# quiz02p - S at the point P(1-1 2(b Sketch S and the normal...

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The University of Sydney School of Mathematics and Statistics Summer School MATH2061: Vector Calculus Quiz Info 2012 Information The quiz will start at 2pm on Monday 13 February, in the lecture The quiz runs for 45 minutes. The quiz is worth 15% of the Vector Calculus mark The format of the quiz is extended answer questions, working will be marked and full marks may not be awarded if working is not shown. You may use a calculator. A formula sheet with standard integral tables will be provided. The content includes material up to Chapter 5.2 of the course notes. Tutorials 1-4 and Practice Sessions 1-4 are recommended. Practice Questions 1. Let R be the region bounded by the triangle with vertices at (0 , 0), (1 , 0), and (0 , 1). Sketch the region R and describe it using set notation. 2. Let C be the helix given by x = t, y = sin t, z = cos t, t : - π π. Find the length of the helix. 3. Evaluate the work integral Z C F · d r where F = xy i + z j - x k and C is given by x = t, y = 2 t, z = t 2 , t : 0 1 . 4. Let S be the surface z = x 2 + y 2 . (a) Find a unit normal to

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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 4-x 2-y 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 ≤ 1-x } OR R = { ( x,y ) ∈ R 2 | ≤ y ≤ 1 , ≤ x ≤ 1-y } 2. 2 √ 2 π 3. 2 3 4. 2 3 i-2 3 j-1 3 k , outwards; OR-2 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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quiz02p - S at the point P(1-1 2(b Sketch S and the normal...

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