Multivariable Calculus (Math 214) Assignment

due December 6, 1999

Stokes Theorem: Generalizations of the Fundamental Theorem

Each of the following three statements is a generalization of the fundamental theorem of calculus. Functions involved are generally assumed to be differentiable.

A. If C is a parametrized path from the point A to the point B in n-space, and f is a scalar-valued function of n variables, then

INT[_{C} \nabla f ] = f(B) - f(A) .

B. If S is a piece of parameterized surface in 3-space with (parameterized) boundary \partial S and F is a vector-field in 3-space that is well-behaved near S, then

INT[INT[_{S} curl F ]] = INT[_{\partial S} F ].

C. If D is a solid domain in 3-space with boundary surface \partial D, oriented by its normal pointing outside D, and F is a vector-field in 3-space that is well-behaved in and around D, then

INT[INT[INT[_{D} div F ]]] = INT[INT[_{\partial D} F ]].

Exercises

Let F be the vector field in space that is given by

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

  1. Apply statement (A) above to evaluate the integral of F over

    1. the line segment from the origin to the point (1, 2, 3).

    2. either semi-circle in the plane x + y - z = 0 traversed from the origin to the point (1, 2, 3).

  2. Apply statement (C) above to evaluate the integral of F over the sphere of radius a > 0 with center at the origin when the sphere is parameterized using ``longitude'' and ``co-latitude''.

  3. How, if at all, can any of the above statements be an aid in working with the surface integral of the vector field F over the hemisphere of radius a > 0 with center at the origin in the upper half z > 0 of space ?

  4. How does ``Green's Theorem'' fit into this outline?

  5. What structural similarities do the three statements A, B, and C share?

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