PP# 4: Can you help me solve and understand this practice problem, please? A clear step by step explanation would be appreciated. Thank you! The temporatureby TOx,y)= x?+y?at a point on a metal plute is given2reantest increase of heata.) Find the direction otgrat the point(-3,-1), V=?

PART A The maximum value of the temperature occurs when ∇T and the unit vector are in the same…

Answered: The temporature by TOx,y)= x?+y? at a…

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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PP# 4: Can you help me solve and understand this practice problem, please? A clear step by step explanation would be appreciated. Thank you!

The temporature
by TOx,y)= x?+y?
at a point on a metal plute is given
2
reantest increase of heat
a.) Find the direction otgr
at the point(-3,-1), V=?

Expert Solution

Step 1

PART A

The maximum value of the temperature occurs when $\nabla T$ and the unit vector are in the same direction.

$\nabla T (x, y) = \frac{d}{d x} (\frac{x}{x^{2} + y^{2}}) \hat{i} + \frac{d}{d y} (\frac{x}{x^{2} + y^{2}}) \hat{j} = (\frac{y^{2} - x^{2}}{{(x^{2} + y^{2})}^{2}}) \hat{i} + (\frac{- 2 x y}{{(x^{2} + y^{2})}^{2}}) \hat{j}$

Put the point (-3,-1)

$\nabla T (- 3, - 1) = - 0.08 \hat{i} - 0.06 \hat{j}$

$\begin{array}{rcl} |\nabla T| & = & \sqrt{{(- 0.08)}^{2} + {(- 0.06)}^{2}} \\ = & \sqrt{0.0100} \\ = & 0.1 \end{array}$

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