1.  Consider the function:

(x,y) =〈  ,  〉

Preform the following tasks:

(a) Compute the divergence of  .

(b) Compute ?C  · d⃗r  Where C is the circle of radius 1 centered at the origin oriented counterclockwise.

(c) Determine whether  conservative.

(d) Determine whether there is a conservative function  : R3  → R3  defined everywhere except (0, 0, 0) with continuous partial derivatives so that for all (x,y)  0 we have:

(x,y,0) =〈  ,  , 0〉

2. Consider a force with an inverse square law, given as follows:

(x,y,z) =

Where C is a constant.

Show that for any closed piece-wise smooth surface S that is oriented outward we have:

\\S  · dσ =

3. Find the area of the region in R2  bound by the curve ⃗r(t) = ⟨sin4 (t), cos (t)⟩, the4 x-axis, and the y-axis.

4. In the study of electricity and magnetism Amp´ere’s law was originally given by: ∇ ×  = µ0 J⃗

Where  is the magnetic field, µ0  is the magnetic constant, and J⃗ is the current density1 . We will use some facts about basic circuits and Stoke’s Theorem to show that this form of Amp´ere’s law is false.

A capacitor is two metal plates separated by a material that prevents charge from moving through. Let’s say that we have a capacitor in R3  set up with two circular disks with radii of 1 meter the lower one centered on the point (0, 0, 1) the upper one centered at (0, 0, 2) (both disks are parallel to the xy-plane). Suppose that we have a wire moving down from the lower disk and a wire and a wire moving up from the upper disk.  Suppose that at time t0  there is a current I0  > 0 moving through both wires.  Notice that at this time there is no current passing between the disks. Here is a simplified diagram: