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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.
 Fall '08
 STAFF
 Math, Calculus

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