Multi-Variable Calculus Assignment

due Tuesday, May 1, 2001

The Fundamental Theorem of Multi-Variable Calculus

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

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

∫_{C} ∇f = f(B) − f(A) .

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

∫ ∫_{S} curl F = ∫_{∂S} F .

C. If D is a solid domain in 3-space with boundary surface ∂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

∫ ∫ ∫_{D} div F = ∫ ∫_{∂D} F .

Exercises

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

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

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

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

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

2. Let F (x, y, z) = (x + y^{2}, y + z^{3}, z + x^{4}). 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, as usual, using “longitude” as first parameter and “co-latitude” as second parameter.

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

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

Compute

∫ ∫_{S} curl F

when S is the upper-half of the sphere x^{2} + y^{2} + z^{2} = 25 parameterized using “longitude” and “co-latitude”, as above.