Multivariable Calculus (Math 214) Assignment

due November 17, 1999

1. Definitions and Terminology

Let S: (x, y, z) = r(u, v) be a parameterized piece of surface in space with (u, v) varying in a plane region R.

Surface integral of a scalar-valued function.

The integral of a scalar-valued function f over S is defined as an ordinary double integral over the parameter region R: INT[INT[_{S} f ]] = INT[INT[_{R} f(r(u, v)) ||{\partial r}/{\partial u}\times {\partial r}/{\partial v}|| du dv]] .

Surface integral of a vector-valued function.

The integral of a vector-valued function F over S is also defined as an ordinary double integral over the parameter region R: INT[INT[_{S} F ]] = INT[INT[_{R} F(r(u, v)) . ({\partial r}/{\partial u}\times {\partial r}/{\partial v}) du dv]] .

2. Exercises

1. What term is used for the integral INT[INT[_{S} 1]], i.e., the integral over a surface S of the constant scalar 1.

2. Recall that the sphere of radius a with center at the origin is parameterized by longitude and colatitude:

   r(u, v) = (a cos u sin v, a sin u sin v, a cos v)

   for (u, v) in a suitable rectangle in the (u, v) plane.

   1. Describe with inequalities a rectangle in the (u, v) plane for which r(u, v) covers the sphere.

   2. Find the integral over the sphere of the function f when the expression f(x, y, z) is given generally by the formula:

      1. 1

      2. x

      3. x^{2} + y^{2} + z^{2}.

   3. Find the integral over the sphere of the vector field F that is given by

      F(x, y, z) = (yz, zx, xy) .

3. Recall that the volume of a solid domain D in space is given by

   vol(D) = INT[INT[INT[ 1 ]]] .

   1. Find the volume of the tetrahedron with vertices at the points (0, 0, 0), (2, 0, 0), (7, 3, 0), and (0, 0, 5).

   2. Use the definition of volume as a triple integral to obtain a formula as a double integral over the plane region S for the volume of the solid domain D that is described by:

      0 <= z <= f(x, y) where (x, y) varies in the region S .