(a) Explanation Given data: Vector field is F ( x , y , z ) = y z e x z i + e x z j + x y e x z k . Consider ∇ f = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k . Write the relation between the potential function f and vector field F . ∇ f = F Substitute f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k for ∇ f , F = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k Compare the equation F = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k with F ( x , y , z ) = y z e x z i + e x z j + x y e x z k . f x ( x , y , z ) = y z e x z (1) f y ( x , y , z ) = e x z (2) f z ( x , y , z ) = x y e x z (3) Integrate equation (1) with respect to x . f ( x , y , z ) = ∫ ( y z e x z ) d x = y z ∫ e x z d x = y z ( e x z z ) + g ( y , z ) { ∵ ∫ e a t d t = e a t a } f ( x , y , z ) = y e x z + g ( y , z ) (4) Apply partial differentiation with respect to y on both sides of equation (4) (b) To determine The value of ∫ C ∇ f ⋅ d r along the curve C .

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Vector Calculus

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Question

Chapter 13.3, Problem 15E

(a)

To determine

To find: The potential function f such that F=∇f .

(b)

To determine

The value of ∫C∇f⋅dr along the curve C.

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Chapter 13.3, Problem 14E

Chapter 13.3, Problem 16E

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Chapter 13 Solutions

Essential Calculus: Early Transcendentals

Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Prob. 9E Ch. 13.1 - Sketch the vector field F by drawing a diagram...

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The potential function f such that F = ∇ f .

Concept explainers

To find: The potential function f such that F=∇f .

The value of ∫C∇f⋅dr along the curve C.

Want to see the full answer?

Chapter 13 Solutions