Consider a 2D plane. The most common choice of coordinates are the Cartesian and Polar coordinate. \[ x = r\cos \theta, \;\; y= r\sin\theta \] The Jacobian matrix is as follows: \[ J = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{bmatrix} = \begin{bmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{bmatrix} \]
Consider a vector \(\mathbf{v}\): \[ \mathbf{v} = v^i e_i \] Consider that we have chosen a different basis \(e'\): \[ e_j' = A_{j}^{i}e_i, \;\;\; e_i = B^{j}_{i}e_j' \] We first have the following equation: \[ e'_j = A^{i}_{j}e_i = A^{i}_{j} B^{k}_{i}e'_{k}=\delta^{j}_ke'_{k}, \;\;\; A^{i}_{j}B^{k}_{i} = \delta_{j}^{k} \] Since the vector itself does not change under the choice of basis: \[ \mathbf{v} = v^{i}e_i = w^{j}e'_j, \;\;\; v^{i}e_i = v^{i} B^{j}_{i}e'_{j} = w^{j}e'_{j}, \;\;\; B^{j}_{i}v^i = w^{j} \] And this is contravariant.