Suppose I have a real vector $x \in \mathbb{R}^n$ and a real, nonlinear vector-valued function $\sigma(x) \in \mathbb{R}^n$. I denote the matrix Jacobian
$$\partial_x \sigma(x)$$
Suppose I have a linear transformation $P: \mathbb{R}^n \rightarrow \mathbb{R}^k$. What is the Jacobian of the projected state space with respect to the projected input? In other words, what is
$$\partial_{Px} P\sigma(x)$$
How would I go about calculating this?