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ps9 - v F Which portions contribute negatively b Find the...

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18.02 Problem Set 9 Due Thursday 11/6/08, 12:45 pm in 2-106. Part A (5 points) Hand in the underlined problems only; the others are for more practice. Lecture 25. Thu Oct. 30 Flux. Normal form of Green’s theorem. Read: Notes V3, V4. Work: 4E/ 1ac , 2 , 3, 4 , 5 (don’t use Green’s theorem); 4F/ 3, 4 . Lecture 26. Fri Oct. 31 Simply-connected regions. Review. Read: Notes V5. Lecture 27. Tue Nov. 4 Exam 3 covering lectures 18–26. Part B (12 points) Directions: Attempt to solve each part of each problem yourself. If you collaborate, solutions must be written up independently. It is illegal to consult materials from previous semesters. With each problem is the day it can be done. Write the names of all the people you consulted or with whom you collaborated and the resources you used. Problem 1. (Thursday, 6 points: 2+2+2) a) Let C be the unit circle, oriented counterclockwise, and consider the vector field vector F = xy ˆ ı + y 2 ˆ . Which portions of C contribute positively to the flux of
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Unformatted text preview: v F ? Which portions contribute negatively? b) Find the ²ux of v F through C by direct calculation (evaluating a line integral). Explain your answer using (a). c) Find the ²ux of v F through C using Green’s theorem. Problem 2. (Thursday, 2 points) A harmonic function f ( x, y ) is one satisfying Laplace’s equation: f xx + f yy = 0. Prove that if the ±eld v F is the gradient of a harmonic function, then the ²ux of v F across any simple closed curve is zero. Problem 3. (Thursday, 4 points) Let C be a di³erentiable curve contained in the portion 0 ≤ x ≤ 1 of the ±rst quadrant, starting somewhere on the positive y-axis and ending somewhere on the line x = 1. Show that the ²ux of the vector ±eld v F = (2 xy-2 x 4 y )ˆ ı + (4 x + 4 x 3 y 2-y 2 )ˆ across C is always equal to-2. (Hint: apply Green’s theorem to the region between C and the x-axis). 1...
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