Find the partial derivatives of the function given by where is the open set of all such that x;SPMgt;0, y;SPMgt;0,z;SPMgt;0.
At what points are the following functions continuous? (Prove using theorems from class.)
Show that is continuous at , but that
(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 spanned by
[Express the problem as a system of linear equations.]
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 ).
In special relativity theory, particles are represented by points in timespace where x,y,z are cartesian coordiantes in space, t is time. An inertial observer 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: . The spacetime coordinate system of one inertial observer A is related to the coordinate system of another inertial observer B by a Lorentz transformation, which is a linear transformation satisfying
If B travels at speed v along the x axis relative to A, then the Lorentz transformation is
is called the Lorentz factor.
(a) Verify that the Lorentz transformation defined above satisfies (1).
(b) Considering v,c constants (which they are), the Lorentz transformation is a linear map . Find its matrix (with respect to standard bases).
(c) Using symmetry, find the Lorentz transformation in the general case when B travels in space at constant vector speed relative to A.
(d) A space station 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 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 in coordinates relative to the space station A. At what time, and with what coordinates will the explosion occur relative to B? [Use a calculator.]