Metric tensor in cylindrical coordinates

Question

In a system of curvilinear coordinates, $(q^1,q^2,q^3)$, basis vector can be derived by differentiating a generic position $M(q^1,q^2,q^3)$ with respect to the coordinates, ie $$\mathbf{e}_i=\frac{\partial M}{\partial q^i}$$ and the expression of the metric tensor is $$g_{ik}=\mathbf{e}_i\cdot \mathbf{e}_k$$ Can I find the expression of this metric tensor in a system of cylindrical coordinates without using any other system of coordinates?

Answer 1

It all depends on what you mean by the dot product in your defining equation for $g_{ik}$. If you are defining an abstract Riemannian metric, then your manifold is an abstract thing. But, ordinarily, when you refer to cylindrical coordinates, you're thinking of $\Bbb R^3$ as your ambient space and you're using "$\cdot$" to represent the dot product on $\Bbb R^3$. In that case, you're going to need to compute $\mathbf e_i$ as vectors in $\Bbb R^3$, and for that you'll need specific coordinates in $\Bbb R^3$ for $M(r,\theta,z)$.