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 \frac{∂F}{∂s} and \frac{∂F}{∂t} in terms of \frac{∂f}{∂x}, \frac{∂f}{∂y}, \frac{∂X}{∂s}, \frac{∂X}{∂t}, \frac{∂Y}{∂s} , \frac{∂Y}{∂t} .

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

\frac{∂^{2}F}{∂s^{2}} = \frac{∂f}{∂x} \frac{∂^{2}X}{∂s^{2}} + \frac{∂^{2}f}{∂x^{2}} (\frac{∂X}{∂s} )^{2} + 2\frac{∂X}{∂s} \frac{∂Y}{∂s} \frac{∂^{2}f}{∂x∂y} + \frac{∂f}{∂y} \frac{∂^{2}Y}{∂s^{2}} + \frac{∂^{2}f}{∂y^{2}} (\frac{∂Y}{∂s} )^{2} .

(c) Find similar formulas for the partial derivatives \frac{∂^{2}F}{∂s∂t} and \frac{∂^{2}F}{∂t^{2}} .

2. Let, in this problem, all functions have continuous partial derivatives of all orders. For a function f : ℝ^{3} → ℝ, a function grad(f) : ℝ^{3} → ℝ^{3} is given by

grad(f) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \frac{∂f}{∂x} ; \frac{∂f}{∂y} ; \frac{∂f}{∂z} ) .

For a function g : ℝ^{3} → ℝ^{3} where

g(x) = ( \begin{matrix}g_{1}(x) \\ g_{2}(x) \\ g_{3}(x) \end{matrix} ) ,

a function curl(g) : ℝ^{3} → ℝ^{3} is given by

curl(g) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}\frac{∂g_{3}}{∂y} − \frac{∂g_{2}}{∂z} \\ \frac{∂g_{1}}{∂z} − \frac{∂g_{3}}{∂x} \\ \frac{∂g_{2}}{∂x} − \frac{∂g_{1}}{∂y} \end{matrix} ) .

Finally, for a function h : ℝ^{3} → ℝ^{3} where

h(x) = ( \begin{matrix}h_{1}(x) \\ h_{2}(x) \\ h_{3}(x) \end{matrix} ) ,

a function div(h) : ℝ^{3} → ℝ is given by

div(h) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = \frac{∂h_{1}}{∂x} + \frac{∂h_{2}}{∂y} + \frac{∂h_{3}}{∂z} .

Prove that curl(grad(f)) = 0, div(curl(g)) = 0.

3. Let

F (( \begin{matrix}x \\ y \\ z \end{matrix} )) = ( \begin{matrix}xy^{2}z^{2} \\ z^{2} sin y \\ x^{2}e^{y} \end{matrix} )).

Calculate div(F), curl(F).

4. Consider the function f : ℝ^{2} → ℝ given by f(x, y) = x^{2} + y^{2}. Determine the tangent plane to the graph of the function f (which is a surface in ℝ^{3} at a point

( \begin{matrix}x \\ y \\ z \end{matrix} ).

Challenge problem:

5. In the xz plane in ℝ^{3}, consider the circle C given by th equation

(x − 2)^{2} + z^{2} = 1.

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 ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ M.

(c) Consider the function f : M → M given as follows: Let C_{α} ⊂ M be the image of C under rotation by the angle α around the z-axis. Then f maps C_{α} to itself by rotating it by the same angle α. (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 +\frac{π}{2} .) Calculate Df.