Explanation on Lawson's Spin Geometry book

Question

This is from Lawson's book "Spin Geometry", specifically the beginning of chapter 3. He gives us the definition of $\textbf{differential operator of order m}$, then he makes an observation about change of local coordinates, but I don't understand what is $[\partial \tilde{x}/\partial x]_{*}^{*}$ He says that is the $\textbf{symmetrization of the mth tensor power of the Jacobian matrix}$, but I don't know exactly what that is.

I know about symmetrization of a tensor of order m, but what is confusing me is the part of the $\textbf{mth tensor power of the Jacobian matrix}$.

Could anyone explain me what is the author talking about?

Thanks

Answer 1

If you put in all the indices, it will look like this: $$\tilde A{}^{\alpha_1\dots\alpha_m} = \sum A^{\beta_1\dots\beta_m}\frac{\partial \tilde x^{\alpha_1}}{\partial x^{\beta_1}}\dots\frac{\partial\tilde x^{\alpha_m}}{\partial x^{\beta_m}}.$$