ps9 - v F ? Which portions contribute negatively? b) Find...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ±eld v F = xy ˆ ı + y 2 ˆ . Which portions of C contribute positively to the ²ux of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: v F ? Which portions contribute negatively? b) Find the ux of v F through C by direct calculation (evaluating a line integral). Explain your answer using (a). c) Find the ux of v F through C using Greens theorem. Problem 2. (Thursday, 2 points) A harmonic function f ( x, y ) is one satisfying Laplaces equation: f xx + f yy = 0. Prove that if the eld v F is the gradient of a harmonic function, then the ux of v F across any simple closed curve is zero. Problem 3. (Thursday, 4 points) Let C be a dierentiable 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 ux of the vector eld v F = (2 xy-2 x 4 y ) + (4 x + 4 x 3 y 2-y 2 ) across C is always equal to-2. (Hint: apply Greens theorem to the region between C and the x-axis). 1...
View Full Document

This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

Ask a homework question - tutors are online