Module II

Lecture-11: Exterior Calculus∗

1 Cartan’s exterior calculus

There an alternative to vector calculus (in Euclidean space or Minkowski spacetime or more general
spacetimes) which exclusively uses antisymmetric tensors of rank (0, p) (aka differential p-forms)
and is called (Cartan’s) exterior calculus. The ingredients for this are

• Differential p-forms (ω(p) )

• Wedge product operation (Λ)

• Exterior derivative operation (d)

• (Hodge) star operation (∗)

We have already discussed about differential p-forms. Now we will discuss about the rest.

1.1 Exterior algebra of forms

The direct sum space of differential forms,

Ω ≡ Ω0 + Ω1 + Ω2 + . . . + Ωp

in a p-dimensional space (say Rp or Rd,1 with p = d + 1), gets turned into an algebra1 under the
wedge product. One can check this explicitly for a few cases, e.g. consider the wedge product of
∗
Please report typos and errors to sroy@phy.iith.ac.in / abhattacharyya@iitgn.ac.in.
1
If a vector space, say V is closed under a new bilinear binary operation (product) whereby the product of two
vectors is another vector,
u · v = w, u, v, w ∈ V
we call that special vector space an algebra.

1
two 1-forms, say ω and ξ:
ω ∧ ξ = (ωµ dxµ ) ∧ (ξν dxν ) ,
= (ωµ ξν ) dxµ ∧ dxν ,
 
ωµ ξν + ων ξµ ωµ ξν + ων ξµ
= + dxµ ∧ dxν ,
2 2
   
ωµ ξν + ων ξµ µ ν ωµ ξν − ων ξµ
= dx ∧ dx + dxµ ∧ dxν ,
2 2
| {z }
=0 (Contraction of sym. and antisym. tensors vanish)
1
= Bµν dxµ ∧ dxν , Bµν ≡ ωµ ξν − ων ξµ .
2
Clearly the result is a 2-form,
1
B= Bµν dxµ ∧ dxν , Bµν = −Bνµ .
2!
Similarly one can check the wedge product of a 1-form and 2-form, say ω(1) and B(2) as follows:
 
λ

1 µ ν
ω(1) ∧ B(2) = ωλ dx ∧ Bµν dx ∧ dx ,
2!
1
= (ωλ Bµν ) dxλ ∧ dxµ ∧ dxν ,
2!
11
= (ωλ Bµν − ωµ Bλν − ων Bµλ ) dxλ ∧ dxµ ∧ dxν ,
2! 3
1
= (ωλ Bµν + ωµ Bνλ + ων Bλµ ) dxλ ∧ dxµ ∧ dxν ,
3!
1
= Hλµν dxλ ∧ dxµ ∧ dxν , Hλµν = ωλ Bµν + ωµ Bνλ + ων Bλµ .
3!


Evidently the basis dxλ ∧ dxµ ∧ dxν is completely antisymmetric in λ, µ, ν, while the component
(ωλ Bµν ) is only antisymmetric in µ, ν but not in λ, ν or λ, µ. So in going from the second step to
the third step, we replaced (ωλ Bµν ) with the a completely antisymmetric combination as follows,
1
ωλ Bµν → (ωλ Bµν − ωµ Bλν − ων Bµλ ) .
3
(The symmetric parts will anyway drop out upon contraction with the basis). Finally in going
from the third to fourth step we have used the antisymmetry of B to make the replacements,
Bλν → −Bνλ , and Bµλ → −Bλµ . In the final line we recognize that the RHS is now a 3-form with
the basis and the component being totally antisymmetric in all three indices as well as having the
correct normalization factor ( 3!1 ). Thus we conclude that direct sum space Ω (of all differential
forms of all ranks) is closed under wedge product operator, thereby turning Ω into an algebra
(Exterior algebra).

