Problem: A metal plate whose temperature at the point (x,y) is given $T(x,y)=200-5{x}^{2}-3xy-{y}^{2}$. 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 $\mathrm{\nabla}T(x,y)$

Question 1: $\mathrm{\nabla}$$T(x,y)=<-10x-3y,-3x-2y>$. Is it right?

Problem 2: Denote the position at time t by $r\left(t\right)=(x\left(t\right),y\left(t\right))$. 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.