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. (7,6,7) [ y 18x² dx + 5z² y (7,1,2) dy + 10z In y dz A general expression for the infinitely many potential functions is f(x,y,z) =
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. (7,6,7) [ y 18x² dx + 5z² y (7,1,2) dy + 10z In y dz A general expression for the infinitely many potential functions is f(x,y,z) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Finding a Potential Function for a Given Field**
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.
\[ \int_{(7,1,2)}^{(7,6,7)} 18x^2 \, dx + \frac{5z^2}{y} \, dy + 10z \ln y \, dz \]
A general expression for the infinitely many potential functions is \( f(x,y,z) = \boxed{\phantom{answer}} \).
---
**Explanation**
In the given field, we are dealing with partial derivatives. A potential function \(f(x,y,z)\) is required such that:
\[ \frac{\partial f}{\partial x} = 18x^2 \]
\[ \frac{\partial f}{\partial y} = \frac{5z^2}{y} \]
\[ \frac{\partial f}{\partial z} = 10z \ln y \]
To find this potential function, we integrate each component partially and then combine the results into one expression up to an arbitrary constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd49838c-9fe9-4c2b-b877-1f4ac641c60c%2F8f9e7d51-e0d9-492b-9cf7-3895d71615a4%2Fm7rj8wq_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding a Potential Function for a Given Field**
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.
\[ \int_{(7,1,2)}^{(7,6,7)} 18x^2 \, dx + \frac{5z^2}{y} \, dy + 10z \ln y \, dz \]
A general expression for the infinitely many potential functions is \( f(x,y,z) = \boxed{\phantom{answer}} \).
---
**Explanation**
In the given field, we are dealing with partial derivatives. A potential function \(f(x,y,z)\) is required such that:
\[ \frac{\partial f}{\partial x} = 18x^2 \]
\[ \frac{\partial f}{\partial y} = \frac{5z^2}{y} \]
\[ \frac{\partial f}{\partial z} = 10z \ln y \]
To find this potential function, we integrate each component partially and then combine the results into one expression up to an arbitrary constant of integration.
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