Page:EB1911 - Volume 14.djvu/581

Rh integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see ). The method of contour integration involves the introduction of complex variables (see : § Complex Variables).

A few results are added

53. The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the article. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z)

through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by

$\iint f(x,y)dxdy.$

The notation refers to a method of evaluating the integral. We may suppose the area divided into a very large number of very small rectangles by lines parallel to the axes. Then we multiply the value of ƒ at any point within a rectangle by the measure of the area of the rectangle, sum for all the rectangles, and pass to a limit by increasing the number of rectangles indefinitely and diminishing all their sides indefinitely. The process is usually effected by summing first for all the rectangles which lie in a strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is equivalent to integrating ƒ(x, y) with respect to y, keeping x constant, and taking certain functions of x as the limits of integration for y, and then integrating the result with respect to x between constant limits. The integral obtained in this way may be written in such a form as

$\int_a^b dx \left \{\int_{f_1(x)}^{f_2(x)}f(x,y)dy \right \},$

and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(x, y)dxdy, with a repeated integral cannot always be made, but implies that the function satisfies certain conditions of continuity. In the same way volume integrals are usually evaluated by regarding them as repeated integrals, and a volume integral is written in the form

$\iiint f(x,y,z)dxdydz.$

Integrals such as surface and volume integrals are usually called “multiple integrals.” Thus we have “double” integrals, “triple” integrals, and so on. In contradistinction to multiple integrals the ordinary integral of a function of one variable with respect to that variable is called a “simple integral. ”

A more general type of surface integral may be defined by taking an arbitrary surface, with or without an edge. We suppose in the first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, and draw the tangent planes at all these points. These

tangent planes form a polyhedron having a large number of faces, one to each marked point; and we may choose the marked points so that all the linear dimensions of any face are less than some arbitrarily chosen length. We may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedra indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function ƒ at a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface integral, and is written

$\iint fdS.$

The extension to the case of an open surface bounded by an edge presents no difficulty. A line integral taken along a curve is defined in a similar way, and is written

$\int fds.$

where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually present themselves in the form

$\int \left( u\frac{dx}{ds}+v\frac{dy}{ds}+w\frac{dz}{ds} \right)$ or $\int_s (udx+vdy+wdz).$

In like manner surface integrals usually present themselves in the form

$\int \left( l\xi+m\eta+n\zeta \right)dS.$

where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.

The area of a bounded portion of the plane of (x, y) may be expressed either as

$\tfrac{1}{2} \int (xdy-ydx),$

or as

$\iint dxdy,$

the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left hand.

53a. We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation

$\iiint \left(\frac{\partial \xi}{\partial x}+\frac{\partial \eta}{\partial y}+\frac{\partial \zeta}{\partial z} \right) dxdydz=\iint (l\xi +m\eta+n\zeta)dS,$

where the volume integral on the left is taken through the volume within a closed surface S, and the surface integral on the right is taken over S, and l, m, n denote the direction cosines of the normal to S drawn outwards. There is a corresponding theorem for a closed curve in two dimensions, viz.,

$\iint \left(\frac{\partial \xi}{\partial x}+\frac{\partial \eta}{\partial y} \right) dxdy=\int \left( \xi\frac{dy}{ds}+\eta\frac{dx}{ds} \right) ds,$

the sense of description of s being the positive sense. This theorem is a particular case of a more general theorem called “Stokes’s theorem.” Let s denote the edge of an open surface S, and let S be covered with a network of curves so that the meshes of the network are nearly plane, then we can choose a sense of description of the edge of any mesh, and a corresponding sense for the normal to S at any point within the mesh, so that these senses are related like the directions of rotation and translation in a right-handed screw. This convention fixes the sense of the normal (l, m, n) at any point on S when the sense of description of s is chosen. If the axes of x, y, z are a right-handed system, we have Stokes’s theorem in the form

$\int_s (udx+vdy+wdz)=\iint \left \{l\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z} \right)+m\left(\frac{\partial u}{\partial x}-\frac{\partial w}{\partial x} \right) \right. \left. +n\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right) \right \}d\text{S},$

where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green’s theorem the differential coefficients ∂/∂x, ∂/∂y, ∂/∂z must be continuous within S. Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green’s theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several physical theories.

54. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two stages. It is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the

second set of variables. It is necessary in the second place to determine the limits of integration which must be employed when the integral in terms of the new variables is evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by x1, x2,, xn, and those of the other set by u1, u2,, un, we have the relation

$dx_1dx_2 \ldots dx_n=\frac{\partial(x_1,x_2,\ldots,x_n)}{\partial(u_1,u_2,\ldots,u_n)}du_1 du_2 \ldots du_n.$