Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/65

 100. On the interpretation of the preceding formulæ.— Infinite values of the quantity which occurs in a volume-integral as the coefficient of the element of volume will not necessarily make the value of the integral infinite, when they are confined to certain surfaces, lines, or pointa Yet these surfaces, lines, or points may contribute a certain finite amount to the value of the volume-integral, which must be separately calculated, and in the case of surfaces or lines is naturally expressed as a surface-or line-integral. Such cases are easily treated by substituting for the surface, line, or point, a very thin shell, or filament, or a solid very small in all dimensions, within which the function may be supposed to have a very large value.

The only cases which we shall here consider in detail are those of surfaces at which the functions of position ($$u$$ or $$\omega$$) are discontinuous, and the values of $$\nabla u, \nabla \times \omega, \nabla. \omega$$ thus become infinite. Let the function $$u$$ have the value $$u_{1}$$ on the side of the surface which we regard as the negative, and the value $$u_{2}$$ on the positive side. Let $$\Delta u = u_{2} - u_{1}.$$ If we substitute for the surface a shell of very small thickness $$a,$$ within which the value of $$u$$ varies uniformly as we pass through the shell, we shall have $$\nabla u = \nu \frac{\nabla u}{a}$$ within the shell, $$\nu$$ denoting a unit normal on the positive side of the surface. The elements of volume which compose the shell may be expressed by $$a[d\sigma]_{0},$$ where $$[d\sigma]_{0}$$ is the magnitude of an element of the surface, $$d\sigma$$ being the vector element. Hence, Hence, when there are surfaces at which the values of $$u$$ are discontinuous, the full value of $$\text{Pot }\nabla u$$ should always be understood as including the surface-integral  relating to such surfaces. ($$\Delta u'$$ and $$d\sigma '$$ are accented in the formula to indicate that they relate to the point $$\rho '.$$)

In the case of a vector function which is discontinuous at a surface, the expressions $$\nabla. \omega \, dv$$ and $$\nabla \times \omega \, dv,$$ relating to the element of the shell which we substitute for the surface of discontinuity, are easily transformed by the principle that these expressions are the direct and skew surface-integrals of $$\omega$$ for the element of the shell. (See Nos. 65, 56.) The part of the surface-integrals relating to the edge of the element may evidently be neglected, and we shall have