Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/142

If $l,m,n$ are the direction-cosines of the normal drawn inwards from the bounding surface at any point, and $dS$ an element of that surface, then we may write

$M=-\iint V(lu+mv+nw)dS-\iiint V({\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}})\,dx\,dy\,dz$;

the integration of the first term being extended over the bounding surface, and that of the second throughout the entire space.

For all spaces within which $u,v,w$ are continuous, the second term vanishes in virtue of equation (1). If for any surface within the space $u,v,w$ are discontinuous but subject to equation (2), we find for the part of $M$ depending on this surface,

$M_{1}=\iint V_{1}(l_{1}u_{1}+m_{1}v_{1}+n_{1}w_{1})dS_{1}$,

$M_{2}=\iint V_{2}(l_{2}u_{2}+m_{2}v_{2}+n_{2}w_{2})dS_{2}$;

where the suffixes $_{1}$ and $_{2}$, applied to any symbol, indicate to which of the two spaces separated by the surface the symbol belongs.

Now, since $V$ is continuous, we have at every point of the surface,

$V_{1}=V_{2}=V\,\!$;

we have also

$dS_{l}=dS_{2}=dS\,\!$;

but since the normals are drawn in opposite directions, we have

$l_{1}=-l_{2}=l,{\color {White}xxxx}m_{1}=m_{2}=m,{\color {White}xxxx}n_{1}=-n_{2}=n;$

so that the total value of M, so far as it depends on the surface of discontinuity, is

$M_{1}+M_{2}=-\iint V(l(u_{1}-u_{2})+m(v_{1}-v_{2})+n(w_{1}-W_{2}))dS$.

The quantity under the integral sign vanishes at every point in virtue of the superficial solenoidal condition or characteristic (2).

Hence, in determining the value of $M$, we have only to consider the surface-integral over the actual bounding surface of the space considered, or

$M=-\iint V(lu+mv+nw)dS$.

Case 1. If V is constant over the whole surface and equal to $C$,

$M=-C\iint (lu+mv+nw)dS$.

The part of this expression under the sign of double integration represents the surface-integral of the flux whose components are $u,v,w,$ and by Art. 21 this surface-integral is zero for the closed surface in virtue of the general and superficial solenoidal conditions (1) and (2).