Explanation 1) Concept: i) If F is a vector field defined on all of R 3 whose component function have continuous partial derivative and c u r l F = 0 , then F is a conservative vector field. ii) If F x , y , z = P □ i + Q □ j + R □ k is a vector field on R 3 and the partial derivatives of P , Q a n d R exists, then the curl of F is vector field on R 3 defined by, c u r l F = ∂ R ∂ y - ∂ Q ∂ z i + ∂ P ∂ z - ∂ R ∂ x j + ∂ Q ∂ x - ∂ P ∂ y k c u r l F = ∇ × F iii) Let C be the smooth curve given by the vector function r t , a ≤ t ≤ b . let f be a differentiable function of two or three variables whose gradient vector ∇ f is continues on C . then ∫ C □ ∇ f · d r = f r ( b ) - f ( r ( a ) ) 2) Given: F x , y , z = 3 x 2 y z - 3 y i + x 3 z - 3 x j + x 3 y + 2 z k C is the curve with the initial point 0 , 0 , 2 and terminal point 0,3 , 0 as shown in the figure, 3) Calculations: The given vector field is, F x , y , z = 3 x 2 y z - 3 y i + x 3 z - 3 x j + x 3 y + 2 z k Let P = 3 x 2 y z - 3 y , Q = x 3 z - 3 x , & R = x 3 y + 2 z Notice that, component function of F has mixed partial derivatives By concept ii) c u r l F = ∂ R ∂ y - ∂ Q ∂ z i + ∂ P ∂ z - ∂ R ∂ x j + ∂ Q ∂ x - ∂ P ∂ y k = x 3 - x 3 i + 3 x 2 y - 3 x 2 y j + 3 x 2 z - 3 x 2 z k = 0 And the domain of F is R 3 Therefore, by concept i) F is a conservative vector field Therefore, their exists f such that F = ∇ f Therefore, f x x , y , z = 3 x 2 y z - 3 y , f y x , y , z = x 3

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ISBN: 9781337051545

Author: Stewart

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Chapter 16.R, Problem 37E

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∫C□F·dr

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Chapter 16.R, Problem 36E

Chapter 16.R, Problem 38E

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Chapter 16 Solutions

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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Basic Differentiation Rules For Derivatives; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=IvLpN1G1Ncg;License: Standard YouTube License, CC-BY

∫ C □ F · d r

Solution Summary: The author explains that if F is a vector field defined on all of R3 whose component function have continuous partial derivative and curlF=0, it is conservative.

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To evaluate:

∫C□F·dr

Want to see the full answer?

Chapter 16 Solutions

∫ C □ F · d r

Solution Summary: The author explains that if F is a vector field defined on all of R3 whose component function have continuous partial derivative and curlF=0, it is conservative.

Videos

To evaluate: ∫C□F·dr

Want to see the full answer?

Chapter 16 Solutions

To evaluate:

∫C□F·dr