Let $U_{i,j}$ be a second order tensor. Show that $\frac{\partial U_{i,j}}{\partial x_{k}}$ is a third order tensor.
I know how to prove that the gradient of a scalar field (which is a tensor of order 0) is a vector (which is a tensor of order 1). So, derivation of the entries of a tensor, which are scalar fields, always produce a vector, thus increasing the order of the tensor by one. But how to prove it in general?
For example, Let V be a vector field. Then $T_{i,j} = \frac{\partial V_{i}}{\partial x_{j}}$, where $V_{i}$ are scalar fields. This is the Jacobian matrix and a second order tensor. We can see this \begin{bmatrix} \nabla V_{1} \\ \nabla V_{2} \\ \vdots \\ \nabla V_{n} \end{bmatrix} So in my thoughts it is only necessary to prove that $\nabla V_{n}$ is a vector, which I know how to do, and then we can think of it as a vector in each line, which gives us a matrix. The rationale is the same for the transition order 2 -> order 3, having a matrix, each entry is the gradient of the scalar field and extending the vector gives us another dimension. But is it proven? How to do it in general?