Find the gradient vector field (F(x, y, z)) of f(x, y, z) = y cos (5²) F(x, y, z) =
Find the gradient vector field (F(x, y, z)) of f(x, y, z) = y cos (5²) 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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Question
6.1.9
![**Gradient Vector Field Calculation**
The task is to find the gradient vector field, denoted as \(\vec{F}(x, y, z)\), for the function \(f(x, y, z) = y \cos\left(\frac{5z}{x}\right)\).
To express the gradient vector field, it is written as:
\[
\vec{F}(x, y, z) = \left\langle \boxed{\phantom{a}}, \boxed{\phantom{a}}, \boxed{\phantom{a}} \right\rangle
\]
### Explanation:
The gradient vector field represents the vector of the partial derivatives of the function \(f(x, y, z)\) with respect to the variables \(x\), \(y\), and \(z\). In this case, it involves calculating:
1. The partial derivative of \(f(x, y, z)\) with respect to \(x\),
2. The partial derivative of \(f(x, y, z)\) with respect to \(y\),
3. The partial derivative of \(f(x, y, z)\) with respect to \(z\).
Each of these derivatives will populate the respective components of the gradient vector \(\vec{F}(x, y, z)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2Fa3a7be08-3b2a-4392-ae06-b9daaa4f8e0c%2Fu7ryya_processed.png&w=3840&q=75)
Transcribed Image Text:**Gradient Vector Field Calculation**
The task is to find the gradient vector field, denoted as \(\vec{F}(x, y, z)\), for the function \(f(x, y, z) = y \cos\left(\frac{5z}{x}\right)\).
To express the gradient vector field, it is written as:
\[
\vec{F}(x, y, z) = \left\langle \boxed{\phantom{a}}, \boxed{\phantom{a}}, \boxed{\phantom{a}} \right\rangle
\]
### Explanation:
The gradient vector field represents the vector of the partial derivatives of the function \(f(x, y, z)\) with respect to the variables \(x\), \(y\), and \(z\). In this case, it involves calculating:
1. The partial derivative of \(f(x, y, z)\) with respect to \(x\),
2. The partial derivative of \(f(x, y, z)\) with respect to \(y\),
3. The partial derivative of \(f(x, y, z)\) with respect to \(z\).
Each of these derivatives will populate the respective components of the gradient vector \(\vec{F}(x, y, z)\).
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