mid2-b - Math 31BH Winter 2011 Midterm II Name Student ID...

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Math 31BH - Winter 2011 - Midterm II Name: Student ID: Instructions: Please print your name and student ID. During the test, you may not use books or notes. Read each question carefully, and show all your work. Answers with no explanation will receive no credit, even if they are correct. There are 4 questions which are worth 40 points, and a bonus question. You have 50 minutes to complete the test. Question Score Maximum 1 8 2 9 3 10 4 13 EXTRA CREDIT 5 Total 40+5
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Problem 1. [ 8 points ] Consider the function f ( x, y ) = 3 y 2 - 2 y 3 - 3 x 2 + 6 xy. Find the critical points of the function and determine their nature.
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Problem 2. [ 9 points ] Assume that u ( x, y ) and v ( x, y ) are harmonic conjugates i.e. they satisfy the Cauchy-Riemann equations u x = v y , u y = - v x . Show that the functions f ( x, y ) = u ( x 2 - y 2 , 2 xy ) , g ( x, y ) = v ( x 2 - y 2 , 2 xy ) are also harmonic conjugates.
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Problem 3. [ 10 points ] Calculate the total derivative of the function f : Mat n × n Mat n × n given by f ( A ) = AA T . Justify your answer rigorously.
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