Explanation Given information: The vector field is. ∮ C ( f ∇ → g ) . d R → = ∬ S ( ( ∇ → f + ∇ → g ) . N ^ ) d S …. ( 1 ) Calculation: The stokes’ theorem states that: Let S is an oriented surface with unit normal vector field N ^ , and let S be bounded by a smooth curve C compatible with S . If V → is a vector field. ∮ C V → . d R → = ∬ S ( c u r l V → . N ^ ) d S Using equation ( 1 ) in stokes theorem. ∮ C ( f ∇ → g ) . d R → = ∬ S ( ∇ → + f ∇ → g ) . N ^ d S Calculate the value ∇ → + f ∇ → g . ∇ → + f ∇ → g = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z f ∂ g ∂ x f ∂ g ∂ y f ∂ g ∂ z | = i ^ ( ∂ f ∂ y ∂ g ∂ z + f ∂ 2 g ∂ y z − ∂ f ∂ z ∂ g ∂ y − f ∂ 2 g ∂ z y ) − j ^ ( ∂ f ∂ x ∂ g ∂ z + f ∂ 2 g ∂ x z − ∂ f ∂ z ∂ g ∂ x − f ∂ 2 g ∂ z x ) + k ^ ( ∂ f ∂ x ∂ g ∂ y + f ∂ 2 g ∂ x y − ∂ f ∂ y ∂ g ∂ x − f ∂ 2 g ∂ y x ) = i ^ ( ∂ f ∂ y ∂ g ∂ z − ∂ f ∂ z ∂ g ∂ y ) − j ^ ( ∂ f ∂ x ∂ g ∂ z − ∂ f ∂ z ∂ g ∂ x ) + k ^ ( ∂ f ∂ x ∂ g ∂ y − ∂ f ∂ y ∂ g ∂ x ) Use the second order derivative

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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 13.6, Problem 59PS

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To find: The value of ∮C(f ∇ →g).dR→ is equal to ∬S( ∇ →f+ ∇ →g).N^dS prove it.

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Chapter 13.6, Problem 58PS

Chapter 13.6, Problem 60PS

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The value of ∮ C ( f ∇ → g ) . d R → is equal to ∬ S ( ∇ → f + ∇ → g ) . N ^ d S prove it.

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To find: The value of ∮C(f ∇ →g).dR→ is equal to ∬S( ∇ →f+ ∇ →g).N^dS prove it.

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Chapter 13 Solutions