Page:EB1911 - Volume 14.djvu/579

and

$d^{n}f(a+\theta h,b+\theta k)=\left\lbrack \left(h{\frac {\partial }{\partial x}}+k{\frac {\partial }{\partial y}}\right)^{n}f(x,y)\right\rbrack _{x=a+\theta h,y=b+\theta k}.$

The last expression is the remainder after n terms, and in it θ 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).

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.Plane curves.

(i.) If ψ 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, θ and if p denotes the perpendicular from the origin to the tangent, then

cos ψ = dr/ds,   sin ψ = rdθ/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 ψ for the pedal is the same as the angle ψ 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 φ denotes the angle which the tangent at any point makes with a fixed line, we have

r ² = p² + (dp/dφ)².

(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotient

$\left|{\frac {\Delta \phi }{\Delta s}}\right|$

where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ 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

$\left|{\frac {d\phi }{ds}}\right|$

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 dφ 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” ρ is the radius of this circle. We have ${\frac {1}{\rho }}=\left|{\frac {d\phi }{ds}}\right|$. 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:—

${\frac {1}{\rho }}=\left|\left\{1+\left({\frac {dy}{dx}}\right)^{2}\right\}^{-{\frac {3}{2}}}{\frac {d^{2}y}{dx^{2}}}\right\vert ,$

$\left|r{\frac {dr}{dp}}\right\vert =\left|p+{\frac {d^{2}p}{d\phi ^{2}}}\right\vert$

(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 ƒ(x) = 0, the function ƒ(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 ƒ(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 r th 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 = 0 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

$p_{1}x+{\frac {1}{2}}q_{1}x^{2}+\ldots ,$ or $p_{2}x+{\frac {1}{2}}q_{2}x^{2}+\ldots ,$

where q₁, . . . and q₂, . . . are determined without ambiguity. If p₁ and p₂ are real the two branches have radii of curvature ρ₁, ρ₂ determined by the formulae

${\frac {1}{\rho _{1}}}=\left|\left(1+p_{1}^{2}\right)^{-{\frac {3}{2}}}q_{1}\right\vert ,\quad {\frac {1}{\rho _{2}}}=\left|\left(1+p_{2}^{2}\right)^{-{\frac {3}{2}}}q_{2}\right\vert .$

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 = L₁L₂ . . . L_n, where L₁, . . . are linear in x, y, we may resolve u_n−1/u_n into partial fractions according to the formula

${\frac {u_{n-1}}{u_{n}}}={\frac {{\text{A}}_{1}}{{\text{L}}_{1}}}+{\frac {{\text{A}}_{2}}{{\text{L}}_{2}}}+\ldots +{\frac {{\text{A}}_{n}}{{\text{L}}_{n}}},$

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 + y ² − a²) = a⁴, where x = 0 is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions.

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 Integral calculus. 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.

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

From the equation d(uv) = udv + vdu we deduce the equation

$\int u{\frac {dv}{dx}}dx=uv-\int v{\frac {du}{dx}}dx,$

or, as it may be written

$\int uwdx=u\int wdx-\int {\frac {du}{dx}}\left\{\int wdx\right\}dx.$

This is the rule of “integration by parts.”

As an example we have

$\int xe^{ax}dx=x{\frac {e^{ax}}{a}}-\int {\frac {e^{ax}}{a}}dx=\left({\frac {x}{a}}-{\frac {1}{a^{2}}}\right)e^{ax}.$

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

$dx={\frac {dx}{dz}}dz$