2. Find a function f(r, y, 2) which makes (yz?, rz?, f(r, y, z)) conservative. Show that your answer works.
2. Find a function f(r, y, 2) which makes (yz?, rz?, f(r, y, z)) conservative. Show that your answer works.
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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![**Problem Statement:**
2. Find a function \( f(x, y, z) \) which makes \( \langle y z^2, x z^2, f(x, y, z) \rangle \) conservative. Show that your answer works.
**Explanation:**
The task is to determine a function \( f(x, y, z) \) such that the vector field \( \langle y z^2, x z^2, f(x, y, z) \rangle \) is conservative. A vector field is conservative if it can be expressed as the gradient of some scalar potential function \( \phi(x, y, z) \). In other words, we need to find a potential function such that:
\[ \nabla \phi = \langle \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \rangle = \langle y z^2, x z^2, f(x, y, z) \rangle. \]
To show that your answer works, verify the necessary conditions for the field to be conservative:
1. Check if the mixed partial derivatives are equal (Clairaut's Theorem on equality of mixed partials).
2. Ensure the line integral of the vector field over any closed path is zero, if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71469e61-2ae3-431f-826d-cc2a14e0ebcf%2F2312dd92-4084-4555-8c95-067c701cae71%2F1wvf48c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
2. Find a function \( f(x, y, z) \) which makes \( \langle y z^2, x z^2, f(x, y, z) \rangle \) conservative. Show that your answer works.
**Explanation:**
The task is to determine a function \( f(x, y, z) \) such that the vector field \( \langle y z^2, x z^2, f(x, y, z) \rangle \) is conservative. A vector field is conservative if it can be expressed as the gradient of some scalar potential function \( \phi(x, y, z) \). In other words, we need to find a potential function such that:
\[ \nabla \phi = \langle \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \rangle = \langle y z^2, x z^2, f(x, y, z) \rangle. \]
To show that your answer works, verify the necessary conditions for the field to be conservative:
1. Check if the mixed partial derivatives are equal (Clairaut's Theorem on equality of mixed partials).
2. Ensure the line integral of the vector field over any closed path is zero, if needed.
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