Proving the curl is independent of the system

Question

Q: Prove that the differential operators of the divergence and the curl are independent of the system

My lecturer provided the proof for the divergence using index notation, but I wasn't sure how to prove the curl; could someone clarify whether my proof is correct?

Divergence:

$$ \begin{align} \nabla' \cdot \textbf{A}' &= \frac{\partial A'_i}{\partial x'_i} \\ &= \frac{\partial x_j}{\partial x'_i} \frac{\partial}{\partial x_j} L_{ik} A_k \\ &= L_{ij} L_{ik} \frac{\partial A_k}{\partial x_j} \\ &= \delta_{jk} \frac{\partial A_k}{\partial x_j} \\ &= \frac{\partial A_j}{\partial x_j} \\ &= \nabla \cdot \textbf{A} \end{align} $$

Therefore proven.

My proof for the application of this similarly to the curl is as follows:

$$ \begin{align} \nabla' \times \textbf{A}' &= \epsilon_{ijk} \frac{\partial A'_k}{\partial x'_j} \\ &= \epsilon_{ijk} \frac{\partial x_l}{\partial x'_j} \frac{\partial}{\partial x_l} L_{mi} A'_m \\ &= \epsilon_{ijk} L_{jm} L_{jl} \frac{\partial A_m}{\partial x_l} \\ &= \epsilon_{ijk} \delta_{ml} \frac{\partial A_m}{\partial x_l} \\ &= \epsilon_{ijk} \frac{\partial A_l}{\partial x_l} \\ &= \nabla \times \textbf{A} \end{align} $$

Is this a correct proof? Have I made any mistakes and have basically just fudged it? Or am I entirely correct? Any help appreciated!

Answer 1

There are some minor mistakes. The first mistake is the following. You substitute $A'_k$ with $L_{mi} A_m$. This is incorrect for two reasons. The first is that it has to depend on $k$, since $A'_k$ does. The second is that you expressed the matrix multiplication part incorrectly (i.e., you got rows and columns reversed).

More importantly, the fact that you "basically just fudged it" indicates that you do not understand what you are doing. I am not trying to place blame here. (I have "fudged" my way through many concepts the first time I encountered them.) But my point is this: Even if your proof is correct, it is useless to you and your education, because you do not understand why it is correct.

I would recommend two steps. First, make sure you actually understand all of the following five facts. Next, go through the proof of the divergence again and see if you can justify each step using one of these five facts. Then you will be ready to correct your proof of the curl.

1. $x' = L x$
2. $A' = L A$
3. When multiplying a matrix $L$ and a vector $x$, you get (using Einstein notation) $(Lx)_i = L_{ij} x_j$.
4. An orthogonal matrix satisfies $L^T L = I$. (Try writing this as a sum over matrix elements.)
5. If $x'$ is a function of $x$, then we can write $$ \frac{\partial}{\partial x'_i} = \frac{\partial x_j}{\partial x'_i} \frac{\partial}{\partial x_j} $$