# Math 296 problems 10

Igor Kriz

Regular problems:

1.

The equations u=f(x,y), x=X(s,t) and y=Y(s,t) define u as a function of s and t, say u=F(s,t).

(a) Use an appropriate chain rule to express the partial derivatives of and in terms of , , , , , .

(b) If f has continuous partial derivatives of second order, prove that

(c) Find similar formulas for the partial derivatives and .

2.

Let, in this problem, all functions have continuous partial derivatives of all orders. For a function , a function is given by

For a function where

a function is given by

Finally, for a function where

a function is given by

Prove that

3.

Let

Calculate , .

4.

Consider the function given by . Determine the tangent plane to the graph of the function f (which is a surface in at a point .

Challenge problem:

5.

In the xz plane in , consider the circle C given by th equation

Let M be the set obtained by rotating the circle C around the z axis (``the surface of a donut'').

(a) By finding local coordinate (=parametrizing) functions at all points, prove that M is a manifold.

(b) Using your parametrizations, calculate the vector tangent space to M at a point .

(c) Consider the function given as follows: Let be the image of C under rotation by the angle around the z-axis. Then f maps to itself by rotating it by the same angle . (In measuring these angles, the angles from the positive part of the x axis to the positive part of the y axis and to the positive part of the z axis is considered to be .) Calculate Df.