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18.02 Pset 6

# 18.02 Pset 6 - Part C For each statement below say whether...

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18.02 HOMEWORK #6, DUE NOVEMBER 4, 2010 BJORN POONEN 1. Part A (After Oct. 28) p. 1045: 9, 29* (8 pts.), 38* (8 pts.) (After Oct. 28) 4D-1c, 4D-3, 4D-6* (5 pts.) (assume that C is positively oriented), 4D-7* (5 pts.) (After Oct. 29) 4E-1, 4E-2, 4E-3, 4E-6* (5 pts., there are at least four ways to do this problem), 4E-7* (7 pts.) (After Oct. 29) 4F-2, 4F-3, 4F-6* (8 + 5 pts.) (After Nov. 2) 4G-1, 4G-6 2. Part B B.1) (After Oct. 29) Let F = - 4 xy 2 i + (4 y - x 2 y ) j . (a) (12 pts.) Find the positively oriented simple closed curve C that maximizes the ﬂux of F across C . (b) (8 pts.) Find the maximum value of the ﬂux. B.2) (After Nov. 2, 5 pts.) Let F be a vector ﬁeld deﬁned on R 2 except at the origin, and suppose that curl F = 0 wherever F is deﬁned. For t > 0, let C t be the circle of radius t centered at the origin, oriented counterclockwise. Explain why the value H C t F · d r is the same for every t > 0. (If your argument says that the value is 0 for every t > 0, then something is wrong with your argument!) 3.
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Unformatted text preview: Part C For each statement below, say whether it is TRUE or FALSE. (Please do not use the abbreviations T and F.) If it is true, explain why; if it is false, give an example demonstrating that it is deﬁnitely false. C.1) (After Oct. 26, 8 pts.) If F and G are conservative vector ﬁelds on a region R , then F + G is conservative on R too. C.2) (After Oct. 26, 8 pts.) If P and Q are diﬀerentiable functions on the region R deﬁned by 4 < x 2 + y 2 < 9, and Q x (2 , 1) 6 = P y (2 , 1), then the vector ﬁeld h P ( x,y ) ,Q ( x,y ) i is not the gradient ﬁeld of any diﬀerentiable function on R . C.3) (After Oct. 29, 8 pts.) If R 1 and R 2 are simply connected regions, then so is their union R 1 ∪ R 2 . Reminder: Please write “Sources consulted: none” at the top of your homework, or list your (animate and inanimate) sources. See the course information sheet for details. 1...
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