Let's say $\mathbf{x}$ is a uniform $d$-dimensional random vector, whose support set is $[-1,1]^d$.
Let's say $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is an arbitrary differentiable bijective function, and the random vector $f(\mathbf{x})$ has the same distribution as $\mathbf{x}$.
Can we conclude that $\frac{\partial f_i}{\partial x_j}=0$ if $i \ne j$ ?
Edited - Additional condition:
Assume that $f$ is not a coordinate-exchanging function (i.e. $f$ is not a function that only re-permutes different components of $\mathbf{x}$)