MA 51000 HOMEWORK ASSIGNMENT #11 SOLUTIONS
Problem 1. Sec. 6.1, pg. 375; prob. 8 In Exercises 8 and 9, let T (x) = A x, where A is a 2 2 matrix. Show that T is one-to-one if and only if the determinant of A is not zero. Solution: First assume that the det
MA 51000 HOMEWORK ASSIGNMENT #7 SOLUTIONS
Problem 1. Sec. 4.1, pg. 274; prob. 18 Let c be a path in R3 with zero acceleration. Prove that c is a straight line or a point. Solution: Write c = c(t). The acceleration is the second derivative: a = c (t). Sinc
MA 51000 HOMEWORK ASSIGNMENT #13 SOLUTIONS
Problem 1. Sec. 7.4, pg. 472; prob. 16 Prove Pappus' theorem: Let c : [a, b] R2 be a C 1 path whose image lies in the right half plane and is a simple closed curve. The area of the lateral surface generated by ro
MA 55300 HOMEWORK ASSIGNMENT #5 SOLUTIONS
Problem 1. Sec. 3.3, pg. 225; prob. 42 Suppose that a pentagon is composed of a rectangle topped by an isosceles triangle (see Figure 3.3.9). If the length of the perimeter is fixed, find the maximum possible area
MA 51000
GOINS
MIDTERM EXAM
This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not
allowed, but you may use your own personal notes; the lecture notes posted on the course
web site; homework solutions posted on the course we
MA 51000 MIDTERM EXAM SOLUTIONS
Problem 1. Let A = aij be an m n matrix, and x be a vector in Rn . Prove the following generalization of the Cauchy-Buniakovsky-Schwarz inequality:
m n
|A x| |A| |x|
where
|A| =
i=1 j =1
aij 2 .
Solution: Express the matrix
MA 51000
GOINS
SAMPLE MIDTERM EXAM
This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not
allowed, but you may use your own personal notes; the lecture notes posted on the course
web site; homework solutions posted on the co
MA 51000 SAMPLE MIDTERM EXAM SOLUTIONS
Problem 1. Let a, b, and c be three vectors in R3 . Assuming that a = 0, show that ab=ac and ab=ac
if and only if b = c. (Hint: Reduce to the case where c = 0.) Solution: (=) If b = c the identities a b = a c and a b
MA 51000 HOMEWORK ASSIGNMENT #12 SOLUTIONS
Problem 1. Sec. 6.4, pg. 415; prob. 5 dA , where D is the unit disk in R2 . (a) Evaluate (x2 + y 2 )2/3 D (b) Determine the real numbers for which the integral is the unit disk.
D
dA is convergent, where again D
MA 51000 HOMEWORK ASSIGNMENT #10 SOLUTIONS
Problem 1. Sec. 5.4, pg. 354; prob. 7 Compute the volume of an ellipsoid with semiaxes a, b, and c. (Hint: Use symmetry and first find the volume of one half of the ellipsoid.) Solution: Define the ellipsoid by t
MA 51000 HOMEWORK ASSIGNMENT #9 SOLUTIONS
Problem 1. Sec. 5.1, pg. 326; prob. 5 A lumberjack cuts out a wedge-shaped piece W of a cylindrical tree of radius r obtained by making two saw cuts to the tree's center, one horizontally and one at an angle . Com
2. Find the minimum and maximum values of f(:n, y)=.x +21312 2_ in - cfw_Jam~
boundedbyz=0,x=3andy=0,y=3. \-%:(i*\jv+(3") ",2
1: '2122 I "325*235'2 va'-
W (3475+ W1
PA
('9
1.
m
+
\n/
4. Let x, y, and 2 be positive numbers Show that (mg/2W3 < (a: + y + 21
MA 51000 HOMEWORK ASSIGNMENT #8 SOLUTIONS
Problem 1. Sec. 4.3, pg. 294; prob. 15 In Exercises 13 to 16, show that the given curve c(t) is a flow line of the given velocity vector field F(x, y, z). c(t) = sin t, cos t, et ; F(x, y, z) = y, -x, z .
Solution
MA 51000 HOMEWORK ASSIGNMENT #6 SOLUTIONS
Problem 1. Sec. 3.5, pg. 254; prob. 10 (a) Define x : R2 R by x(r, ) = r cos and define y : R2 R by y(r, ) = r sin . Show that (x, y) (r, ) = r0 .
(r0 ,0 )
(b) When can we form a smooth inverse function r(x, y), (