- #1

Vanilla Gorilla

- 45

- 7

- TL;DR Summary
- Suppose we have an operator that, in Cartesian Coordinates, represents the sum of all the UNMIXED Nth partial derivatives. How would we write a more general formula for this operator that works in any coordinate system?

__Preface__

Generalizing these formulas, let's construct an operator ##\nabla^{n}##, such that $$\nabla^{n}=\frac{\partial^n f}{\partial^n x} + \frac{\partial^n f}{\partial^n y} + \frac{\partial^n f}{\partial^n z}$$ even more generally $$\nabla^{n} =\sum_{i=1}^{m} \frac{\partial^n f}{\partial^n x_{i}}$$ when there are ##m## dimensions.

__My Question__

My first guess would probably be $$\nabla^{n} = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k^{n-1} F \right)$$ but past that, I'm not sure.

Any help is much appreciated!

P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear. Also, I know this isn't high school material, but I am currently in high school, which is why I made my level "Basic/high school level."

Note to moderators: I think that is the most appropriate forum for this post. However, if not, apologies, and please feel free to move at your own discretion :)