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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![**Problem Statement:**
Find the gradient vector field \(\vec{F}(x, y, z)\) of \(f(x, y, z) = \tan(x + 5y + z)\).
**Solution:**
\[
\vec{F}(x, y, z) = \left \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right \rangle
\]
- The empty boxes should be filled by calculating the partial derivatives of the function \( f(x, y, z) = \tan(x + 5y + z) \):
- \(\frac{\partial f}{\partial x} = \sec^2(x + 5y + z)\)
- \(\frac{\partial f}{\partial y} = 5 \sec^2(x + 5y + z)\)
- \(\frac{\partial f}{\partial z} = \sec^2(x + 5y + z)\)
Thus, the gradient vector field will be:
\[
\vec{F}(x, y, z) = \left \langle \sec^2(x + 5y + z), 5 \sec^2(x + 5y + z), \sec^2(x + 5y + z) \right \rangle
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ed8060-678c-4d36-a5c7-0e7a65a577a0%2Ffd655c03-a6d8-4064-9979-ae1c44e571bc%2Fnhcix8_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:**
\[
\vec{F}(x, y, z) = \left \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right \rangle
\]
- The empty boxes should be filled by calculating the partial derivatives of the function \( f(x, y, z) = \tan(x + 5y + z) \):
- \(\frac{\partial f}{\partial x} = \sec^2(x + 5y + z)\)
- \(\frac{\partial f}{\partial y} = 5 \sec^2(x + 5y + z)\)
- \(\frac{\partial f}{\partial z} = \sec^2(x + 5y + z)\)
Thus, the gradient vector field will be:
\[
\vec{F}(x, y, z) = \left \langle \sec^2(x + 5y + z), 5 \sec^2(x + 5y + z), \sec^2(x + 5y + z) \right \rangle
\]
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