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Although it is not defined on all of space R³, the field associated with the line integral below is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. (6,2,2) S 15x² dx +- (6,1,2) 4z² dy + 8z In y dz

Although it is not defined on all of space R³, the field associated with the line integral below is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. (6,2,2) S 15x² dx +- (6,1,2) 4z² dy + 8z In y dz

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
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### Line Integrals, Conservative Fields, and Potential Functions Although it is not defined on all of space \( \mathbb{R}^3 \), the field associated with the line integral below is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. #### Given Line Integral \[ \int_{(6,1,2)}^{(6,2,2)} \left( 15x^2 \, dx + \frac{4z^2}{y} \, dy + 8z \ln y \, dz \right) \] ### Explanation of the Problem 1. **Verifying the Field is Conservative:** - To establish that the field is conservative, one can use the component test for a simply connected domain. - A vector field \( \mathbf{F} = (P, Q, R) \) is conservative if there exists a scalar potential function \( f \) such that \( \mathbf{F} = \nabla f \). 2. **Finding the Potential Function:** - Compute the potential function \( f(x, y, z) \) by integrating each component of the field with respect to its respective variable, ensuring consistency across mixed partial derivatives. 3. **Evaluating the Integral:** - Once the potential function \( f \) is identified, evaluate the potential function at the bounds of the line integral, i.e., \( f(6, 2, 2) - f(6, 1, 2) \). This approach simplifies the computation of the line integral, avoiding direct path integration by leveraging potential functions for conservative fields.
Transcribed Image Text:### Line Integrals, Conservative Fields, and Potential Functions Although it is not defined on all of space \( \mathbb{R}^3 \), the field associated with the line integral below is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. #### Given Line Integral \[ \int_{(6,1,2)}^{(6,2,2)} \left( 15x^2 \, dx + \frac{4z^2}{y} \, dy + 8z \ln y \, dz \right) \] ### Explanation of the Problem 1. **Verifying the Field is Conservative:** - To establish that the field is conservative, one can use the component test for a simply connected domain. - A vector field \( \mathbf{F} = (P, Q, R) \) is conservative if there exists a scalar potential function \( f \) such that \( \mathbf{F} = \nabla f \). 2. **Finding the Potential Function:** - Compute the potential function \( f(x, y, z) \) by integrating each component of the field with respect to its respective variable, ensuring consistency across mixed partial derivatives. 3. **Evaluating the Integral:** - Once the potential function \( f \) is identified, evaluate the potential function at the bounds of the line integral, i.e., \( f(6, 2, 2) - f(6, 1, 2) \). This approach simplifies the computation of the line integral, avoiding direct path integration by leveraging potential functions for conservative fields.
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