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I have two problems.

I'm going through the Classical Theory of Fields by Landau/Lifshitz and in Section 32 they're deriving the energy-momentum tensor for a general field. We started with a generalized action (in 4 dimensions) and ended up with the definition of a tensor:

[itex]

T^{k}_i =q_{,i} \frac{\partial \Lambda}{\partial q_{,k}}-\delta^{k}_i \Lambda

[/itex]

Where [itex]q_{,i} \equiv \frac{\partial q}{\partial x^i}[/itex] and [itex]\Lambda[/itex] is the Lagrangian density of the field. This led to the conclusion that:

[itex]

\frac{\partial T^{k}_i}{\partial x^k}=0

[/itex]

Which is thefirstthing I'm confused about.

Second, using previous results about four divergences:

[itex]

\frac{\partial A^k}{\partial x^k} = 0

[/itex]

If this is true, then it is equivalent to saying [itex]\int A^k dS_k[/itex] is conserved. This led to:

[itex]

P^i = const. \int T^{ik}dS_k

[/itex]

The constant was determined to be [itex]\frac{1}{c}[/itex] but that is unimportant to my question for now. They say the defintion of [itex]T^{ik}[/itex] is not unique since we can add a 2nd rank tensor to this and still retrieve the same result:

[itex]

T^{ik}+\frac{\partial \psi^{ik\ell}}{\partial x^{\ell}}

[/itex]

Where [itex]\psi^{ik\ell}=-\psi^{i\ell k}[/itex]. This apparently still yields:

[itex]

\frac{\partial T^{ik}}{\partial x^k}=0

[/itex]

(for now let's ignore the switching between mixed/contravariant). In other words, the symmetric operator [itex]\frac{\partial^2}{\partial x^k \partial x^{\ell}}[/itex] applied to the antisymmetric (in [itex]k[/itex] and [itex]\ell[/itex]):

[itex]

\frac{\partial^2 \psi^{ik\ell}}{\partial x^k \partial x^{\ell}} = 0

[/itex]

This is mysecondquestion. Why is this zero? Can anyone show me the math behind this? I'm having trouble sorting it out.

Thank you!

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# Second derivative of tensor

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