I recently came across a youtube video (https://www.youtube.com/watch?v=6O6iZug6e6Y) on transformation electromagnetism (yes this is physics) and some of the math equations that was postulated did not have any references. I have been searching the web to find references but with out any luck. As my mathematical understanding is limited to math used in physics I simply do not really know how to formulate my search correctly.
missing reference number 1
In the youtube video it was postulated that a function $\vec{E}$ in $R^3$ transformed as:
$\vec{E}(\vec{r})=A^T\vec{E'}(\vec{r}')$
under coordinate transformation where ' indicates the transformed coordinate system and A is the jacobian matrix for the given coordinate transformation. I believe this is simply the invers function theorem? https://en.wikipedia.org/wiki/Inverse_function_theorem
missing reference number 2
This is my real missing reference and it is postulated at 8:52 in the video.
A linear operator $F$ is transformed under coordinate transformation as:
$F'(\vec{r}')= \frac{AF(\vec{r})A^T}{\text{det}A}$.
Again A is the jacobian for the given coordinate transformation and the ' indecates the transformed coordinate system.
I apologise for my sloopy physics-formulation of the question.