Find the gradient vector field (F(x, y, z)) of f(x, y, z) = ye 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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![**Gradient Vector Field Calculation**
To find the gradient vector field \(\vec{F}(x, y, z)\) of the function \(f(x, y, z) = ye^{-4xz}\), we start by determining partial derivatives with respect to each variable \(x\), \(y\), and \(z\).
**Task:**
Calculate the components of \(\vec{F}(x, y, z)\) as follows:
- \(\frac{\partial f}{\partial x}\)
- \(\frac{\partial f}{\partial y}\)
- \(\frac{\partial f}{\partial z}\)
**Fill in the blanks with the corresponding components:**
\[
\vec{F}(x, y, z) = \left( \boxed{\phantom{x}}, \boxed{\phantom{y}}, \boxed{\phantom{z}} \right)
\]
This exercise involves applying the concept of the gradient, which is a vector pointing in the direction of the greatest rate of increase of the function with each component being the partial derivative of \(f\) with respect to each corresponding variable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07dc3289-c262-41f1-9fc5-a8d18500de1b%2F1be7c817-6f12-4869-89a1-8b2665b3782e%2Fndf6g9_processed.png&w=3840&q=75)
Transcribed Image Text:**Gradient Vector Field Calculation**
To find the gradient vector field \(\vec{F}(x, y, z)\) of the function \(f(x, y, z) = ye^{-4xz}\), we start by determining partial derivatives with respect to each variable \(x\), \(y\), and \(z\).
**Task:**
Calculate the components of \(\vec{F}(x, y, z)\) as follows:
- \(\frac{\partial f}{\partial x}\)
- \(\frac{\partial f}{\partial y}\)
- \(\frac{\partial f}{\partial z}\)
**Fill in the blanks with the corresponding components:**
\[
\vec{F}(x, y, z) = \left( \boxed{\phantom{x}}, \boxed{\phantom{y}}, \boxed{\phantom{z}} \right)
\]
This exercise involves applying the concept of the gradient, which is a vector pointing in the direction of the greatest rate of increase of the function with each component being the partial derivative of \(f\) with respect to each corresponding variable.
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