MATH 296 PROBLEMS 9

IGOR KRIZ

Regular problems:

1. Consider the function f : ℝ^{2} → ℝ^{2} given by f(x, y) = (x^{2} + y, xy). Suppose g : U → ℝ^{2} is an inverse function to f, which is defined and has total differential on an open set U ⊂ ℝ^{2}. Find the total differential of g at the point (x^{2} + y, xy).

2. Find all the partial derivatives of the function f(x, y) = e^{xy} cos(x + y) up to order 3.

3. Find the tangent plane to the surface in ℝ^{3} given in parametric form by

( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}s^{2} − t^{3} \\ s + t \\ 2st \end{matrix} )

at the point s = 2, t = 3.

4. Let f : ℝ^{3} → ℝ^{2} and g : ℝ^{3} → ℝ^{3} be two functions given by:

f(( \begin{matrix}x \\ y \\ z \end{matrix} )) = ( \begin{matrix}x^{2} + y + z \\ 2x + y + z^{2} \end{matrix} ) ,

g(( \begin{matrix}u \\ v \\ w \end{matrix} )) = ( \begin{matrix}uv^{2}w^{2} \\ w^{2} sin v \\ u^{2}e^{v} \end{matrix} )

Compute Df, Dg, g ∘ f,D(g ∘ f).

Challenge problems:

5. If f(0, 0) = 0 and

f(x, y) = \frac{xy}{x^{2} + y^{2}} if (x, y) ≠ (0, 0),

prove that the partial derivatives of f exist at every point of ℝ^{2}, although f is not continuous at (0, 0).

6. Suppose f : U → ℝ where U is an open set in ℝ^{n}. Prove that if all partial derivatives exist and are bounded in U (i.e. are all in an interval (−M, M) for some constant M > 0), then f is continuous. [Hint: We proved in 295 that if ϕ is a one-variable function in an interval satisfying |f′(x)| < M everywhere, then |f(x) − f(y)| < M|x − y|. Use this fact.]