Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/48

55. Surface integrals.—The integral $\iint \omega .d\sigma ,$ in which $d\sigma$ represents an element of some surface, is called the surface-integral of $\omega$ for that surface. It is understood here and elsewhere, when a vector is said to represent a plane surface (or an element of surface which may be regarded as plane), that the magnitude of the vector represents the area of the surface, and that the direction of the vector represents that of the normal drawn toward the positive side of the surface. When the surface is defined as the boundary of a certain space, the outside of the surface is regarded as positive.

The surface-integral of any given space (ie., the surface-integral of the surface bounding that space) is evidently equal to the sum of the surface-integrals of all the parts into which the original space may be divided. For the integrals relating to the surfaces dividing the parts will evidently cancel in such a sum.

The surface-integral of $\omega$ for a closed surface bounding a space $dv$ infinitely small in all its dimensions is

$\nabla .\,\omega \,dv.$

This follows immediately from the definition of $\nabla \omega ,$ when the space is a parallelopiped bounded by planes perpendicular to $i,j,k.$ In other cases, we may imagine the space—or rather a space nearly coincident with the given space and of the same volume $dv$—to be divided up into such parallelopipeds. The surface-integral for the space made up of the parallelopipeds will be the sum of the surface-integrals of all the parallelopipeds, and will therefore be expressed by $\nabla .\,\omega dv.$ The surface-integral of the original space will have sensibly the same value, and will therefore be represented by the same formula. It follows that the value of $\nabla .\,\omega$ does not depend upon the system of unit vectors employed in its definition.

It is possible to attribute such a physical signification to the quantities concerned in the above proposition, as shall make it evident almost without demonstration. Let us suppose $\omega$ to represent a flux of any substance. The rate of decrease of the density of that substance at any point will be obtained by dividing the surface-integral of the flux for any infinitely small closed surface about the point by the volume enclosed. This quotient must therefore be independent of the form of the surface. We may define $\nabla .\,\omega$ as representing that quotient, and then obtain equation (1) of No. 54 by applying the general principle to the case of the rectangular parallelopiped.

56. Skew surface-integrals.—The integral $\iint d\sigma \times \omega$ may be called the skew surface-integral of $\omega .$ It is evidently a vector. For a closed surface bounding a space $dv$ infinitely small in all dimensions this integral reduces to $\nabla \times \omega dv,$ as is easily shown by reasoning like that of No. 55.