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DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS
FORREST GORDON
Abstract. As they are classically written, Maxwell’s Equations
~ and the magnetic field B
~ as vector
describe the electric field E
fields on R3 which are determined by a time-dependent smooth
function ρ called the charge density and a time-dependent vector
field J~ called the current density. These equations can be rewritten
more elegantly in the language of differential geometry, which we
introduce herefor the special case R3 . We will describe the exterior
algebra Ω R3 of differential forms on R3 and the exterior derivative on this space. Along the way, we will encounter new analogues
of a few familiar tools from elementary vector calculus, and for fun
we will conclude by viewing Maxwell’s Equations written in terms
of differential forms.
1. Introduction
We begin with a quick review of vector calculus on R3 . For neatness,
we will denote the coordinate partial derivative operators by ∂x , ∂y ,
∂
, etc. We will use ∇ to denote the del operator,
and ∂z , instead of ∂x
which is the formal vector whose components are these three partial
derivatives: ∇ ≡ h∂x , ∂y , ∂z i. This operator will allow us to describe
the way smooth functions and vector fields are changing at each point
in R3 .
Let f : R3 → R be a smooth function. The gradient of f is the
vector field on R3 determined by the operation of del on f :
grad f = ∇f = h∂x f, ∂y f, ∂z f i.
The value of grad f at a point p ∈ R3 is a vector (∇f )|p whose three
components are the rates at which the value of f is increasing in the
three coordinate directions. This vector points in the direction of the
greatest rate of increase of f , with magnitude equal to this rate.
Now let F : R3 → R3 be a vector field, written in coordinate form as
F = hf (x, y, z), g(x, y, z), h(x, y, z)i, where f , g, and h are differentiable functions. The curl of F is the vector field on R3 that we obtain
when we take the formal cross product of the del operator and F :
Date: Spring, 2010.
1
2
FORREST GORDON
curl F = ∇ × F = (∂y h − ∂z g), (∂z f − ∂x h), (∂x g − ∂y f ) .
Perhaps the physical meaning of the curl is obscured by this strange
arrangement of partial derivatives, but the curl operator does just what
its name suggests: it tells us how tightly and in which direction the
vector field F is “curling” at each point p ∈ R3 . Specifically, the
vector (∇ × F ) |p points along the axis of this local rotation, and the
magnitude of this vector is proportional to the local rate of rotation
around this axis (with sign determined by the Right Hand Rule).
The divergence of the vector field F is the real-valued function that
we obtain when we take the formal dot product of del and F :
div F = ∇ · F = ∂x f + ∂y g + ∂z h.
Thus the divergence of F at a point p ∈ R3 is a measure of how F
“diverges” from p. If (∇ · F )(p) > 0, then p is a source of F ; if
(∇ · F )(p) < 0, then p is a sink of F . With grad, curl, and div fresh
in mind, we behold Maxwell’s Equations, shown here with all physical
constants set to 1:
Maxwell’s Equations
(1.1)
(1.2)
(1.3)
(1.4)
~ =0
∇·B
~
~ + dB = 0
∇×E
dt
~ =ρ
∇·E
~
~ − dE = J~
∇×B
dt
Equation (1.1), Gauss’ Law for Magnetism, states that the diver~ is zero at every point in space. In other
gence of the magnetic field B
words, the magnetic field has no sources or sinks; there are no magnetic
monopoles. Of course, there are electric monopoles, such as protons
and electrons, and the distribution of these monopoles in space is described by the charge density function ρ. Equation (1.3), Gauss’
~ is equal to ρ
Law, states that the divergence of the electric field E
3
everywhere on R . Equation (1.2), Faraday’s Law of Induction, states
that an electric field is induced whenever the magnetic field changes.
On the other hand Equation (1.4), Ampere’s Law with Maxwell’s Correction, states that a magnetic field is induced whenever the electric
field changes and whenever electric charge moves around in space. The
flow of electric charge through space is described by the current density field J~ in Equation (1.4).
DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS 3
These equations encapsulate our understanding of electromagnetism.
