Consider a function real valued function $f$ whose expression is given in terms of rectangular coordinates $(x,y,z)$ and the function $\tilde f$ such that: $$ \tilde f(r,\theta,z)=f\circ g(r,\theta,z) $$ where $g$ is the change of variables from the cylindrical coordinates to rectangular coordinates.

Let's apply the chain's rule, we get $$ \nabla \tilde f[r,\theta,z]^\intercal = \nabla f[g(r,\theta,z)]^\intercal\times Jac_g(r,\theta,z) \qquad (\star) $$ where $Jac_g$ is the Jacobian of $g$.

  1. Cylindrical coordinates

It is well known (see Del operator in Cylindrical coordinates (problem in partial differentiation)) that: $$ \frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k = \frac{\partial f}{\partial r}e_r+\frac{1}{r}\frac{\partial f}{\partial \theta}e_\theta+\frac{\partial f}{\partial z}e_z \qquad (\star \star) $$ where $(i,j,k)$ is the orthonormal basis related to rectangular coordinates and $(e_r,e_\theta,e_z)$ is the orthonormal basis related to the cylindrical coordinates. This equation results from a change of variables and a change of basis but in the first point, I only did a change of variables ... I don't like how they usually proof $(\star \star)$ (see for instance the accepted answer in the referred post), because they use the same notation $f$ for the function of different variables.

My question is the following: what are the (straighforward) connections between $(\star)$ and $(\star \star)$ ?

More precisely I wonder the following questions: in the equation $(\star)$, what are the basis of vectors assocaited to this matrices ? Is it $(e_r,e_\theta,e_z)$ ?

  1. The first point to determine gradient in cylindrical coordinates may be fastidious for other coordinates system (spherical), that's why I am thinking to use $(\star)$. Also I am wondering if it's possible to do the same in the Frenet frame ? that is to express the gradient in the Frenet coordinates ?

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