Chapter 6.6, Problem 13P

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 $3\text{D}$ graphs and contour plots.

Let $\varphi =e^x$ cos y. Let $\varphi$ represent either temperature or electrostatic potential. Refer to Problem 11 for definitions and find:

(a) The direction in which the temperature is increasing most rapidly at $(1,-\pi /4)$ and the magnitude of the rate of increase.

(b) The rate of change of temperature with distance at $(0,\pi /3)$ in the direction $i+j\sqrt{3}$

(c) The direction and magnitude of the electric field at $(0,\pi ).$

(d) The magnitude of the electric field at $x=-1,$ any $y$.