By Paulo Ney de Souza, Jorge-Nuno Silva (auth.)

In 1977 the maths division on the collage of California, Berkeley, instituted a written exam as one of many first significant standards towards the Ph.D. measure in arithmetic. Its goal was once to figure out even if first-year scholars within the Ph.D. software had effectively mastered simple arithmetic with the intention to proceed within the application with the chance of good fortune. given that its inception, the examination has develop into a huge hurdle to beat within the pursuit of the measure. the aim of this publication is to publicize the cloth and reduction within the practise for the exam throughout the undergraduate years on account that a) scholars are already deeply concerned with the fabric and b) they are going to be ready to take the examination in the first month of the graduate software instead of within the center or finish of the 1st yr. The publication is a compilation of roughly 9 hundred difficulties that have seemed at the initial checks in Berkeley over the past two decades. it's a useful resource of difficulties and options for each arithmetic scholar who plans to go into a Ph.D. software. scholars who paintings via this e-book will strengthen challenge fixing talents in parts akin to actual research, multivariable calculus, differential equations, metric areas, advanced research, algebra, and linear algebra. the issues are equipped through topic and ordered in an expanding point of hassle. Tags with the precise examination yr give you the chance to rehearse whole examinations. The appendix comprises directions on having access to digital models of the checks in addition to a syllabus, facts of passing ratings, and a Bibliography used through the recommendations. This new version includes nearly one hundred twenty new difficulties and two hundred new options. it truly is an amazing potential for college kids to reinforce their beginning in easy arithmetic and to arrange for graduate stories.

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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.