In this post: Clarification of Tensor, Jacobian
it is written "The Jacobian is in fact an altogether more simple construct. We talk about it in the context of maps from $\mathbb{R}^n \to \mathbb{R}^n$. It is indeed a tensor field (since it's a matrix defined at every point on the manifold $\mathbb{R}^n)$."
I'm wondering whether the Jacobian matrix is truly a tensor. Possible pro-arguments: It can be written as a matrix. Its components are dependent on the choice of a coordinate system.
Possible counter-arguments: Its components are in fact dependent on the choice of two coordinate systems. It cannot be simply written as sum of products of components times basis vectors.
If the Jacobian matrix were a real tensor, its determinant were a tensor density?
