I would be greatful for your comments.

Question.

A function is defined by

.

(a) Find the set of critical points of .

Roughly sketch the u and v contours in the -plane together with the critical points of . Inlcude and label the contours u=0 and v=0.

(b) Write down the derivative of the function and the derivative of the local inverse at a non-critical point .

Deduce an expression for and evaluate it at the point =(2,1).

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

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

(d) Find directly from the local inverse function you found in part (c).

Evaluate it at the image point of =(2,1) in the -plane and verify that this agrees with the result in part (b).

(e) Write down the equations of the tangent flat to at the point =(2,1).

I'll post my work in a separate post.