**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*.

Wed Mar 18 20:39:27 EST 1998