MATH 296 PROBLEMS 8

IGOR KRIZ

Regular problems:

1. Find the partial derivatives of the function f : U → ℝ given by f(x, y, z) =

x^{(y^{z})} where U ⊂ ℝ^{3} is the open set of all ( \begin{matrix}x \\ y \\ z \end{matrix} ) such that x > 0, y > 0, z > 0.

2. At what points ( \begin{matrix}x \\ y \end{matrix} ) are the following functions continuous? (Prove using

theorems from class.)

(a) f(x, y) = ln(cos(x^{2} + y^{2}))

(b) F (x, y) = y^{2} + arctan \frac{x+y}{1−xy}.

3. Let

f(x, y) = \{ \begin{matrix}x sin \frac{1}{y} & if & y ≠ 0, \\ 0 & if & y = 0. \end{matrix}

Show that f : ℝ^{2} → ℝ is continuous at ( \begin{matrix}0 \\ 0 \end{matrix} ), but that

lim_{y→0}( lim_{x→0} f(x, y)) ≠ lim_{x→0}(lim_{y→0} f(x, y)).

4.

(a) Prove that an intersection of subspaces of a vector space U is a subspace.

(b) Find a basis of the intersection of the subspaces of ℝ^{3} spanned by

( \begin{matrix}1 \\ 2 \\ −1 \end{matrix} ) , ( \begin{matrix}2 \\ 2 \\ 1 \end{matrix} )

and

( \begin{matrix}0 \\ 1 \\ 1 \end{matrix} ) , ( \begin{matrix}1 \\ 1 \\ 1 \end{matrix} )

.

[Express the problem as a system of linear equations.] Challenge problems:

5. In a metric space X, prove that: (a) An intersection of two open sets is open.

(b) A union of arbitrarily (possibly infinitely) many open sets is open.

(c) A set is open if and only it is a union of (possibly infinitely many) open balls (sets of the form U_{ϵ}(x) = {y ∈ X|ρ(x, y) < ϵ}).

6. In special relativity theory, particles are represented by points in timespace

( \begin{matrix}x \\ y \\ z \\ t \end{matrix} ) where x, y, z are cartesian coordiantes in space, t is time. An intertial ob-

server travels at constant speed in space (possibly 0, but that is relative). Einstein’s basic principle says that all laws of physics are equal for all inertial observers. In particular, the speed of light is the same: c = 3.10^{8}m/s. The spacetime coordi-

nate system ( \begin{matrix}x \\ y \\ z \\ t \end{matrix} ) of one inertial observer A is related to the coordinate system

( \begin{matrix}x′ \\ y′ \\ z′ \\ t′ \end{matrix} ) of another inertial observer B by a Lorenz transform, which is a linear

transformation satisfying x^{2} + y^{2} + z^{2} − c^{2}t^{2} = x^{′2} + y^{′2} + z^{′2} − c^{2}t^{′2}. If B travels at speed v along the x axis relative to A, then the Lorenz transform is

x′ = γ(x − vt), y′ = y, z′ = z, t′ = γ(t − vx/c^{2})

where

γ = \frac{1}{(1 − v^{2}/c^{2})^{1/2}}

is called the Lorenz factor.

(a) Considering v, c constants (which they are), the Lorenz transform is a linear map ℝ^{4} → ℝ^{4}. Find its matrix (with respect to standard bases).

(b) Using symmetry, find the Lorenz transform in the general case when B travels

in space at constant vector speed ( \begin{matrix}v_{1} \\ v_{2} \\ v_{3} \end{matrix} ) relative to A.

(c) A spacestation A carries a gyroscopic device which specifies Cartesian coordinates in space with origin A. At time 0, a rocket B departs from A at constant speed 10^{6}m/s in the direction of the x coordinate. The rocket also carries a gyroscope which allows it to maintain a coordinate system parallel to that of A. Now

an explosion occurs at time t = 2.1˙10^{5}s in coordinates ( \begin{matrix}2.5˙10^{5}m \\ 3˙10^{4}m \\ 6˙10^{5}m \end{matrix} ) relative to

the space station A. At what time, and with what coordinates will the explosion occur relative to B? [Use a calculator.]