Calculate the differential of $\mathbb{R}^3$ function

Question

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?

Answer 1

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