Still, if we take a moment to look at all four of them at once, we may
find our mathematical sensibilities a little offended. For one thing,
Equations (1.1) and (1.2) look almost exactly like Equations (1.3) and
~ and E
~ switching roles. The only differences are the minus
(1.4), with B
sign in Equation (1.4) and the non-zero right hand sides in the second
pair of equations. We might infer that there is some redundancy or
indirectness in this presentation of Maxwell’s Equations, and since we
have heard on the street that electricity and magnetism are different
~ and E
~
faces of the same force, we might wonder why separate fields B
are needed at all, instead of some single unified field.
Additionally, Equations (1.2) and (1.4) involve the cross product.
This operation requires an arbitrary choice of handedness on R3 and is
only defined for pairs of vectors in R3 , and so for instance we cannot
comfortably work with these equations on Minkowski spacetime, which
is a special version of R4 . There is some property of the cross product
which is essential to these equations, though; we might wonder if we
can isolate this property “behind” the handedness convention.
One last complaint we might raise - and this is a more subtle thing
- is that these equations are written in a way which depends on global
coordinates on R3 . This is undesirable because, in addition to more
technical reasons, “the universe does not come equipped with any standard set of coordinates”. Thus we would like to rewrite these equations
in some elegant way which does not depend on a global coordinate system and involves only one unified field. But there is no obvious way
for us to do this; our best lead right now is that we would like to get
rid of the cross product in these equations. With this goal in mind, we
set out to develop some new machinery.
2. The Exterior Algebra over a Vector Space
Let us consider the characteristics of the cross product as it operates
on the vector space R3 before we attempt to replace it with a more
general product defined on arbitrary vector spaces, which we will denote by ∧ and refer to as the wedge product. For starters, the cross
product is bilinear and distributive over vector addition on R3 , and R3
is closed under the cross pruduct. We will want our wedge product to
share these three properties.
The real distinguishing characteristic of the cross product, though,
is that it is antisymmetric. That is,
~v × w
~ = −w
~ × ~v
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FORREST GORDON
for all vectors ~v and w
~ in R3 . This is ordinarily presented as a consequence of the Right Hand Rule: if ~v and w
~ are linearly independent
3
vectors in R , then they span a plane in R3 , and the cross products
~v × w
~ and w
~ × ~v are both normal to this plane but have opposite magnitudes, according to the Right Hand Rule. But antisymmetry would
also follow from a Left Hand Rule convention on R3 , and since we are
only interested in what is happening behind the handedness convention, we simply make our wedge product antisymmetric without any
concern for generalizing the Right Hand Rule. This means, though,
that if we would like to take a wedge product of two vectors on some
vector space V , we are left without a natural way to place this product
in V . Our solution to this apparent dilemma is simple, and perhaps
surprising: we do not require wedge products to reside in V at all! Instead, we treat them simply as formal products, and rather than give
them geometric meaning inside of V (as cross products have in R3 ), we
deal with them outside of V via the properties of the wedge product
which we have chosen above, along with one last property (which is not
a property of the cross product): we decide that our wedge product will
be associative.
In vector calculus we often work with expressions like ~u + (~v × w),
~
where the addition of ~u and ~v × w
~ makes sense because both of these
summands are elements of V . We will now be working with expressions
like ~u + (~v ∧ w),
~ even though ~u is an element of V and ~v ∧ w
~ is not.
We will also encounter expressions like ~u ∧ (~v ∧ w).
~
Manipulating
wedge products “alongside” the vectors in V in this way will generate
a superspace of V called the exterior algebra over V , denoted ΛV .
We pause to note that including associativity may seem like a strange
and unwarranted step: why make the wedge product associative when
the cross product - our starting point for creating the wedge product - is
not associative itself? In fact, it is easily shown that the cross product
could not have been associative and still satisfied the Right Hand Rule.
Since we have abandoned the Right Hand Rule, we have no reason
not to assume associativity. The major benefit of an associative wedge
product is that we will be able to deal with longer wedge products, such
as the 3-fold wedge product ~u ∧~v ∧ w,
~ without worrying about the order
in which wedge products are evaluated, since (~u ∧ ~v ) ∧ w
~ = ~u ∧ (~v ∧ w).
