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 R → = x i ^ + y j ^ + z k ^ and the sphere S : x 2 + y 2 + z 2 = a 2 with constants a > 0 . Here, ‖ R → ‖ R → = x 2 + y 2 + z 2 〈 x , y , z 〉 F → = 〈 x a , y a , z a 〉 The value of the integral ∬ S ‖ R → ‖ R → ⋅ N → d s can be evaluated as, ∬ S F → ⋅ N → d s = ∭ R <

6th Edition

ISBN: 9781465208880

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 43PS

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To calculate: The value of ∬S‖R→‖R→⋅N→ds ; where R→=xi^+yj^+zk^ and the sphere S:x2+y2+z2=a2 with constants a>0 .

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Chapter 13.7, Problem 42PS

Chapter 13.7, Problem 44PS

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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 ∬ S ‖ R → ‖ R → ⋅ N → d s ; where R → = x i ^ + y j ^ + z k ^ and the sphere S : x 2 + y 2 + z 2 = a 2 with constants a > 0 .

Solution Summary: The author explains how the divergence theorem states that, Let S be a smooth, orientable surface that encloses an open set containing R.

Concept explainers

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To calculate: The value of ∬S‖R→‖R→⋅N→ds ; where R→=xi^+yj^+zk^ and the sphere S:x2+y2+z2=a2 with constants a>0 .

Want to see the full answer?

Chapter 13 Solutions

The value of ∬ S ‖ R → ‖ R → ⋅ N → d s ; where R → = x i ^ + y j ^ + z k ^ and the sphere S : x 2 + y 2 + z 2 = a 2 with constants a &gt; 0 .

Solution Summary: The author explains how the divergence theorem states that, Let S be a smooth, orientable surface that encloses an open set containing R.

Concept explainers

Videos

To calculate: The value of ∬S‖R→‖R→⋅N→ds ; where R→=xi^+yj^+zk^ and the sphere S:x2+y2+z2=a2 with constants a>0 .

Want to see the full answer?

Chapter 13 Solutions

The value of ∬ S ‖ R → ‖ R → ⋅ N → d s ; where R → = x i ^ + y j ^ + z k ^ and the sphere S : x 2 + y 2 + z 2 = a 2 with constants a > 0 .