the gradient vector field of f(z, y) = In(x + 4y) %3D Work
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 the Gradient Vector Field**
In this exercise, we aim to find the gradient vector field of the function \( f(x, y) = \ln(x + 4y) \).
### Function:
\[ f(x, y) = \ln(x + 4y) \]
### Gradient Vector Field:
The gradient vector field \( \nabla f \) of a function \( f(x, y) \) is defined as:
\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]
### Steps:
1. **Partial Derivative with respect to \( x \):**
\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left[ \ln(x + 4y) \right] \]
2. **Partial Derivative with respect to \( y \):**
\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left[ \ln(x + 4y) \right] \]
### Input Fields:
There are two input fields to represent the components of the gradient vector:
\[ \left( \ \ \quad , \ \ \ \right) \]
Insert the solutions from the steps above into the fields to complete the problem.
### Buttons:
- **Add Work:** This button allows you to input additional detailed steps to show your work.
---
This component is designed to facilitate the understanding and calculation of gradient vectors, a fundamental concept in multivariable calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdec92e-028d-4f37-af60-841922a4cee5%2F7e35b847-2e13-4483-b8dd-47c5614087f3%2Fju2ysg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Gradient Vector Field**
In this exercise, we aim to find the gradient vector field of the function \( f(x, y) = \ln(x + 4y) \).
### Function:
\[ f(x, y) = \ln(x + 4y) \]
### Gradient Vector Field:
The gradient vector field \( \nabla f \) of a function \( f(x, y) \) is defined as:
\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]
### Steps:
1. **Partial Derivative with respect to \( x \):**
\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left[ \ln(x + 4y) \right] \]
2. **Partial Derivative with respect to \( y \):**
\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left[ \ln(x + 4y) \right] \]
### Input Fields:
There are two input fields to represent the components of the gradient vector:
\[ \left( \ \ \quad , \ \ \ \right) \]
Insert the solutions from the steps above into the fields to complete the problem.
### Buttons:
- **Add Work:** This button allows you to input additional detailed steps to show your work.
---
This component is designed to facilitate the understanding and calculation of gradient vectors, a fundamental concept in multivariable calculus.
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