362-5

# 362-5 - MAT 362 Answers to selected exercises week of Sept...

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MAT 362 Answers to selected exercises, week of Sept. 28 Graded problems 1. The work done is the di f erence in the potential function: W = f ( r 2 ) - f ( r 1 ) , where f ( r ) = - GMm / r . We have G = 6 . 7 × 10 - 8 cm 3 / s 2 g, m = 6 × 10 27 g, M = 3 . 3 × 10 5 m , r 1 = 1 . 5 × 10 12 cm, and r 2 = r 1 + 1. Since r 2 = r 1 + 1 is only slightly larger than r 1 , Taylor’s theorem implies that, to a good approximation, W = - GMm 1 r 2 - 1 r 1 ! GMm 1 r 1 ! 2 . Substituting in the values for the constants yields W 3 . 5 × 10 31 dynes = 3 . 5 × 10 22 J . 2. F is the curl of another vector field only if ∇ · F = 0. (a) ∇ · F = 3, so F is not the curl of another vector field. (b) ∇· F = 2 x - 2 x + 0 = 0, so F is the curl of another vector field. To find it, use the recipe in Exercise 4. This gives G 1 = Z z 0 ( t - 2 xy ) dt - Z y 0 0 dt = z 2 / 2 - 2 xyz G 2 = - Z z 0 ( x 2 + 1) dt = - z ( x 2 + 1) G 3 = 0 . It is straightforward to check that

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362-5 - MAT 362 Answers to selected exercises week of Sept...

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