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Extra resources for Berkeley Problems in Mathematics

Example text

Prove that f(x, t) > Ofor all x and t. •• , Xn ), j = 1, ... , n. Prove that If(x) - f(y)1 ::: (where ~ lIuli = Jur + ... + u~). JiZKllx - yll 34 2. 22 (Fa83, Sp87) Let f : ~n \ {OJ -+ ~ be a function which is continuously differentiable and whose partial derivatives are uniformly bounded: for all (XI, ... , xn) ~ (0, ... , 0). Show that if n ~ 2, then f can be extended to a continuousfunctiondefined on all of~n. Show that this isfalse ifn 1 by giving a counterexample. 23 (Sp79) Let f : ~n \ {OJ -+ ~ be differentiable.

13 (Sp86) For). a real number, find all solutions of the integral equations qJ(x) =ex +). fox /x-Y)qJ(y)dy, O::sx::s 1, c +). fol e(x-Y)1/I(y)dy, O::s x::s 1. 14 (Sp86) Let V be a finite-dimensional vector space (over C) of coo complex valued functions on ~ (the linear operations being defined pointwise). , f(x + a) belongs to V whenever f (x) does, for all real numbers a). 15 (Fa99) Describe all three dimensional vector spaces V of coo complex valued functions on ~ that are invariant under the operator of differentiation.

Prove that and hence that A < O. 11 (Sp92) Let f be a one-to-one C 1 map of]R3 into ]R3, and let J denote its Jacobian determinant. Prove that if Xo is any point of]R3 and Qr(XO) denotes the cube with center Xo, side length r, and edges parallel to the coordinate axes, then IJ(xo)1 Here, II . II . = r-l>O hmr -3 . vol (f(Qr(XO))) =:: hmsup is the Euclidean norm in ]R 3. 1 (Fa87) Find a curve C in JIt2, passing through the point (3, 2), with the following property: Let L(xo, YO) be the segment of the tangent line to Cat (xo, YO) which lies in the first quadrant.