Multivariable Calculus (Math 214) Assignment

due December 1, 1999

1. Compute the following multi-variable derivatives:

   1. Find div(yz, zx, xy).

   2. Find curl(yz, zx, xy).

   3. Find div(curl(yz, zx, xy)).

   4. Find \nabla arctan (y/x).

   5. Find div(\nabla arctan (y/x)).

   6. Find (for f a general scalar function): div(\nabla f).

   7. Find for n = 2 (general f): curl(grad(f)).

   8. Find for n = 3 (general f): curl(grad(f)).

2. Attempt to Apply Green's Theorem to evaluate the path integral over the circle of radius a with center at the origin, when traversed once counter-clockwise, of the vector fields

   1. F(x, y) = (0, x).

   2. F(x, y) = (-y/rho^{2} , x/rho^{2}), where rho^{2} = x^{2} + y^{2}.

3. Let T be the transformation of the plane that is defined by formula

   T(u, v) = (1 - u - v + u v, uv) .

   1. Verify that for all u, v one has T(v, u) = T(u, v).

   2. Compute the Jacobian determinant J_{T} of T.

   3. Describe the set U of pairs (u, v) where J_{T}(u, v) = 0.

   4. Describe concretely the set T(U) consisting of all points T(u, v) as the point (u, v) varies in U.

   5. Let D be the unit square

      D : 0 <= u <= 1 , 0 <= v <= 1 .

      Observing that T is not bi-single-valued on D since T(v, u) = T(u, v), find an explicit subset Delta of D on which T is bi-single-valued with T(Delta) = T(D).

   6. Use the substitution rule for double integrals with the transformation T to find the area of T(Delta).

   7. Sketch the set T(D).

   8. Give analytic descriptions of each boundary curve of T(D).