Explanation Given information: The given function is. ∬ S f ∇ f . N d S = ∭ D ‖ ∇ f ‖ 2 d V … ( 1 ) 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 Calculate the left hand side of the equation ( 1 ) using ∇ f = ( f x i + f y j + f z k ) . ∬ S f ∇ f . N d S = ∬ S f ( f x i + f y j + f z k ) . N d S = ∬ S ( f f x i + f f y j + f f z k ) . N d S Using divergence theorem. ∬ S ( f f x i + f f y j + f f z k ) . N d S = ∭ D d i v ( f f x i + f f y j + f f z k ) d V = ∭ D ( f f x x + ( f x ) 2 + f f y<

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Chapter 13.7, Problem 45PS

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To find: The value of div(∇f) .

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Chapter 13.7, Problem 44PS

Chapter 13.7, Problem 46PS

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Chapter 13 Solutions

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 d i v ( ∇ f ) .

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

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To find: The value of div(∇f) .

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Chapter 13 Solutions