Introduction to differential forms

As an application, Green’s theorem shows that the area of D can be computed
as line integral on the boundary
∫ ∫ ∫
dxdy = ydx
D
If S is a closed oriented surface in R 3 , such as the surface of a sphere, Stoke’s
theorem show that any exact 2-form integrates to 0. To see this write S as the
union of two surfaces S 1 and S 2 with common boundary curve C. Orient C
using the right hand rule with respect to S 1 , then orientation comming from S 2
goes in the opposite direction. Therefore
∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫
dω = dω + dω = ω − ω = 0
S
S 1 S 2 C
C
In vector notation, Stokes’ theorem is written as
∫ ∫
∫
∇ × F · n dS = F · ds
S
where F is a C 1 -vector field.
In physics, there a two fundamental vector fields, the electric field E and the
magnetic field B. They’re governed by Maxwell’s equations, one of which is
C
∇ × E = − ∂B
∂t
where t is time. If we integrate both sides over S, apply Stokes’ theorem and
simplify, we obtain Faraday’s law of induction:
∫
E · ds = − ∂ ∫ ∫
B · n dS
C ∂t
To get a sense of what this says, imagine that C is wire loop and that we are
dragging a magnet through it. This action will induce an electric current; the
left hand integral is precisely the induced voltage and the right side is related to
the strength of the magnet and the rate at which it is being dragged through.
Stoke’s theorem works even if the boundary has several components. However,
the inner an outer components would have opposite directions.
S
C
C
2
S
C
1
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