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20e_04s_1s

# 20e_04s_1s - f with ∇ f = F but ±rst we should check the...

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Math 20E First Midterm Solutions April 21, 2004 1. Since curl, divergence and cross product require that the functions be vectors, (a), (b) and (d) make no sense. (c) zero since the cross product is perpendicular to F and the dot product of perpen- dicular vectors is zero. (e) ∇ · F = 2 x + 0 + 1 = 2 x + 1. (f) zero since the cross product of a vector with itself is zero. 2. (a) is a ball with a circular hole. It is open and connected but not simply connected. (b) is a cylinder with a ball removed. It is open, connected and simply connected. (c) is the xy -plane with the vertical strip 1 < x < 1 removed. It is not open, not connected and (therefore) not simply connected. 3. We could try to find f with
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Unformatted text preview: f with ∇ f = F , but ±rst we should check the equations ∂F 1 /∂y = ∂F 2 /∂x ∂F 1 /∂z = ∂F 3 /∂x and ∂F 2 /∂z = ∂F 3 /∂y . Since they fail, the vector ±eld is not conservative. 4. We can parameterize the line segment by R = t Q + (1 − t ) P = (1 − t, 1 + t, t ) for 0 ≤ t ≤ 1. Then d R /dt = ( − 1 , 1 , 1) and F ( R ) = (1 − t, 1 − t 2 , 0). Thus the answer is i 1 (1 − t, 1 − t 2 , 0) · ( − 1 , 1 , 1) dt = i 1 (( t − 1) + (1 − t 2 )) dt = 1 2 / 2 − 1 3 / 3 = 1 / 6 . 5. Since the ±eld is conservative and we are integrating around a closed curve, the integral is zero....
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