# How to find the mixed tensor, contravariant tensor and tensor trace of $F$

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

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?

• Can you please fix your MathJax so that it is readable? – G. Smith Oct 9 '19 at 15:35

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