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

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.

$\iint d\sigma \,u=\iiint dv\,\nabla u,$

(1)

$\iint d\sigma \,\omega =\iiint dv\,\nabla \omega ,$

(2)

$\iint d\sigma .\omega =\iiint dv\,\nabla .\omega ,$

(3)

$\iint d\sigma .\Phi =\iiint dv\,\nabla .\Phi ,$

(4)

$\iint d\sigma \times \omega =\iiint dv\,\nabla \times \omega ,$

(5)

$\iint d\sigma \times \Phi =\iiint dv\,\nabla \times \Phi .$

(6)

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.)

$\int d\rho \,u=\iint d\sigma \times \nabla u,$

(1)

$\int d\rho \,\omega =\iint d\sigma \times \nabla \omega ,$

(2)

$\int d\rho .\omega =\iint d\sigma .\nabla \times \omega ,$

(3)

$\int d\rho .\Phi =\iint d\sigma .\nabla \times \Phi ,$

(4)

$\int d\rho \times \omega =\iint \nabla \omega .d\sigma -\iint d\sigma \,\nabla .\omega .$

(5)

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,

$\nabla [u\omega ]=\nabla u\omega +u\nabla \omega ,$

(1)

$\nabla [\tau \times \omega ]=\nabla \tau \times \omega -\nabla \omega \times \tau ,$

(2)

$\nabla \times [\tau \times \omega ]=\omega .\nabla \tau -\nabla .\tau \omega -\tau .\nabla \omega +\nabla .\omega \tau ,$

(3)

$\nabla (\tau .\omega )=\nabla \tau .\omega +\nabla \omega .\tau ,$

(4)

$\nabla .\{\tau \omega \}=\nabla .\tau \omega +\tau .\nabla \omega ,$

(5)

$\nabla \times \{\tau \omega \}=\nabla \times \tau \omega -\tau \times \nabla \omega ,$

(6)

$\nabla .\{u\Phi \}=\nabla u.\Phi +u\nabla .\Phi ,$

(7)

${\text{etc.}}$

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