Math 250 quad Fall 2006 Selected Solutions HH 3.7.10: Computing Operator Norm Let A be an k matrix. Let AT denote the transpose of A. (a) Show that AT A is symmetric. (b) Show that all eigenvalues of AT A are non-negative, and that they are all posi
Mathematics 250 Selected Solutions
Fall 2008 Assignment # 7
MVM 5.2.10: Find the the isosceles trapezoid with maximum area, given the total length of the two sides and the bottom. Response: There are three variables involved in this maximization problem:
Mathematics 250 Assignment # 6 Selected Soltuions
Fall 2008
Exercises; MVM 3.6.3, 3.6.6, 3.6.7, 3.6.8. MVM 3.6.6: Let f be u x v u g = v u y v ascalar-valued function on R2 , and let g be a mapping from R2 to itself. Write . Set F = f g. Compute
2F uv .
Mathematics 250 Selected Solutions
Fall 2008 Assignment #2
MM, Exercise 3.2.1: The equation of the tangent plane to the graph of a scalar-valued function f at a given point a is Df (a)(x - a) = 0. x -1 b) If f ( ) = x2 + y 2 , then Df = [ 2x 2y ]. At the
p po MVM 6.2.8: Show that if f ( V ) is a smooth function of 3 variables, and Vo is a point where none T T o po of the three partial derivatives vanish, so that it the level surface of f through Vo can be parametrized To by solving for any one of the var
Mathematics 250
Fall 2008
Proof Practice # 7 Given open domains U Rm , V R , and functions F : U V and G : V Rk , show that if F and G are continuous, then the composition G F is also continuous. Response: Some of you approached this through the - definit
Mathematics 250
Fall 2008
Proof Practice # 4 Suppose that D and E are open domains in Rk , and that F : D E is a function defined on D and taking values in E. We say that F is invertible iff there is a function G : E D, so that G F = ID and F G = IE . Her
Mathematics 250 Fall 2008 Proof Practice # 3 For a continuous real-valued function f defined on an interval [a, b] in R, and a partition P of [a, b], let RP (f ) denote the lower Riemann sum for f and P , obtained by evaluating f at its minimum point in e
Mathematics 250
Fall 2008
Proof Practice # 8 Let U be an open domain in Rm . Let f : U Rk and g : U R be functions. Define f g to be the function from U to Rk+ whose first k coordinates are the coordinates of f , and whose last coordinates are f (u) the c
Mathematics 250
Fall 2008
Proof Practice # 2 a) Let F : D Rm be a vector-valued function defined on an open domain D Rn . Let's call F locally Lipschitz if, given any point x in D, we can find a ball Br (x) around x, such that F is Lipschitz on Br (x). (N
Mathematics 250 Selected Solutions
Fall 2008 Assignment # 9
MVM, 8.3.2: Calculate the following line integrals. x a) C xy 3 dx, where C is the unit circle, C = : x2 + y 2 = 1 , oriented counterclockwise. y 0 1 b) C zdx + xdy + ydz, where C is the line seg
Mathematics 250 Selected Solutions
Fall 2008 Assignment # 8
AS8.1: What is the relation between 7.5.8 and 7.5.19 a)? 1 1 1 Response: In problem 7.5.8, it is shown that 1 det a1 b1 c1 is the area of the triangle with vertices 2 a2 b2 c2 a1 b c A= , B = 1 ,
Math 250, Fall 2003 Selected Solutions, Assignment # 1 HH 1.5.5 d) Is the set Q rational numbers open or closed in R, the real numbers? Response; The set of rational numbers Q is neither open nor closed in the real numbers R. Rather, both Q and R - Q
Selected Solutions, Assignment 3, Fall 2006 AS3.1 a) Compute the derivative of the identity mapping. Response: The identity is a linear transformation. We will compute the derivative of any linear transformation. Suppose that L : Rk Rm is a linear m
Math 250 Fall 2006 Selected Solutions, Assignment #2 HH 1.7.16 c): Interpret the mapping A A2 on 2 2 matrices as a mapping on R4 by identifying a a a b b matrix with the column vector . Compute the derivative of the mapping on R4 , and show tha
Mathematics 250 Fall 2006 Selected solutions, Assignments #4 and #5 HH 2.7.9: Find a number > 0 such that the set of equations x + y2 = a y + z2 = b z + x2 = c has a unique solution near 0 when |a|, |b|, |c| < . Response: We think of the left h
Mathematics 250 Fall 2006 Selected Solutions, Assignment #6
Hubbard and Hubbard, Problem 3.3.5: Find the cardinality of the set of multiexponents of degree m in with n entries. (Equivalently, find the number of monomials of degree m in n variables.
Mathematics 250 Fall 2006 Selected Solutions Assignment #8 Proof Practice # 8: Suppose that A is an invertible k k matrix, and that B is another k k matrix |A-1 | such that |B - A| < |A1 | . Show that B is invertible, and that |B -1 | 1-|B-A|A-1 |
Mathematics 250 Fall 2007 Selected Solutions, Assignment 4
1 MVM 4.5.1: a) The equation xy = 1 can be solved explicitly to find the expression y = x , on the domain R - cfw_0 of non-zero real numbers. b) The equation 2 sin(xy) = 1 can be converted to sin(
Mathematics 250 Fall 2008 Selected solutions, Assignment #1 Assignment # 1 MVM, Exercise 2.2.1: Are the following sets open? Closed? a) A = cfw_x : 0 < x 2 bf R. This set is neither open nor closed. It does not contain an interval around the upper boundar
Mathematics 250 Fall 2008 Selected Solutions, Assignment 3 MVM, Exercise 7.3.11: Derive Euler's formula for homogeneous functions. A function f : Rm R is called homogeneous of degree k if , for any vector x and any scalar t, we have the equality f (tx) =
Mathematics 250
Fall 2008
Proof Practice # 1 MM, Exercise 2.5 of Chapter 2: Show that a closed ball B(a, r) is indeed a closed set. Argument 1: We first argue directly in terms of the definition of closed. A set X is said to be closed if every sequence cf