I am trying to understand how the $\nabla$ operator changes whent the coordinates change (note that I have read other similar questions here and don't really understand what is happening):
In $x,y,z$ coordinates of course $\nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})$
Why then in cylindrical coordinates we have $\nabla f=(\frac{\partial f}{\partial ρ},\frac{1}{ρ}\frac{\partial f}{\partial θ},\frac{\partial f}{\partial z})$?
Ι really can't understand why the $\frac{1}{ρ}$ infront of $\frac{\partial f}{\partial θ}$.
I am familiary with the the chain rule and how to write $\frac{\partial}{\partial x^i}$ in other coordinates but this doesn't seem to help me.
Question: Can you explain how to go from the nabla from one set of coordinates to another? In particular from cartesian to cylindrical or to spherical.
As this is tagged with
differential-geometryI'll follow a more geometric approach, if this doesn't suit, I am happy to tag this answer ascommunity-wiki.The computation you seek follows from the general definition of $\nabla$ on a manifold, $M$.
Thence \begin{align} \langle\nabla f(p)|v\rangle&=d_pf(v)\\ &=\sum_i\left.\frac{\partial f}{\partial x^i}\right|_pdx^i(v) \end{align} Here, $p$ is a point in some manifold $M$, and $v \in T_p M$.
Summation here is over basis vectors of the tangent space. Expand this in order to find component $i$ $$ (\nabla f)_i=\frac{1}{k_i}\frac{\partial f}{\partial x^i} $$ Where $k_i$ is the modulus of the $i$th tangent vector.