Comment: Note that there is an upper limit to the rank of p-forms in a D-dimensional space.
In a D-dimensional space, the points are labeled by D coordinates, say {xµ } , µ = 1, ..., D (or
µ = 0, . . . , D − 1). Hence the number of 1-form basis elements, namely dxµ is D. We can only
wedge D number of 1-forms, since the resulting D-form
dx1 ∧ dx2 ∧ . . . dxD

2
has exhausted all possible distinct 1-forms. If we try to construct a (D + 1)-form by taking wedge
product of this D-form with another 1-form, say dxp , the result will vanish since the index p will
appear twice while being antisymmetrized in itself!

2 Exterior derivative (d)

The exterior derivative turns a p-form into a (p + 1)-form using a derivative operation:

d : Ωp → Ωp+1 .

Consider a p-form
1
ωp =ωµ µ ...µ dxµ1 ∧ dxµ2 ∧ . . . ∧ dxµp .
p! 1 2 p
The exterior derivative of this p-form, denoted by dωp is defined by,
1
dωp ≡ ∂ν ωµ1 µ2 ...µp dxν ∧ dxµ1 ∧ dxµ2 ∧ . . . ∧ dxµp . (1)
p!
Since the basis is antisymmetric under the swap of ν with any of the µ’s we can rewrite the RHS
as,
1
∂ν ωµ1 µ2 ...µp − ∂µ1 ων µ2 ...µp − ∂µ2 ωµ1 ν µ3 ...µp . . . dxν ∧ dxµ1 ∧ dxµ2 ∧ . . . ∧ dxµp .


dωp =
(p + 1)!
Example: Consider the Maxwell 1-form potential,

A = Aµ dxµ

The exterior derivative of this Maxwell 1-form potential, using the formula (1), is

dA = ∂ν Aµ dxν ∧ dxµ

or, antisymmetrizing the component ∂ν Aµ in the indices ν, µ, we get,

1
dA = (∂ν Aµ − ∂µ Aν ) dxν ∧ dxµ .
2
We recognize the term ∂ν Aµ − ∂µ Aν as the Maxwell field strength, Fνµ . Thus we see that,
1
dA = Fνµ dxν ∧ dxµ ,
2
= F.

Similarly using the exterior derivative, the homogeneous Maxwell equations (Faraday-Lenz law
and the “no sources or sinks law for magnetic field” )

∂λ Fµν + ∂µ Fνλ + ∂ν Fλµ = 0

can be expressed together in the simple form,

dF = 0.

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Leibniz rule:
The exterior derivative satisfies the following identity,

d (αp ∧ βq ) = dαp ∧ βq + (−)p αp ∧ dβq (2)

which is the Leibniz rule for taking exterior derivatives of wedge products of two forms, αp and βq .
Here p is not necessarily equal to q so these are forms of different rank/order.

2.1 Important Identity

The exterior derivative can act on a form only once. No higher order exterior derivatives exist.
This is due to the fundamental identity,
d2 = 0,
or more explicitly,
d 2 ωp = 0
for an arbitrary p-form field, ωp . This can be easily proven as follows. First we recall that (1),
1
dωp = ∂ν ωµ1 µ2 ...µp dxν ∧ dxµ1 ∧ dxµ2 ∧ . . . ∧ dxµp .
p!
Then,
 
2 1
d ωp = d (dωp ) = d ∂ν ωµ1 µ2 ...µp dxν ∧ dxµ1 ∧ dxµ2 ∧ . . . ∧ dxµp
,
p!
 
1 λ ν µ1 µ2 µp
= ∂λ ∂ν ωµ1 µ2 ...µp dx ∧ dx ∧ dx ∧ dx ∧ . . . ∧ dx ,
p!
= 0.

Here the last expression vanishes because it is a contraction of object symmetric in λ, ν, to wit
∂λ ∂ν (. . .), with an object which is antisymmetric in λ, ν, namely dxλ ∧ dxν .

