![]() Problem 19 Suppose we wish to find the extrema of F : Rn R on some compact domain given where G (G1. You may describe Tp C and Np C as a span or as in R4 given cartesian equations, your choice. We have aimed at presenting the broadest range of problems that you are. Let X : R3 R4 be defined (R cos sin sin R sin sin sin R cos sin R cos Let X(R3 ) V. problems covers elementary and intermediate calculus, and much of advanced calculus. Problem 18 Suppose R R is a fixed, positive constant. If G(T (x), T (y)) G(x, y) for all x, y Rn then what condition does this force A to satisfy? Problem 16 Calculate dG(v,w) (H, K) for G given in the previous problem. Suppose a linear transformation T : Rn Rn has T (x) Ax for all x Rn. Problem 15 Let G : Rn Rn R be defined G(x, y) xT Jy for some invertible matrix J. Show that F G is also differentiable at p. Assume F, G are differentiable functions from V to W which are (Frechet) differentiable at p. ![]() If p is a critical point then use the theory of quadratic forms to classify the extrema. In each case, identify (p) and decide if p is a critical point. Problem 13 Suppose functions of the form f : R3 R have multivariate power series expansions centered at p as given below. Derive the tangent and normal spaces to C at that is calculate Tp C and Np C. Studying Advanced Calculus MA11003 at Indian Institute of Technology Kharagpur On Studocu you will find 81 practice materials, 81 lecture notes, 28 tutorial work. Find a parametrization(s) of the curve(s) F Problem 12 Let (t, t2, t3, t4 ) for all t R. Problem 8 Let F (x, y, z) (x2 y 2 z 2, x2 ). a Use the fact given above to derive a nice formula for Z 2 x4 dx. y Problem 7 If a (0, then it can be shown that r Z e dx. Problem 6 Suppose w x2 y 2 z 2 and z x2 y 2. Problem 5 Find the standard matrix of T (x, y) (x 2y, 3x 4y). However, be sure the entries in the matrices are correct. You may leave your answer in terms of the product of two appropriate matrices. Problem 3 Define what it means for U V to be an open set in V where V is a normed linear space with norm Problem 4 Suppose F (x, y) (xy, x2 y 2, x3 y 3 ) and G(a, b, c) (a b, b c). Problem 2 Give an example of a function which is differentiable, but not continuously differentiable at a point. Preview text Math 332: Fall 2013 Test 1 Show your work and justify steps.
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