Chapter 6.6, Problem 10P

For Problems 10 to 14, use a computer as needed to make plots of the given surfaces and the isothermal or equipotential curves. Try both 3 D graphs and contour plots.

If the temperature in the $(x,y)$ plane is given by $T=xy-x,$ sketch a few isothermal curves, say for $T=0,1,2,-1,-2.$ Find the direction in which the temperature changes most rapidly with distance from the point $(1,1),$ and the maximum rate of change. Find the directional derivative of $T$ at $(1,1)$ in the direction of the vector $3i-4j.$ Heat flows in the direction $-\nabla T$ (perpendicular to the isothermals). Sketch a few curves along which heat would flow.