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Unformatted text preview: S at the point P (1 ,1 , 2). (b) Sketch S and the normal. Is the normal pointing inwards or outwards? 5. Consider the vector ﬁeld F = xy i + y j . (a) Calculate ∇ · F . (b) Let C be the square with vertices (0 , 0), (2 , 0), (2 , 2), (0 , 2). Calculate the ﬂux of F outwards across C . 6. Consider a solid whose base is the circle x 2 + y 2 ≤ 1 and whose height at ( x,y ) is given by 4x 2y 2 . Find the volume of this solid. 7. Let S be the surface given by z = x + y + 3 with 0 ≤ x ≤ 1 , ≤ y ≤ 2. If the cost of painting S at the point ( x,y,z ) is $( x 2 + y 2 ) per unit square, calculate the total cost of painting. Short Answers 1. R = { ( x,y ) ∈ R 2  ≤ x ≤ 1 , ≤ y ≤ 1x } OR R = { ( x,y ) ∈ R 2  ≤ y ≤ 1 , ≤ x ≤ 1y } 2. 2 √ 2 π 3. 2 3 4. 2 3 i2 3 j1 3 k , outwards; OR2 3 i + 2 3 j + 1 3 k , inwards 5. (a) y + 1 (b) 8 6. 7 π 2 7. 10 √ 3 3...
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 Three '09
 NOTSURE
 Statistics, Vector Calculus, University of Sydney School of Mathematics, vector calculus quiz, normal pointing inwards, Vector Calculus mark

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