Carole Juarez

2022-02-22

Problem: A metal plate whose temperature at the point (x,y) is given $T\left(x,y\right)=200-5{x}^{2}-3xy-{y}^{2}$. Given a function f(x,y), the vector $\mathrm{\nabla }f=<{f}_{x},{f}_{y}>$ points in the direction of greatest increase for f.
Problem 1: Compute $\mathrm{\nabla }T\left(x,y\right)$
Question 1: $\mathrm{\nabla }$$T\left(x,y\right)=<-10x-3y,-3x-2y>$. Is it right?
Problem 2: Denote the position at time t by $r\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)$. We want ${r}^{\prime }\left(t\right)=\mathrm{\nabla }T\left(r\left(t\right)\right)$ for all t to ensure the temperature is increasing as much as possible. Interpret this equation as a system of linear ODEs with the form ${r}^{\prime }=Ar$
Question 2: not sure how to proceed.

aksemaktjya

(i) yes (ii)
$\left[\begin{array}{c}\stackrel{˙}{x}\\ \stackrel{˙}{y}\end{array}\right]=\left[\begin{array}{c}-10-3\\ -3-2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

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