| ∫∫∫ ( | ∂ξ | + | ∂η | + | ∂ζ | ) dx dy dz = ∫∫ (lξ + mη + nζ) dS, |
| ∂x | ∂y | ∂z |
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.,
| ∫∫ ( | ∂ξ | + | ∂η | ) dx dy = ∫ ( ξ | dy | − η | dx | ) ds, |
| ∂x | ∂y | ds | 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
| ∫s (u dx + v dy + w dz) = ∫∫ { l ( | ∂w | − | ∂v | ) + m ( | ∂u | − | ∂w | ) + n ( | ∂v | − | ∂u | ) } dS, |
| ∂y | ∂z | ∂z | ∂x | ∂x | ∂y |
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 ∂ξ/∂x, ∂η/∂y, ∂ζ/∂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 Change of Variables in a Multiple Integral. 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
| dx1 dx2 ... dxn = | ∂ (x1, x2, ..., xn) | du1 du2 ... dun. |
| ∂ (u1, u2, ..., un) |
In regard to the second stage of the process the limits of integration must be determined by the rule that the integration with respect to the second set of variables is to be taken through the same domain as the integration with respect to the first set.