Tutorial 9

# Tutorial 9 - , and h .) (c) Find R C F · d s , where C is...

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MATH1211/10–11(1)/Tu9 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010–11 First Semester: Tutorial 9 1. Let r = x i + y j be the position vector of any point in the plane. Show that the ﬂux of F = r across any simple closed curve C in R 2 is twice the area inside C . 2. Determine whether the given vector ﬁeld F is conservative. If it is, ﬁnd a scalar potential function for F . (a) F = 2 x sin y i + x 2 cos y j . (b) F = yz 2 i + ( x + xz 2 ) j + 2 xyz k . 3. 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
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Unformatted text preview: , 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 ). 4. Find RR S ( x 2 + y 2 ) dS , where S is the lateral surface of the cylinder of radius a and height h whose axis is the z-axis. 5. Verify Stoke’s Theorem for the vector ﬁeld F = z i + x j + y k and the surface S parametrized by X ( s,t ) = ( s cos t, s sin t, t ), 0 ≤ s ≤ 1 , ≤ t ≤ π/ 2. 6. Verify Gauss’s theorem for the vector ﬁeld F = x i + y j + z k and the 3-dimensional region D = { ( x,y,z ) | ≤ z ≤ 9-x 2-y 2 } . 1...
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