I'm trying to create some notes on multivariable calculus, but It has been hard to find the geometric point of view of this:
- Jacobian Matrix (Derivative of a VF)
- Laplacian
- Nabla operator
- Curl
- Divergence
And most commonly some useful identities such as the curl of a gradient, divergence of a gradient.
Many of the book that I have look up are theorical based. And yep, many videos on YouTube show kind of the overall the physical phenomena but they assume some background on physics. :(
I'm interested in explaining this operators with the help of Mathematica or any kind of graphics to show how they work on the vector field.
For example, I do know that the divergence is a measurement of how much flow enters on the neighborhood of P compared to how much it leaves. And the curl measures the rotation of the vector field in the neighborhood around P. But I can't think of a graph to explain this. Neither I have found such a explanation for the Jacobian matrix besides being the linear map which works as a the derivative nor the laplacian or nabla operators.
I would love if you can share some of your favorite books or any idea, having in mind that my knowledge in physics is merely basic.