Difference between the Jacobian matrix and the metric tensor

Question

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial e_j}+\dfrac{\partial y}{\partial e_i}\dfrac{\partial y}{\partial e_j}+\dfrac{\partial z}{\partial e_i}\dfrac{\partial z}{\partial e_j}$). As far as I understand it is used when transforming the arc element ds from one coordinate system e.g Cartesian to another one e.g. cylindrical polars. Since the coordinates in the cylindrical polars are orthogonal only $g_{11},g_{22},g_{33}$ are non zero. And we can see that the non zero components of the metric tensor are actually the same as the magnitude of metric coefficients $magnitude(h_i)=g_{ii}$. But the metric coefficients are also present in the Jacobian matrix as collumns of the Jacobian matrix. But you can also use the Jacobian matrix to do the coordinate transformation.

So based on that I am wondering whether there is a relation between the Jacobian matrix and the metric tensor? e.g. Jacobian matrix is used when we transform in the coordinate system with the locally perpendicular axis, but the metrix tensor is used more generally?

Thank you for all the answeres in advanced

Answer 1

A metric tensor takes two tangent vectors and returns a number, their inner product. Under a coordinate transformation or a map between manifolds, tangent vectors $u$ are transformed (pushed-forward) by the differential of the map represented by the Jacobian matrix: $u\mapsto Ju$, and the Euclidean inner product $u^Tu\mapsto(Ju)^T(Ju)=u^TGu$, where $G=J^TJ$ is the matrix of the new metric tensor. From a relevant question about volume forms, you may see sometimes that the new volume element gets a factor written either $|\det J|=\sqrt{|\det G|}$. Answering your 2nd question, neither matrix has to be diagonal.