Consider the inner product defined as $\langle x,y \rangle = \sum_{i=1}^n x_i y_i$ on a manifold $ \subset \mathbb{R}^n$. I know that gradient and divergence are physical quantities that have coordinate-independent meanings. Taking two different coordinate systems on a Riemannian manifold in $\mathbb{R}^3$ with the inner product as defined for $n=3$, we have for instance,
$ \nabla a = \dfrac{\partial a}{\partial x} \hat{x} + \dfrac{\partial a}{\partial y} \hat{y} + \dfrac{\partial a}{\partial z} \hat{z} = \dfrac{\partial a}{\partial r} \hat{r} + (1/r)\dfrac{\partial a}{\partial \theta} \hat{\theta} + \dfrac{\partial a}{\partial z} \hat{z}, $
where the gradient of a scalar function $a$ is expressed in Cartesian and cylindrical coordinates respectively in the two equalities. In fact, we can use the fact that writing the gradient field in cylindrical coordinates produces the same vector field, to derive the formula for a gradient field in cylindrical coordinates from that in Cartesian coordinates.
What is the equivalent invariance in the Jacobians of the same vector field expressed in different coordinate systems? The determinant of the Jacobian is the same but rows of the Jacobian seem to express different quantities. For example, each row of the Jacobian matrix is the gradient vector of the corresponding Cartesian component of the vector field in Cartesian coordinates but what is the intuition for the Jacobian matrix in cylindrical coordinates? All I know about the Jacobian matrix is that it's a linear operator that can be applied on a vector in the tangent space at a point on the manifold to result in the change in the different components (expressed in local coordinates) of the vector field at the point. How do I use this to derive what the Jacobian matrix looks like in cylindrical coordinates (or some other local coordinates)?
Really sorry if this is a silly and vague question - I have never studied manifolds formally. Thank you very much for your time!