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I want to calculate the differential of the function $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $$F(r, \theta, \phi)=\left (r\sin \left (\theta\right )\cos \left (\phi \right ), r\sin \left (\theta\right )\sin \left (\phi \right ), r\cos \left (\theta\right )\right )$$ For that do we have to calculate the Jacobi matrix? Or how is in this case the differential defined?

Mary Star
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I'm not sure that a vector field has a specific 'differential', not as simply as 1-dimensional functions anyway. You can find for example the partial derivatives of it with respect to each of the 3 variables or you can find the $\mathit{divergence}$ of the field, both are related to derivatives.

This article goes into more detail about things like "total derivatives" and more if that's what you mean. Meaning of derivatives of vector fields

jcneek
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  • What about the notions of "total differential", "Fréchet derivative", etc.? – azif00 May 05 '21 at 21:06
  • I was just giving some examples if we are talking about more basic operations, the article I tagged talked about total differentials and more – jcneek May 05 '21 at 21:08
  • So the total differential is the Jacobi matrix for such a function, or not? – Mary Star May 06 '21 at 13:22
  • There are similar in their calculation but the total differential is dependent on a basis and it is a linear map – jcneek May 06 '21 at 13:30