Last Name First Name MATH 23b SPRING 2004 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Final Exam May 28 2004 Directions You have three hours for this exam No calculators notes books etc are allowed Please answer on the pages provided There are blank pages included after problems 2 and 6 for extra work Show all work Problem Points points by part Score 1 40 2 each 2 20 10 each 3 20 4 4 12 4 10 5 15 5 each 6 20 10 each Total 125 125 1 True or False T or F All bounded sets in Rn are either open or compact T or F Discrete sets in Rn have no accumulation points T or F T or F If A Rn is open then every x A is an interior point of A 2 If Mn R Rn has the usual Euclidean topology T or F then GLn R A Mn R det A 6 0 is an open set 2 If Mn R Rn has the usual Euclidean topology then SLn R A Mn R det A 1 is a compact set T or F Let A Rn Rm be a linear map and let b Rm If f Rn Rm is given by f x Ax b then Jf A T or F If f Rn R is differentiable at a Rn then all of its directional derivatives exist at a T or F If f R2 R is in the class C 3 then T or F Let f R2 R be in the class C 3 If f a 0 and 2 f a 0 then f has a local minimum at a T or F According to the Inverse Function Theorem the function f x y xy x2 y 2 is locally invertible at every point of its domain T or F The set A x 0 x R R2 has measure zero T or F The set of all real numbers with no 8 s in their decimal expansions has measure zero considered as a subset of R T or F Let A Rn be a closed rectangle and let f A R be bounded If P is a partition of A and P 0 is a refinement of P then L f P L f P 0 T or F Let f Rn R If o f a 0 then f is continuous at a T or F Let f Rn R If o f a 0 then f is discontinuous at a 2f y x 2f x y T or F If f Rn R is continuous then f is integrable on any compact set A Rn T or F If f a b R is increasing then f is integrable on a b T or F If A Rn is a closed rectangle and f A R is R bounded then L A f the lower integral of f exists T or F Let U Rn be an open ball and let x1 x2 U Let C be a piece wise smooth curve in U from x1 to x2 and let 1 a b C and 2 c d C be two parametrizations of C Then Rd Rb 0 0 a F 1 t 1 t dt c F 2 t 2 t dt T or F Let F R2 R2 be a vector field with coordinate functions F P Q If P and Q are both continuously differentiable then F is conservative 2 Optimization Problems a Theoretical Let g Rn R be a continuously differentiable function and let S x Rn g x 0 and g x 6 0 i Use the Implicit Function Theorem to justify that every point on S has a neighborhood in which one of the variables may be written as a function of the other n 1 variables In particular this implies that S is an n 1 manifold ii Let p Rn be a fixed point not on S and let q be the point on S closest to p We are assuming that such a q exists Prove that the vector p q is orthogonal to any vector tangent to S at q Hint Consider the function f x p x 2 b Practical Use the method of Lagrange multipliers to find the dimensions of the largest rectangle in terms of area that can be x2 y 2 inscribed in the ellipse 2 2 1 a b 3 Integration Theory a Let A Rn be a closed rectangle and let f A R be bounded Define what it means for f to be integrable on A b Let A Rn be a closed rectangle and let f A R be bounded State the theorem that most completely characterizes whether or not f is integrable on A c Let A Rn be a closed rectangle and suppose that f A R is both R bounded and integrable Show that if f x 0 x A then A f 0 4 A Double Integral Let D x y x2 y 2 1 and x y 0 be the portion of the closed unit disk in the first quadrant of R2 and let f D R be defined by f x y 2xy x2 y 2 0 if x y 6 0 0 if x y 0 0 Z f Justify that f is integrable on D and compute D 5 Line Integrals a Let F x y z y x 2z and let C be the straight line segment from a 1 2 3 to b 2 4 7 Z F using the definition Parametrize C and compute C b State the Fundamental Theorem of Line Integrals c Could you have used the Fundamental Theorem of Line Integrals from part b to evaluate the line integral from part a Why or why not 6 Stokes Theorem Let S be a parametrized surface in R3 with oriented boundary C That is suppose there exists some D R2 that is open connected and simply connected with a boundary D that is a piece wise smooth positively oriented curve and a function D S that is continuous on D and continuously differentiable except possibly on a set of measure zero and bijective on D By convention denote the independent variables as u and v Let F U R3 be a vector field that is class C 1 and represented by the components F x y z P x y z Q x y z R x y z where U R3 is some open set containing S Z Z Theorem Stokes Theorem Z curl F S F C Note that the right hand side is a standard line integral around the piece wise smooth positively oriented with respect to a right hand coordinate system curve C The left hand side is a flux integral which by definition is evaluated as follows If G R3 R3 is any continuous vector field then with the parametrization above Z Z Z Z G G n dS S S Z Z G u v D u v du dv where n is the normal vector to the surface …