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I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of $F$:

enter image description here Our professor didn't teach us that much about the math of tensor, which makes it very difficult in doing this question.

I have calculated $A_\mu$, $A_\mu A^\mu$, $\partial_\mu A^\mu$, $\partial_\nu(\partial_\mu A^\mu)$, $F_{\mu\nu}$. However, I have trouble dealing with the mixed tensor:

$F_\mu{}^\nu = g^{\rho\nu} F_{\mu\rho}$, which we can get $g^{\rho\nu}$ by $g_{\mu\rho} g^{\rho\nu} = \delta_\mu{}^\nu$.

However, I am confused in how to find $g^{\rho\nu}$ by $g_{\mu\rho} g^{\rho\nu} = \delta_\mu{}^\nu$, since it introduced a new variable of $\rho$. How do we deal with $\rho$?

Is there any kind of rules in dealing with tensor manipulation?

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    $\begingroup$ Can you please fix your MathJax so that it is readable? $\endgroup$ – G. Smith Oct 9 '19 at 15:35
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The inverse metric tensor $g^{\mu\nu}$ (or with any other choice of index naming) is the inverse of $g_{\mu\nu}$, in the matrix sense. Since it's usually $4\times 4$ or larger, I would shug it into Mathematica or Wolfram Alpha to get $g^{\mu\nu}$.

If $g_{\mu\nu}$ is diagonal, then $g^{\mu\nu}$ is element-wise the reciprocal of each element of $g_{\mu\nu}$. Once you have it, you can find your "mixed" tensor via,

$$F^\nu_\mu = g^{\rho\nu}F_{\mu\rho} = \sum_{\rho = 0}^3 g^{\rho\nu} F_{\mu\rho}.$$

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