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Math 296 problems 8

Igor Kriz

Regular problems:


Find the partial derivatives of the function tex2html_wrap_inline130 given by tex2html_wrap_inline132 where tex2html_wrap_inline134 is the open set of all tex2html_wrap_inline136 such that x;SPMgt;0, y;SPMgt;0,z;SPMgt;0.


At what points tex2html_wrap_inline140 are the following functions continuous? (Prove using theorems from class.)

(a) tex2html_wrap_inline142

(b) tex2html_wrap_inline144 .




Show that tex2html_wrap_inline148 is continuous at tex2html_wrap_inline150 , 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 tex2html_wrap_inline156 spanned by





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

Challenge problems:


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 tex2html_wrap_inline164 ).


In special relativity theory, particles are represented by points in timespace tex2html_wrap_inline166 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: tex2html_wrap_inline174 . The spacetime coordinate system tex2html_wrap_inline166 of one inertial observer A is related to the coordinate system tex2html_wrap_inline180 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 tex2html_wrap_inline198 . 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 tex2html_wrap_inline202 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 tex2html_wrap_inline216 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 tex2html_wrap_inline222 in coordinates tex2html_wrap_inline224 relative to the space station A. At what time, and with what coordinates will the explosion occur relative to B? [Use a calculator.]

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Igor Kriz
Sun Mar 8 19:40:58 EST 1998