Find the gradient vector field (F(x, y, z)) of ƒ(x, y, z) = ln(5x + 6y + z) . F(x, y, z) =
Find the gradient vector field (F(x, y, z)) of ƒ(x, y, z) = ln(5x + 6y + 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
6.1.7
![**Find the gradient vector field** \( \vec{F}(x, y, z) \) **of** \( f(x, y, z) = \ln(5x + 6y + z) \).
\[ \vec{F}(x, y, z) = \langle \text{__________} , \text{__________} , \text{__________} \rangle \]
This mathematical problem involves finding the gradient vector field of the function \( f(x, y, z) \). The function \( f(x, y, z) \) is given as the natural logarithm of the expression \( 5x + 6y + z \). To find the gradient vector field \( \vec{F}(x, y, z) \), you need to compute the partial derivatives of the function with respect to each variable \( x \), \( y \), and \( z \). These derivatives will fill in the blanks in the vector notation \( \langle \text{__________} , \text{__________} , \text{__________} \rangle \), reflecting the rate of change of the function in each principal direction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2F9462071c-b1a6-4ffd-9195-816b89d1b4c1%2F2r8epwp_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the gradient vector field** \( \vec{F}(x, y, z) \) **of** \( f(x, y, z) = \ln(5x + 6y + z) \).
\[ \vec{F}(x, y, z) = \langle \text{__________} , \text{__________} , \text{__________} \rangle \]
This mathematical problem involves finding the gradient vector field of the function \( f(x, y, z) \). The function \( f(x, y, z) \) is given as the natural logarithm of the expression \( 5x + 6y + z \). To find the gradient vector field \( \vec{F}(x, y, z) \), you need to compute the partial derivatives of the function with respect to each variable \( x \), \( y \), and \( z \). These derivatives will fill in the blanks in the vector notation \( \langle \text{__________} , \text{__________} , \text{__________} \rangle \), reflecting the rate of change of the function in each principal direction.
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