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32B(W08)_PracticeMidterm2

# 32B(W08)_PracticeMidterm2 - Z C xydx e y dy where C is the...

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MATH 32B - Lecture 4 - Winter 2008 Practice Midterm 2 - March 3, 2008 NAME: STUDENT ID #: DISCUSSION SECTION: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 6 problems for a total of 100 points. 1

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2 1.(15 points) Evaluate the line integral Z C 2 xye x 2 dx + e x 2 dy over a path C from (0,1) to (1,3).
3 2.(15 points) Find the work done by the force field F ( x, y, z ) = y i - x j + 2 z k in moving the particle along the arc of the circular helix r ( t ) = cos t i + sin t j + t k from the point (1 , 0 , 0) to the point (0 , 1 , π 2 )

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4 3.(15 points) Use Green’s Theorem to evaluate the line integral

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Unformatted text preview: Z C xydx + e y dy where C is the path from (0 , 0) to (1 , 1) along the graph of y = x 2 and from (1 , 1) to (0 , 0) along the graph of y = √ x. 5 4.(20 points) Find the area of the part of the surface with parametric equations x = u + v, y = uv, z = u-v, u 2 + v 2 ≤ 2 6 5.(15 points) For F = xy i + ( y 2-x 2 ) j + x 2 z 2 k ﬁnd curl F and div F . 7 6.(20 points) Suppose that f is a scalar function and F is a vector ﬁeld on R 3 . Prove the identity curl( f F ) = f curl F + ∇ f × F assuming that the appropriate partial derivatives exist and are continuous....
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