tutorial9

# tutorial9 - F is conservative. If it is, ﬁnd a scalar...

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MATH1211/10-11(2)/Tu9/TNK THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Tutorial 9 Date of tutorial classes: April 7–8. (The section/problem numbers in the following refer to those in the textbook.) 1. ( § 6.2 no.2) Verify Green’s theorem for the given vector ﬁeld F = M ( x,y ) i + N ( x,y ) j and the region D by calculating both I ∂D M dx + N dy and ZZ D ( N x - M y ) dA : F = ( x 2 - y ) i + ( x + y 2 ) j , D is the rectangle bounded by x = 0 , x = 2 , y = 0, and y = 1. 2. ( § 6.2 no.14) Use Green’s theorem to ﬁnd the area between the ellipse x 2 / 9 + y 2 / 4 = 1 and the circle x 2 + y 2 = 25. 3. Determine whether the given vector ﬁeld
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Unformatted text preview: F is conservative. If it is, ﬁnd a scalar potential function for F . (a) ( § 6.3 no.4) F = 2 x sin y i + x 2 cos y j . (b) F = yz 2 i + ( x + xz 2 ) j + 2 xyz k . 4. ( § 6.3 no.22) Let f,g , and h be functions of class C 1 of a single variable. (a) Show that F = ( f ( x ) + y + z ) i + ( x + g ( y ) + z ) j + ( x + y + h ( z )) k is conservative. (b) Determine a scalar potential for F . (Your answer will involve integrals of f,g , and h .) (c) Find R C F · d s , where C is any path from ( x ,y ,z ) to ( x 1 ,y 1 ,z 1 ). 1...
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## This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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