1873_solutions

B and such that the inequality d t u 4 holds whenever

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Unformatted text preview: k  a k 1  3 a k  1 a k 1  1 a k 1  1 a k  a k 1  2 xk a k1 1 1 1 2a k a k1 1 x1 1 dx k d x 1 , x 2 , , x k  x2 u a k1 1 xk 1 dud x 1 , x 2 , , x k 2a k a k1 1 d x 1 , x 2 , , x k 1 a 1  1 a 2  1  a k 1  1 2  a k  a k 1 a 1  a 2    a k 1  a k  a k 1  1  k  1 2  a k  a k1 B a k  1, a k1  2 1 a k 1  1 xk x 1 1 x 2 2 x k k 11 1   xk  x2 a 1  1 a 2  1  a k 1  1 2  a k  a k 1 a 1  a 2    a k 1  a k  a k 1  1  k  1 a 1  1 a 2  1  a k 1  1 2  a k  a k 1 a 1  a 2    a k 1  a k  a k 1  1  k  1 a 1  1 a 2  1  a k 1  1 a k  1 a k 1  1 a 1  a 2    a k 1  a k  a k 1  1  k  1 Thus the assertion p k  1 is true and the truth of p k for each k follows by mathematical induction.  4. Prove that if k is any positive integer then vol Q k  1 . k! This assertion follows at once from Exercise 3 when we take a j  0 for every j. The hard work has now been done. The rest of the exercises in this section follow quite simply. 5. Prove that if a and b are nonnegative numbers and R2 x S  x, y 0 and y 0 and x 2  y 2 1 then 4 ÞÞ x a y b d x, y  a b 4 2 S Hint: Make the substitutions x  over the standard simplex Q 2 . u and y  v  b 1 2 a 1 2 . v one at a time to reduce the integral to one that is taken 6. Given that a, b and c are nonnegative numbers and that R 3 x 0 and y 0 and z S  x, y, z 0 and x 2  y 2  z 2 then 8 ÞÞÞ x a y b z c d x, y, z  S 412 a 1 2 b 1 2 a b c 5 2 c 1 2 . 1 Deduce that the volume of a ball with radius 1 in R 3 is 4/3. 7. Given that S x  x 1 , x 2 , , x k Rk xj 0 for each j and x 1. and given nonnegative numbers a 1 , a 2 , , a k , express the integral ÞS x a x a x a dx 12 k 1 2 k in terms of gamma functions. Deduce that the volume of the ball in R k that has center O and radius 1 is k 1 2 1 . k 2 Work out this expression for a few values of k. 8. a. Suppose that B is the ball in R k with center O and radius r  0 then vol B  k 1 2 rk 1 k 2 . b. Prove that if B is the ball introduced in part a then, in the event that k is even and k  2n then we have n 2n vol B   r . n! c. Prove that if B is the ball introduced in part a then, in the event that k is odd and k  2n  1 then we have n 2 n 1 2 2 n 1 . vol B   r 2n  1 ! 9. Suppose that r  0 and for each positive integer n, suppose that B n is the ball with center O and radius r in the space R n . Show that if we agree to define m B 0  1 then Ý vol B 2n  exp r 2 n0 and Ý vol B 2n1  n0 sinh 2r  .  Exercises on Total Derivatives 1. For each of the following functions f, determine whether or not f is continuous at the point 0, 0 , whether or not the partial derivatives of f exist at 0, 0 and whether or not f is differentiable at 0, 0 . Use Scientific Notebook to sketch the graph of each of these functions. a. We define x2y2 f x, y  x 2 y 2 0 413 if x, y 0, 0 if x, y  0, 0 . 12 10 8 6 4 2 0 -5 0 y 5 2 4 This function is continuous at 0, 0 . Now we have f...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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