electrodynamic-identities

Useful Identities for Electrodynamics

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Integration by Parts (examples)

$\begin{displaymath}\divg{(f\vA)} = f(\divg{\vA}) + (\vA \cdot \grad)f \end{displaymath}$

$\begin{displaymath}\int_{V/S} \divg{(f\vA)}dV = \int_{V/S} f(\divg{\vA}) + \int_{V/S} (\vA \cdot \grad)f \end{displaymath}$

$\begin{displaymath}\oint_S (f\vA)\cdot d\va = \int_{V/S} f(\divg{\vA}) + \int_{V/S} (\vA \cdot \grad)f \quad\textrm{(divergence theorem)}\end{displaymath}$

$\begin{displaymath}\int_{V/S} f(\divg{\vA}) = \oint_S (f\vA)\cdot d\va - \int_{V/S} (\vA \cdot \grad)f \quad\textrm{(integration by parts, or:)}\end{displaymath}$

$\begin{displaymath}\int_{V/S} (\vA \cdot \grad)f = \oint_S (f\vA)\cdot d\va - \int_{V/S} f(\divg{\vA}) \quad\textrm{(integration by parts)}\end{displaymath}$

$\begin{displaymath}\curl{(f\vA)} = f(\curl{\vA}) - (\vA \times \grad)f \end{displaymath}$

$\begin{displaymath}\int_{S/C} \curl{(f\vA)}\cdot d\va = \int_{S/C} f(\curl{\vA})\cdot d\va - \int_{S/C} (\vA \times \grad)f \cdot d\va \end{displaymath}$

$\begin{displaymath}\oint_C f\vA \cdot d\vell = \int_{S/C} f(\curl{\vA})\cdot d\v... ... (\vA \times \grad)f \cdot d\va \quad\textrm{(Stokes' theorem)}\end{displaymath}$

$\begin{displaymath}\int_{S/C} f(\curl{\vA})\cdot d\va = \oint_C f\vA \cdot d\vel... ... \times \grad)f \cdot d\va \quad\textrm{(integration by parts)}\end{displaymath}$

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Robert G. Brown 2014-08-25