Integration

Line Integrals
Where the curve C is parametrized as r(t)
and we solve this as..
$∫_{a}(F_{oncurve}∣∣dtdr ∣∣dt)$
2D Surface Integrals
Where S could be S = r(t,k) or H (x,y,z) =0
What’s important here, is that you get the normal, N (Usually the gradient)
and we solve this as…
$∫_{a}(F_{onsurface}∣∣n∣∣dA)$
3D Surface Integrals
We solve this one is a similar way!
Theorems (Special cases that give alternative ways to calculate)

1D
FOTOC
$∫_{x=a}(dxdg )=g(b)−g(a)$b) Path Independent Special Case
$∫_{c}∇g⋅T^ds=g(b)−g(a)$another form of that
$∫_{c}(∇g×k^)N^ds=g(b)−g(a)$
2D
Green’s Theorem
Normal/Flux Form, also known as 2D Divergence Theorem.
$∬_{R}∇⋅FdA=∮_{c}F⋅N^ds$Tangential/ Circular form, also known as 2D Stokes theorem
$∬_{R}(∇×F)⋅k^dA=∮_{c}F⋅T^ds$
3D
Stokes Theorem
$∫_{c}(∇×F)N^ds=∮_{c}F⋅T^ds$When Stokes is a flat region, S turns into R< and we just get Green’s theorem again.
Divergence Theorem
$∭_{Vol}(∇⋅F)dVol=∮∮_{S}F⋅N^ds$