UW Math 324, sections A and B, Spring 2013 - Handout #3

The "Big Theorems" of integral calculus

by M. Micheli.- June 7, 2013

1. The Fundamental Theorem of Calculus (from Math 126). Let g(x), x E ~, be a differentiable function of one variable. For. any choice of points a, b E ~ we have: :1

g(b) - g(a) = lb g'(x) dx.

o b

2. The Fundamental Theorem for Line Integrals. For any differentiable scalar function f(x, y, z), (x,y,z) E ~3, and any piecewise-smooth oriented curve C: r(t), t E [a,b] we have:

x

f(r(b)) - f(r(a)) = fc "f, Tds. r(h)

(It also works in two dimensions!).

3. Green's.Theorem. For any closed curve C in ~2, oriented positively (counterclocwise) and any vector differentiable field F(x,y) = (P(x,y),Q(x,y)) that is defined in D (the region enclosed by C) we have:

C

4. Stokes' Theorem. For any oriented surface S in ~3, with a boundary C that is positively oriented with respect to S, and any differentiable vector field F defined in ~3, we have

fc F . TdS = lfs(cUrlF).ndS

(It generalizes Green's Theorem to three dimensions.)

5. The Divergence Theorem. For any solid E in ~3 whose boundary S (which is a closed surface) is oriented outwards, and any differentiable vector field F defined in ~3, we have

lis F· ndS = 1 1 L (divF) dV.