _ _ / ½[pi] / ½[pi] (x.) | sin^(2n+1) x dx = | cos^(2n+1) x dx = _/ 0 _/ 0

2·4 ... (2n) --------------, (n an integer). 3·5 ... (2n+1) _ / dx (xi.) | --------------- can be reduced by one of the substitutions _/ (1 + e cos x)^n

e + cos x e + cos x cos [phi] = -----------, cosh u = -----------, 1 + e cos x 1 + e cos x

of which the first or the second is to be employed according as e < or > 1.

New transcendents.

50. Among the integrals of transcendental functions which lead to new transcendental functions we may notice _ _ / x dx / log x e^z | ----- or | --- dz, _/ 0 log x´ _/ -x z

called the "logarithmic integral," and denoted by "Li x," also the integrals _ _ / x sin x / x cos x | ----- dx and | ----- dx, _/ 0 x _/ [oo] x

called the "sine integral" and the "cosine integral," and denoted by "Si x" and "Ci x," also the integral _ / x | e^-x² dx _/ 0

called the "error-function integral," and denoted by "Erf x." All these functions have been tabulated (see TABLES, MATHEMATICAL).

Eulerian integrals.

51. New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations _ / 1 B(l, m) = | x^(l-1) (1 - x)^(m-1) dx, _/ 0 _ / [oo] [Gamma](n) = | e^-t t^(n-1) dt. _/ 0

When n is a positive integer [Gamma](n + 1) = n!. The Beta function (or "Eulerian integral of the first kind") is expressible in terms of Gamma functions (or "Eulerian integrals of the second kind") by the formula

B(l, m)·[Gamma](l+m) = [Gamma](l)·[Gamma](m).

The Gamma function satisfies the difference equation

[Gamma](x + 1) = x [Gamma](x),

and also the equation

[Gamma](x)·[Gamma](1-x) = [pi]/sin (x[pi]),

with the particular result

[Gamma](½)= [root][pi].

The number _ _ | d | - | -- {log [Gamma](1 + x)} | , or -[Gamma]´(1), |_ dx _|x=0

is called "Euler's constant," and is equal to the limit _ _ | / \ | lim. | ( 1 + ½ + 1/3 + ... + 1/n ) - log n |; n=[oo] |_ \ / _|

its value to 15 decimal places is 0.577 215 664 901 532.

The function log [Gamma](1 + x) can be expanded in the series

/ x[pi] \ log [Gamma](1 + x) = ½ log ( --------- ) \ sin x[pi] / 1 + x - ½ log ----- + {1 + [Gamma]´(1)} x 1 - x

- 1/3 (S3 - 1)x³ - 1/5 (S5 - 1)x^5 - ...,

where

1 1 S_(2r+1) = 1 + -------- + -------- + ..., 2^(2r+1) 3^(2r+1)

and the series for log [Gamma](1 + x) converges when x lies between - 1 and 1.

Definite integrals.