Explanation Given information: The given vector function is. F = ( x y 2 ) i + ( y z 2 ) j + ( x 2 y ) 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 y ) k The surface S bounded above by a sphere ρ = 2 and bounded below by the cone ϕ = π 4 . Using divergence theorem. ∭ R d i v F d V = ∭ R ( y 2 + z 2 ) d V Let the values are. x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ The integral is. ∭ R ( y 2 + z 2 ) d V = ∫ 0 2 π ∫ 0 π 4 ∫ 0 2 ( ρ 2 sin 2 ϕ sin θ + ρ 2 cos 2 ϕ ) ( ρ 2 sin ϕ ) ( d ρ ) ( ρ ϕ ) ( d θ ) = ∫ 0 2 π ∫ 0 π 4 ∫ 0 2 ρ 4 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( d ρ ) ( ρ ϕ ) ( d θ ) = ∫ 0 2 π ∫ 0 π 4 ( ρ 4 5 ) | 0 2 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( ρ ϕ ) ( d θ ) = ∫ 0 2 π ∫ 0 π 4 ( ρ 4 5 ) | 0 2 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( ρ ϕ ) ( d θ ) ∭ R ( y 2 + z 2 ) d V = ( 2 5 5 ) ∫ 0 2 π ∫ 0 π 4 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( ρ ϕ ) ( d θ ) Calculate the value ( 2 5 5 ) ∫ 0 2 π ∫ 0 π 4 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( ρ ϕ ) ( d θ ) . ( 2 5 5 ) ∫ 0 2 π ∫ 0 π 4 ( sin 3 ϕ sin 2 θ + cos 2 ϕ sin ϕ ) ( ρ ϕ ) ( d θ ) = ( 2 5 5 ) ∫ 0 2 π ( sin 2 θ ∫ 0 π 4 ( sin 3 ϕ ) ( d ϕ ) + ∫ 0 π 4 ( cos 2 ϕ sin ϕ ) ( d ϕ ) ) ( d ϕ ) Calculate the first integral of the above equation

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

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

Publisher: Kendall Hunt Publishing

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Chapter 13.7, Problem 37PS

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To find: The value of ∬SF.N dS .

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Chapter 13.7, Problem 36PS

Chapter 13.7, Problem 38PS

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Chapter 13 Solutions

Student's Solution and Survival Manual for Calculus

Ch. 13.1 - Prob. 1PS Ch. 13.1 - Prob. 2PS Ch. 13.1 - Prob. 3PS Ch. 13.1 - Prob. 4PS Ch. 13.1 - Prob. 5PS Ch. 13.1 - Prob. 6PS Ch. 13.1 - Prob. 7PS Ch. 13.1 - Prob. 8PS Ch. 13.1 - Prob. 9PS Ch. 13.1 - Prob. 10PS

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The value of ∬ S F . N d S .

Solution Summary: The author explains the divergence theorem for a continuous vector field F with continuous partial derivatives in an open set containing R.

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To find: The value of ∬SF.N dS .

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Chapter 13 Solutions