Chapter 4.6, Problem 308E

For the following exercises, solve the problem.

308. If the electric potential at a point (x. y) in the

xy-plane is $V\left(x,y\right)=e^{-2x}\cos \left(2y\right)$ . then the electric intensity vector at $\text{E=-}\nabla \text{V}\left(x,y\right)$.

a. Find the electric intensity vector at ( $\frac{\pi }{4},0$ ).

b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector E.

309. In two dimensions, the motion of an ideal fluid is governed by a velocity potential q. The velocity

components of the fluid u in the x—direction and v in the y—direction, are given by ( u. v ) = V q. Find the velocity components associated with the velocity potential p(x, y) = sin .zx sin 2irv.