I would be greatful for your comments.

Question.

A function $\displaystyle f:R^2->R^2$ is defined by

$\displaystyle f(x,y)=(u,v)^T=(x-y,x^2-y^2)$.

(a) Find the set of critical points of $\displaystyle f$.

Roughly sketch the u and v contours in the $\displaystyle (x,y)$-plane together with the critical points of $\displaystyle f$. Inlcude and label the contours u=0 and v=0.

(b) Write down the derivative of the function $\displaystyle f$ and the derivative of the local inverse at a non-critical point $\displaystyle (x,y)$.

Deduce an expression for $\displaystyle \frac{\partial{x}}{\partial{u}}$ and evaluate it at the point $\displaystyle (x,y)$=(2,1).

(c) Find, if possible, an appropriate formula for the local inverse function at each of the points $\displaystyle (x,y)$=(2,1) and (1,1) justifying your answers. Carefully state the region in the $\displaystyle (u,v)$-plane on which the local inverse is defined and its image on the $\displaystyle (x,y)$-plane.

On separate diagrams, sketch the $\displaystyle u$ and $\displaystyle v$ contours through each of the given points. Comment on the relationship between the contours in each diagram.

(d) Find $\displaystyle \frac{\partial{x}}{\partial{u}}$ directly from the local inverse function you found in part (c).

Evaluate it at the image point of $\displaystyle (x,y)$=(2,1) in the $\displaystyle (u,v)$-plane and verify that this agrees with the result in part (b).

(e) Write down the equations of the tangent flat to $\displaystyle f$ at the point $\displaystyle (x,y)$=(2,1).

I'll post my work in a separate post.