Review of Curve and Surface Integrals

for Calculus III (Math 214)


Curve integrals

C:   rt,atb

   Note:  dr=rtdt=Tds;    ds=dr=rtdt

Surface integrals

S:   Wu,v,u,vE     (E a planar region)

   Note:  dW=Wu×Wvdudv=Ndσ;    dσ=dW=Wu×Wvdudv
where Wu=Wu,Wv=Wv

For both curves and surfaces a parameterization determines an orientation. The orientation of a surface in R3 is determined by the two-fold choice of a unit normal to that surface, and the boundary of an oriented surface is given the orientation that is related to the chosen unit normal in a right-handed coordinate system by the “right-hand rule”.

The Fundamental Theorem of Calculus

Theorems of the form Gω=Gdω

dim G
left side
right side
1 f f fbfa Iftdt
Fund. Thm. of Calculus I interval in R from a to b
1 f f fBfA Cgradf·dr
C path in Rn from A to B
2 F·dr FdA RF·dr RcurlFdA
Green's Thm. R region in R2 (R anti-clockwise in right-hand coord. system)
2 F·dr ×F·dW SF·dr ScurlF·dW
Classical Stokes’ Thm. S surface in R3 (with r/l hand rule)
3 F·dW ·FdV DF·dW DdivFdV
The Divergence Thm. D domain in R3 (D with outer normal)