Question

3(a). Let $A^{mu u}$ be an antisymmetric tensor, $S^{ hosigma}$ be a symmetric tensor, and $T^{alphaeta}$ be an arbitrary tensor, with no definite symmetry. Show that $A^{mu u}S_{mu u} = 0$, and that any $T^{alphaeta}$ can be decomposed into symmetric and antisymmetric parts, that is, that $T^{mu u}S_{mu u} = frac{1}{2}(T^{mu u} + T^{ umu})S_{mu u}$ and $T^{mu u}A_{mu u} = frac{1}{2}(T^{mu u} - T^{ umu})A_{mu u}$. 3(b). Derive the continuity equation $partial J^{mu}/partial x^{mu} = 0$ directly from the inhomogeneous Maxwell equation $partial F^{mu u}/partial x^{ u} = mu_0 J^{mu}$.

          3(a). Let $A^{mu
u}$ be an antisymmetric tensor, $S^{
hosigma}$ be a symmetric tensor, and $T^{alphaeta}$ be an arbitrary tensor, with no definite symmetry. Show that $A^{mu
u}S_{mu
u} = 0$, and that any $T^{alphaeta}$ can be decomposed into symmetric and antisymmetric parts, that is, that $T^{mu
u}S_{mu
u} = frac{1}{2}(T^{mu
u} + T^{
umu})S_{mu
u}$ and $T^{mu
u}A_{mu
u} = frac{1}{2}(T^{mu
u} - T^{
umu})A_{mu
u}$.

3(b). Derive the continuity equation $partial J^{mu}/partial x^{mu} = 0$ directly from the inhomogeneous Maxwell equation $partial F^{mu
u}/partial x^{
u} = mu_0 J^{mu}$.

$3(a). Let A^mu u be an antisymmetric tensor, S^ hosigma be a symmetric tensor, and T^alphaeta be an arbitrary tensor, with no definite symmetry. Show that A^mu uSmu u = 0, and that any T^alphaeta can be decomposed into symmetric and antisymmetric parts, that is, that T^mu uSmu u = frac12(T^mu u + T^ umu)Smu u and T^mu uAmu u = frac12(T^mu u - T^ umu)Amu u. 3(b). Derive the continuity equation partial J^mu/partial x^mu = 0 directly from the inhomogeneous Maxwell equation partial F^mu u/partial x^ u = mu0 J^mu.$

Added by Linda C.

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