Find the gradient vector field (F(x, y, z)) of ƒ(x, y, z) = tan(x + 5y + z) . F(x, y, z) =
Find the gradient vector field (F(x, y, z)) of ƒ(x, y, z) = tan(x + 5y + z) . 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
3.1.8
![## Problem Statement
Find the gradient vector field \( \vec{F}(x, y, z) \) of \( f(x, y, z) = \tan(x + 5y + z) \).
## Solution
The gradient vector field \( \vec{F}(x, y, z) \) is given by:
\[
\vec{F}(x, y, z) = \left\langle \text{[Gradient with respect to } x] , \text{[Gradient with respect to } y] , \text{[Gradient with respect to } z] \right\rangle
\]
### Explanation:
- The notation \( \vec{F}(x, y, z) \) represents the vector field formed by computing the gradient of the function \( f(x, y, z) \).
- The angles \( \langle \rangle \) denote that the result is a vector with three components, corresponding to the partial derivatives of the function with respect to each of the variables \( x \), \( y \), and \( z \).
The boxes in the diagram indicate where the calculated partial derivatives with respect to \( x \), \( y \), and \( z \) should be placed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ed8060-678c-4d36-a5c7-0e7a65a577a0%2F47febb9e-97ea-442d-8cb4-65abc49eeaa1%2Ftd2vzpt_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Find the gradient vector field \( \vec{F}(x, y, z) \) of \( f(x, y, z) = \tan(x + 5y + z) \).
## Solution
The gradient vector field \( \vec{F}(x, y, z) \) is given by:
\[
\vec{F}(x, y, z) = \left\langle \text{[Gradient with respect to } x] , \text{[Gradient with respect to } y] , \text{[Gradient with respect to } z] \right\rangle
\]
### Explanation:
- The notation \( \vec{F}(x, y, z) \) represents the vector field formed by computing the gradient of the function \( f(x, y, z) \).
- The angles \( \langle \rangle \) denote that the result is a vector with three components, corresponding to the partial derivatives of the function with respect to each of the variables \( x \), \( y \), and \( z \).
The boxes in the diagram indicate where the calculated partial derivatives with respect to \( x \), \( y \), and \( z \) should be placed.
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