Final Exam Fall 2013 - Math 318 Final 1 2 3 4 Name Student...

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Math 318 Final Dec 18, 2013 1. 2. 3. 4. 5. 6. 7. Total: Name: Student No. 2 hours. No aids are allowed. There are 7 problems and 22 pages, for a total of 100 points. Problem 1. (6+6) Find an equation of the tangent plane to z = f x y at the point 2 3 f 2 3 for each of the following: (a) f x y = x 2 + xy. 1
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(b) f x y = x - y 2 x 2 + y . 2
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Problem 2. (6+6) Let U R n be an open set. Recall that a C 2 function f : U R is harmonic if f satisfies Laplace’s equation n X i =1 2 f ∂x 2 i = 0 . (a) Let f : R 2 R is a quadratic polynomial, i.e. f is a polynomial of degree 2. Show that f is a harmonic, if and only if f is of the form f x y = a ( x 2 - y 2 ) + bxy + cx + dy + e, where a, b, c, d, e are constants. 4
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(b) Recall in the homework that for f : R 2 R , if we let F r θ = f r cos θ r sin θ , then it follows that 2 F ∂r 2 + 1 r ∂F ∂r + 1 r 2 2 F ∂θ 2 = 2 f ∂x 2 + 2 f ∂y 2 .
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  • Spring '14
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  • 2 hours, Inverse function, aij, Inverse Function Theorem

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