Is the Energy Momentum Tensor for Scalar Fields Always Symmetric?

[BillKet]

Homework Statement

Show that if the Lagrangian only depends on scalar fields $\phi$, the energy momentum tensor is always symmetric: $T_{\mu\nu}=T_{\nu\mu}$

Homework Equations

$T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L$

The Attempt at a Solution

So the second therm in the SE tensor is symmetric so we need to prove that the first term is, too. But I am really not sure how to proceed for a general Lagrangian. For example, if we have a term like $(\partial_\mu \phi)^2\partial_\nu \phi$, that wouldn't by symmetric. Am I reading this the wrong way? Thank you!

 

[Orodruin]

Your proposed Lagrangian term is not even Lorentz invariant ...

 

[BillKet]

Your proposed Lagrangian term is not even Lorentz invariant ...

Oh sorry for that, I didn't mean that term is alone, it was just an example. It could be $(\partial_\mu \phi)^2\partial_\nu \phi J^\nu$ where I have an external source. My point was, what terms am I allowed to use (respecting Lorentz invariance) in the general case? I just don't know where to start from in solving this...