Given the function z= f(x, y) = ln(x² + xy - y²), find the following: a) Find gradient at the point (2,3). Vf(2,3)= b) Given the unit vector ū=i-j, find the directional derivative at the point (2,3). Dif (2,3)=
Given the function z= f(x, y) = ln(x² + xy - y²), find the following: a) Find gradient at the point (2,3). Vf(2,3)= b) Given the unit vector ū=i-j, find the directional derivative at the point (2,3). Dif (2,3)=
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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Please only do part c and d
For part a, I got
For part b, I got 29/5
I just need part c and d. Please explain well
![### Problem Set
#### Given the function \( z = f(x,y) = \ln(x^2 + xy - y^2) \), find the following:
**a)** Find **gradient** at the point \((2,3)\).
\[ \nabla f(2,3) = \]
**b)** Given the unit vector \( \vec{u} = \frac{7}{25} \vec{i} - \frac{24}{25} \vec{j} \), find the **directional derivative** at the point \((2,3)\).
\[ D_{\vec{u}} f(2,3) = \]
**c)** In which **unit direction** does the graph of \( z = f(x,y) \) have its **steepest decline** at the point \((2,3)\), and what is this **decline**?
**d)** In which **direction** is the graph of \( z = f(x,y) \) **level** at the point \((2,3)\)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9fe5af2-5e22-4571-b7b9-b6f685526b72%2F3368bd1b-d040-48d0-8a01-416811f310f0%2F75v8t4g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Set
#### Given the function \( z = f(x,y) = \ln(x^2 + xy - y^2) \), find the following:
**a)** Find **gradient** at the point \((2,3)\).
\[ \nabla f(2,3) = \]
**b)** Given the unit vector \( \vec{u} = \frac{7}{25} \vec{i} - \frac{24}{25} \vec{j} \), find the **directional derivative** at the point \((2,3)\).
\[ D_{\vec{u}} f(2,3) = \]
**c)** In which **unit direction** does the graph of \( z = f(x,y) \) have its **steepest decline** at the point \((2,3)\), and what is this **decline**?
**d)** In which **direction** is the graph of \( z = f(x,y) \) **level** at the point \((2,3)\)?
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