How does the Jacobian relate to sketch of x,y coordinates with u,v constant?

Question

T is a non-linear transformation, with the following component functions: x = u/v, y = v

On a sketch of the x-y plane, with u and v constant, how does the Jacobian, J = 1/v, relate to the sketch of y = u/x in the x,y plane?

My thinking is that it's some sort of scale factor of area. Forgive my naivety, I'm struggling with the geometrical interpretation of the Jacobian.

Any help would be appreciated!

Answer 1

If

$$\begin{align} x &= \dfrac{u}{v} \\ y &= v \end{align}$$

Then

$$\mathbf J(u, v) = \begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\[1em] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} \dfrac{1}{v} & -\dfrac{u}{v^2} \\ 0 & 1\end{bmatrix}$$

The Jacobian determinant is equal to $\dfrac{1}{v}$.

Answer 2

A differentiable transform is locally linear and it maps a (small) circle to an ellipse. The ratio of the areas is the Jacobian determinant. The ellipse axis are the Eigenvectors of the matrix.