Chapter 6.6, Problem 11P

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.

(a) Given $\varphi =x^2-y^2,$ sketch on one graph the curves $\varphi =4,\varphi =1,\varphi =0$ $\varphi =-1,\varphi =-4.$ If $\varphi$ is the electrostatic potential, the curves $\varphi =$ const. are equipotentials, and the electric field is given by $E=-\nabla \varphi .$ If $\varphi$ is temperature, the curves $\varphi =$ const. are isothermals and $\nabla \varphi$ is the temperature gradient; heat flows in the direction $-\nabla \varphi$.

(b) Find and draw on your sketch the vectors $-\nabla \varphi$ at the points $(x,y)=(\pm 1,\pm 1)$ (0,\pm2), (t2,0). Then, remembering that $\nabla \varphi$ is perpendicular to $\varphi =$ const., sketch, without computation, several curves along which heat would flow [see(a)].