Page:BatemanElectrodynamical.djvu/6

A term such as $\int \int E_{x}dx\ dt$ may now be understood to mean^[1]

$\int \int E_{x}\left[{\frac {\partial (x,t)}{\partial (y,z)}}{\frac {\partial (y,z)}{\partial (\alpha ,\beta )}}+{\frac {\partial (x,t)}{\partial (z,x)}}{\frac {\partial (z,x)}{\partial (\alpha ,\beta )}}+{\frac {\partial (x,t)}{\partial (x,y)}}{\frac {\partial (x,y)}{\partial (\alpha ,\beta )}}\right]d\alpha \ d\beta$

and a term such as $\int \int \int \rho w_{x}dy\ dz\ dt$ to mean

$\int \int \int \rho w_{x}{\frac {\partial (y,z,t)}{\partial (x,y,z)}}dx\ dy\ dz.$

The equation (II) may now be written

${\begin{array}{ll}\int \int \left[\left(E_{x}-H_{y}{\frac {\partial t}{\partial z}}+H_{z}{\frac {\partial t}{\partial y}}\right)dy\ dz\right.&+\left(E_{y}-H_{z}{\frac {\partial t}{\partial x}}+H_{x}{\frac {\partial t}{\partial z}}\right)dz\ dx\\\\&\left.+\left(E_{z}-H_{x}{\frac {\partial t}{\partial y}}+H_{y}{\frac {\partial t}{\partial x}}\right)dx\ dy\right]\end{array}}$ $=-\int \int \int \left(\rho w_{x}{\frac {\partial t}{\partial x}}+\rho w_{y}{\frac {\partial t}{\partial y}}+\rho w_{z}{\frac {\partial t}{\partial z}}-\rho \right)dx\ dy\ dz.$

Transforming the left-hand side by means of Green's theorem, we get

${\begin{array}{ll}\int \int \int \left[{\frac {d}{dx}}\left(E_{x}-H_{y}{\frac {\partial t}{\partial z}}+H_{z}{\frac {\partial t}{\partial y}}\right)\right.&+{\frac {d}{dy}}\left(E_{y}-H_{z}{\frac {\partial t}{\partial x}}+H_{x}{\frac {\partial t}{\partial z}}\right)\\\\&\left.+{\frac {d}{dz}}\left(E_{z}-H_{x}{\frac {\partial t}{\partial y}}+H_{y}{\frac {\partial t}{\partial x}}\right)\right]dx\ dy\ dz.\end{array}}$

Now

${\frac {dE_{x}}{dx}}={\frac {\partial E_{x}}{\partial x}}+{\frac {\partial E_{x}}{\partial t}}{\frac {\partial t}{\partial x}},$ ${\frac {dH_{y}}{dx}}={\frac {\partial H_{y}}{\partial x}}+{\frac {\partial H_{y}}{\partial t}}{\frac {dt}{dx}};$

hence the above integral may be written

${\begin{array}{l}\int \int \int \left[\left({\frac {\partial E_{x}}{\partial t}}-{\frac {\partial H_{z}}{\partial y}}+{\frac {\partial H_{y}}{\partial z}}\right){\frac {\partial t}{\partial x}}+\left({\frac {\partial E_{y}}{\partial t}}-{\frac {\partial H_{x}}{\partial z}}+{\frac {\partial H_{z}}{\partial x}}\right){\frac {\partial t}{\partial y}}\right.\\\qquad \left.+\left({\frac {\partial E_{z}}{\partial t}}-{\frac {\partial H_{y}}{\partial x}}+{\frac {\partial H_{x}}{\partial y}}\right){\frac {\partial t}{\partial z}}+\left({\frac {\partial E_{x}}{\partial x}}+{\frac {\partial E_{y}}{\partial y}}+{\frac {\partial E_{z}}{\partial z}}\right)dx\ dy\ dz,\right]\end{array}}$

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1. The relation ${\frac {\partial (x,t)}{\partial (\alpha ,\beta )}}={\frac {\partial (x,t)}{\partial (y,z)}}{\frac {\partial (y,z)}{\partial (\alpha ,\beta )}}+{\frac {\partial (x,t)}{\partial (z,x)}}{\frac {\partial (z,x)}{\partial (\alpha ,\beta )}}+{\frac {\partial (x,t)}{\partial (x,y)}}{\frac {\partial (x,y)}{\partial (\alpha ,\beta )}}$ is a particular ease of a general relation given in Scott's Determinants, p. 134.