# 237q3 - âˆ âˆ = C C dy x dx y(b If R d c b a f â†’ Ã— is...

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1. (a) [7] Find the constant 0 k such that 0 ) ( = k r , where r = r , 0 r = ) , , ( z y x . (b) [8] Evaluate C d curl x F ) ( , where ) , , ( ) , , ( z yz y xz z y x - + = F and C is the segment of the curve of intersection of the plane x y = and the paraboloid 2 2 y x z + = from (0,0,0) to (1,1,2). 2

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2. (a) [3] Is the set } 4 0 : ) , {( 2 2 + < = y x y x S a regular region in 2 R ? Justify your answer. (b) [10] Evaluate dy x y y x dx x xy C ) 3 3 ( ) 1 ( 2 2 2 4 2 - + + + + where C is the part of the ellipse 1 4 2 2 = + y x that lies to the left of the y -axis ) 0 ( x with counterclockwise orientation. 3
3. [4 marks each part] Prove or disprove the statements (giving counterexample if applicable). (a) If C is a simple, closed, piecewise smooth curve, then

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Unformatted text preview: âˆ« âˆ« = C C dy x dx y (b) If R d c b a f â†’ Ã— ] , [ ) , ( : is continuous then the function âˆ« = d c dy y x f x F ) , ( ) ( is uniformly continuous on ) , ( b a . (c) If a simple curve C having 1 C parametrization n R R b a â†’ âŠ‚ ] , [ : g has length l and 1 â‰¤ F , then l d C â‰¤ â‹… âˆ« | | x F . Here F denotes a continuous vector field on n R . 4 THIS PAGE IS INTENTIONALLY LEFT BLANK for you to finish/redo any question or for rough work. 5...
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## This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto.

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237q3 - âˆ âˆ = C C dy x dx y(b If R d c b a f â†’ Ã— is...

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