It is often stated and proved in textbooks that the momentum density is also the energy flux.
The explanation is often done using the dust model.
However, it is possible that in a real fluid, there is heat conduction via particle collision. There is energy flux, but since no molecules are ever...
There is no need to use Green's Theorem.
I have just checked Schaum's Advanced Calculus by Wrede & Speigel
They just use the area of a parallelogram with vectors and derived the entire thing simply, geometrically and intuitively. The Jacobian just pops out naturally.
(ref: page 216, Second...
I must apologize.
What I meant is rank 3 and rank 4, and not 3rd and 4th dimension. So sorry.
You have got a proof for the general form of rank 3 or rank 4 isotropic cartesian tensor?
There are 2 approaches to learning tensors. The component approach and the component-free approach.
In general for physics and engineering students, the component approach is easier to start with. The one-form is the component-free approach.
The 2 books I recommend is the component approach. It...
My first tensor book is Tensor Calculus by Kay. Very good for introduction. There are many which starts from even a simpler baseline.
The second one is Synge and Schild.
Try them!
(note: isn't this suppose to be in the calculus page?)