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.