Explanation Given information: The given vector function is. F = ( x y 2 ) i + ( y z 2 ) j + ( x 2 z ) k Calculation: The divergence theorem states that: Let S be a smooth, oriental surface that encloses a solid region R in R 3 . Then for a continuous vector field F with continuous partial derivatives in an open set containing R the following is true: ∬ S F . N d S = ∭ R d i v F d V The vector field is. F = ( x y 2 ) i + ( y z 2 ) j + ( x 2 z ) k The surface S bounded above by a sphere ρ = 2 0 ≤ ϕ ≤ π 4 0 ≤ ϕ ≤ 2 π . The surface with bottom is. 0 ≤ ρ ≤ 2 θ = π 4 0 ≤ θ ≤ 2 π Calculate the value of d i v F = x 2 + y 2 + z 2 . Using the divergence theorem. ∬ S F . N d S = ∭ R ( x 2 + y 2 + z 2 ) d x d y d z Convert into the polar form. ∬ S F . N d S = ∭ R ( x 2 + y 2 + z 2 ) d x d y d z = ∫ 0 2 π ∫ 0 π 4 ∫ 0 2 ρ 2 ( sin 2 ϕ ( cos 2 θ + sin 2 θ ) + cos 2 ϕ ) ρ 2 sin ϕ d ρ d ϕ

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 13.7, Problem 38PS

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To find: The value of ∬SF.N dS .

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Chapter 13.7, Problem 37PS

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The value of ∬ S F . N d S .

Solution Summary: The author explains how the divergence theorem states that S is a smooth, oriental surface that encloses solid region R in R3.

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To find: The value of ∬SF.N dS .

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