Explanation Concept used: The divergence theorem states that, Let S be a smooth, orientable surface that encloses a solid region R in ℝ 3 . If F → is a continuous vector field whose components have continuous partial derivatives in an open set containing R , then ∬ S F → ⋅ N → d s = ∭ R div F → d V ; where N → is the outward unit normal field for the surface S . Consider the vector function ∬ S F → ⋅ N → d s for F → = curl [ y i ^ + x j ^ − z k ^ ] and the hemisphere S : z = 4 − x 2 − y 2 together with the disk x 2 + y 2 ≤ 4 in the x y − plane

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ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 13.7, Problem 32PS

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To calculate: The value of the surface integral ∬SF→⋅N→ds for F→=curl[yi^+xj^−zk^] and the hemisphere S:z=4−x2−y2 together with the disk x2+y2≤4 in the xy− plane.

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Chapter 13.7, Problem 31PS

Chapter 13.7, Problem 33PS

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The value of the surface integral ∬ S F → ⋅ N → d s for F → = curl [ y i ^ + x j ^ − z k ^ ] and the hemisphere S : z = 4 − x 2 − y 2 together with the disk x 2 + y 2 ≤ 4 in the x y − plane.

Solution Summary: The author explains the divergence theorem, where S is a smooth, orientable surface whose components have continuous partial derivatives in an open set.

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To calculate: The value of the surface integral ∬SF→⋅N→ds for F→=curl[yi^+xj^−zk^] and the hemisphere S:z=4−x2−y2 together with the disk x2+y2≤4 in the xy− plane.

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Chapter 13 Solutions