3 Hodge Star (∗) operation and the inner product of two

p-forms
We observe that the vector space of p-forms, Ωp and the vector space of (D − p)-forms, ΩD−p have
the same dimensions,    
D D
= .
p D−p
A fundamental theorem in the theory of vector spaces is that two (finite-dimensional) vector spaces
which have the same dimensions are isomorphic. In the case when the two vector spaces are Ωp
and ΩD−p , this isomorphism is given by the Hodge “∗” operation. The ∗-operation is defined
on a p-form basis element as follows,
1
∗ (dxµ1 ∧ . . . ∧ dxµp ) = ϵµ1 ...µp ν1 ...νD−p dxν1 ∧ . . . ∧ dxνD−p .
(D − p)!

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Evidently the argument/input is the p-form basis element, dxµ1 ∧ . . . ∧ dxµp , while the output is a
(D − p)-form owing to the expansion in the (D − p)-form basis elements dxν1 ∧ . . . ∧ dxνD−p . The
basis coefficients on the RHS (components) are the completely antisymmetric Levi-Civita tensor,
albeit with hybrid indices (both upstairs and downstairs) obtained by raising (lowering) indices by
the (inverse) metric,
ϵµ1 ...µp ν1 ...νD−p = ϵλ1 ...λp ν1 ...νD−p η µ1 λ1 . . . η µp λp .

3.1 Inner Product of two p-forms

The Hodge ∗ operation allows us to define an inner product in the space of all forms of all ranks,
Ω. The inner product of two p-forms, say αp and βp is then defined by,
ˆ
(αp , βp ) = αp ∧ (∗βp ) .
M
In the RHS there is a D-form since it is a wedge product of a p-form (αp ), and a (D − p)-form
(∗βp ). A D-form is always proportional to the volume element (volume form), i.e.,
αp ∧ (∗βp ) = (Function of the x’s) dD x,
and so this is ready made for volume integration! That is why we have the integral sign on the
RHS. One can check this satisfies all the properties of an inner product in a real vector space:
• Symmetry:
(αp , βp ) = (βp , αp )
• Positivity,
(αp , αp ) ≥ 0,
where the equality holds only when αp = 0.
Thus the space Ω is now an inner product space!

Now one can show the following,

αp ∧ ∗βp = βp ∧ ∗αp . (3)
This identity guarantees the symmetry property of the inner product (αp , βp ) = (βp , αp ).

4 Adjoint of exterior derivative (Codifferential) δ

Since the space Ω is an inner product space, one can define the adjoint of every operator. In
particular we can ask what is the adjoint of the exterior derivative, d operator. Call the adjoint
of the d to be δ, dubbed the codifferential. Let’s define it using the definition of the adjoint in
a real inner product space, namely
(βp−1 , δαp ) ≡ (dβp−1 , αp ) .
Note that according this definition, the codifferential actually lowers the rank of the form,
i.e. converts a p-form into a (p − 1)-form! Also we note that, from the definition, δ can be
expressed in terms of a sequence of exterior derivative d and Hodge ∗-operations as follows,
δωp = (−)D p+D+1 ∗ d ∗ ωp .
So in that sense δ is not an entirely new operation.

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4.1 Laplacian
The exterior derivative, d operator increases the rank of form while the codifferential, δ reduces
the rank of the form on which it acts,

d : Ωp → Ωp+1 ,
δ : Ωp → Ωp−1 .

None of these can act twice, since

d2 = 0 = δ 2 .
So it seems like we cannot have second order derivative equations using d or δ. However most
equations in physics are second order, so how will we formulate equations with second order
derivative in the language of differential forms? To this end, we introduce/define the Laplacian
operator,
∆ = (d + δ)2 = d δ + δ d.
Evidently, the ∆ operator does not alter the rank of the form it acts on,

∆ : Ωp → Ωp .

References
[1] Eguchi-Gilkey-Hanson, Sec. 2.3 “Differential Forms”.