Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/93

Rh 164. The following equations between surface-integrals for a closed surface and volume-integrals for the space enclosed seem worthy of mention. One or two have already been given, and are here repeated for the sake of comparison. It may aid the memory of the student to observe that the transformation may be effected in each case by substituting $$\iiint dv \, \nabla $$ for $$\iint d\sigma.$$

165. The following equations between line-integrals for a closed line and surface-integrals for any surface bounded by the line, may also be mentioned. (One of these has already been given. See No. 60.)     These transformations may be effected by substituting $$\iint [d\sigma \times \nabla]$$ for $$\int d\rho.$$ The brackets are here introduced to indicate that the multiplication of $$d\sigma$$ with the $$i, j, k$$ implied in $$\nabla$$ is to be performed before any other multiplication which may be required by a subsequent sign. (This notation is not recommended for ordinary use, but only suggested as a mnemonic artifice.)

166. To the equations in No. 65 may be added many others, as,        The principle in all these cases is that if we have one of the operators $$\nabla, \nabla., \nabla \times$$ prefixed to a product of any kind, and we make any