Chapter 13.1, Problem 69ES

If $T\left(x,y\right)$ is the temperature at a point $\left(x,y\right)$ on a thin metal plate in the xy-plane, then the level curves of T are called isothermal curves. All points on such a curve are at the same temperature. Suppose that a plate occupies the first quadrant and $T\left(x,y\right)=xy.$

(a) Sketch the isothermal curves on which $T=1,T=2,\text{and T=3}\text{.}$

(b) An ant, initially at $\left(1,4\right)$ wants to walk on the plate so generating the graph Off from various view so that the temperature along its path remains constant. What path should the ant take and what is die temperature along that path?