Applied Mathematics and Statisticsmmiche18/MATH324/LECTURES/Math324_Lectures_01-28.pdfUW Math 324,...
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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.