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
| 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 ]]. |
Let F be the vector field in space that is given by
| F(x, y, z) = (yz, zx, xy) . |
the line segment from the origin to the point (1, 2, 3).
either semi-circle in the plane x + y - z = 0 traversed from the origin to the point (1, 2, 3).
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.
Let F be the vector field in space that is given by
| F(x, y, z) = (y + 2 z, z + 2 x, x + 2 y) . |
| INT[INT[_{S} curl F ]] |