Math380_Summer08_Exam2 - Z c F · ds and use it to explain...

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Name: Math 380 – Exam 2 July 16, 2008 50 points possible 1. Find the second-order Taylor formula for the function f ( x,y ) = x y at the point (1 , 2). (There is no need to fully simplify the formula.) 2. Let A = 2 4 - 3 4 1 3 - 3 3 - 1 . (a) Define the quadratic form Q A ( x ). (b) Compute Q A (3 , 0 , 1). (c) Classify the quadratic form. 3. Let f be a function with continuous second partial derivatives. Let the points (1 , 1) and (1 , - 1) be critical points of f . Use the following information to classify the critical points. (a) f xx (1 , 1) = - 12, f yy (1 , 1) = 0 and f xy (1 , 1) = - 1 (b) f xx (1 , - 1) = 12, f yy (1 , - 1) = 2 and f xy (1 , - 1) = 1 2 4. Let F ( x,y,z ) = ( x 2 y, - 2 yz,x 3 y 2 ). (a) Explain why the vector-valued function F is called a vector field. (b) Find the divergence of F . (c) Is the vector field F incompressible? Briefly justify your answer. 5. (a) Evaluate Z c yz dx + z dy + yz dz when c ( t ) = ( t,t 2 ,t 3 ) and t [0 , 1] (b) Give one of the physical interpretation of the vector-line integral
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Unformatted text preview: Z c F · ds and use it to explain the effects of a reparametrization of the path c . 6. Consider the vector-valued function c ( θ ) = (sin 2 θ, cos2 θ,θ ). (a) Sketch the curve traced out by c . (b) Find the length of the curve from the points (0 ,-1 ,π/ 2) to (0 ,-1 , 3 π/ 2). 7. Let F ( x,y,z ) = ± y 2 2 + yz,xy + xz,xy ¶ . (a) Compute ∇ × F . (b) Show that F is a gradient field and find F ’s potential function f . (c) The path r ( θ ) = ˆ-1 + 4 √ 2 3 π θ cos θ, 2 + 4 √ 2 3 π θ sin θ, ! as 0 ≤ θ ≤ 9 π/ 4 is a parametriza-tion of the spiral that starts at the point (-1 , 2 , 0) and ends at the point (2 , 5 , 0). Evaluate the integral Z r F · d s ....
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This note was uploaded on 02/15/2011 for the course MATH 380 taught by Professor Staff during the Summer '08 term at University of Illinois, Urbana Champaign.

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Math380_Summer08_Exam2 - Z c F · ds and use it to explain...

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