Multivariable Calculus (Math 214) Assignment

due November 15, 1999

1. For given positive constants a and h make a qualitative sketch of the right circular conical domain described by the inequalities

   {	0 <= z <= h 0 <= x^{2} + y^{2} <= (a/h)^{2} z^{2}

   and find its volume.

2. Let D be the domain of the previous exercise. Evaluate the triple integral

   INT[INT[INT[_{D} z dx dy dz ]]] .

3. Let C be the path that is given parametrically by

   R(t) = (( cos (theta))t , ( sin (theta))t) for 0 <= t <= r ,

   where r and theta are constants. Evaluate the path integral

   INT[_{C} F]

   when F is the vector field that is defined by

   1. F(x, y)=(y e^{x y}, x e^{x y}) .

   2. F(x, y)=(x - y, x + y) .

   3. F(x, y)=({x}/{SQRT{x^{2} + y^{2}}}, {y}/{SQRT{x^{2} + y^{2}}}) .

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

4. Let g in succession, one at a time, be the function defined by making g(r, theta) the value of the path integral in each of the parts of the previous exercise. Can the function g be made a function f of Cartesian coordinates x, y in the plane if r and theta are regarded as polar coordinates? If so, what is that function f, and what is its gradient \nabla f ?

5. How do the various parts of the previous exercise relate to the formula for path integrals of gradient vector fields?