Midterm exam Math 5710 Fall 2013

Your name.................... October 23, 2013

1. The electrical circuit is given in Figure 1. Find the current between nodes A and B. Show algebraic formulas.

2. Show, computing curl v, that

(a) v_{1} = (y^{2}, x) is not the gradient of any function, but

(b) v_{2} = (\frac{2x}{y}, −\frac{x^{2}}{y^{2}} )

is. Find the potential.

3. Can vector field v_{3} = (xy, x) be represented as v_{3} =curl w where w is some vector field?

4. The force field F depends on the coordinates (x, y, z) as F = (y^{2}z, 2x y z,?). What is the third component of the force, if it is known that F is conservative F = ∇u.

5. In pure rotation, the velocity field v is represented as v = (ω_{1}, ω_{2}, ω_{3})× (x, y, z). Compute curl v, div v. Show that v and ω are perpendicular.

6. Compute ∇ · (u × v), where u, v are two vectors. 4