~
The bilinearity, distributivity, antisymmetry, and associativity of the
wedge product impose a nice, natural structure on the exterior algebra
ΛV , which we now describe.
Let {~v1 , . . . , ~vn } be a basis for V . The bilinearity of ∧ has the important consequence that every k-fold wedge product of vectors in V can
be written as a linear combination of k-fold wedge products of basis
DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS 5
vectors. Furthermore, because ∧ is associative and antisymmetric, every k-fold wedge product of basis vectors may be rearranged so that its
k basis vectors are written out in non-descending order (with respect to
the indices of the basis vectors), and this only involves multiplication
by some power of −1. For example, the 4-fold wedge product
~v1 ∧ ~v4 ∧ ~v2 ∧ ~v7
is equivalent to the non-descending wedge product
−~v1 ∧ ~v2 ∧ ~v4 ∧ ~v7 .
Thus every k-fold wedge product of vectors in V is a linear combination
of non-descending k-fold wedge products of basis vectors:
X
ασ ~vσ(1) ∧ · · · ∧ ~vσ(k) ,
(2.1)
w
~1 ∧ · · · ∧ w
~k =
σ∈L
where
• L is the set of all non-descending lists of k (not necessarily
distinct) basis vectors,
• ~vσ(i) is the ith entry in the list σ, and
• each ασ is a real number.
The bilinearity of ∧ also gives us that w
~ ∧0 = 0∧w
~ = 0 for each w
~ in
V , and this is easily extended to the statement that for all w
~ i in V ,
(2.2)
w
~1 ∧ · · · ∧ 0 ∧ · · · ∧ w
~ k = 0.
The antisymmetry of ∧ gives us that for all w
~ in V ,
(2.3)
w
~ ∧w
~ = 0.
Therefore, in (2.1), for each list σ which is not strictly ascending, we
have by (2.3) that the wedge product ~vσ(1) ∧ · · · ∧ ~vσ(k) contains a zero,
and then by (2.2) that the whole product is zero. Thus the set of all
strictly ascending k-fold wedge products of basis vectors ~vi is a basis for
the vector space (over R) spanned by k-fold wedge products of vectors
k
in V . This space is denoted Λ
V . If we write n = dim V , then it is
n
clear that there are exactly k elements in this basis, and so we have:
n
k
dim Λ V =
.
k
Now an element of Λ0 V - that is, a linear combination over R of 0-fold
wedge products - is not really a wedge product of vectors at all; it is
just some coeffecient in R. Likewise, an element of Λ1 V is really just
an element of V . Thus we have natural isomorphisms Λ0 V ∼
= R and
Λ1 V ∼
V
,
which
agree
with
the
above
equation.
In
general,
an
element
=
of ΛV can be broken down uniquely into (n + 1) components, with one
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FORREST GORDON
component beloning to each Λk V as k varies from 0 to n. That is, the
exterior algebra over V is the direct sum of these vector spaces:
n
M
ΛV =
Λk V .
k=0
Notice that the exterior algebra over V is a finite dimensional vector
space:
n
X
dim ΛV =
dim Λk V = 2n .
k=0
Notice also that ΛV is closed under the wedge product for every vector
space V , just as R3 is closed under the cross product.
3. Differential Forms on R3
We are starting to get somewhere now. In the wedge product and
the exterior algebra it generates over any given vector space, we have
found our sought-after replacement for the cross product. We could
try to use the exterior albegra ΛR3 to rewrite Maxwell’s Equations. It
turns out - for reasons which we will not discuss in this short survey
- that this is not what we want to do. Instead, we will work with the
exterior algebra over the vector space of 1-forms on R3 . These objects
are defined on every smooth manifold, but for concreteness we will
describe them only in the special case of R3 .
Let C ∞ (R3 ) denote the set of all smooth (that is, infinitely differentiable) functions on R3 , and let Vect(R3 ) denote the set of all smooth
vector fields on R3 . Notice that if f is an element of C ∞ (R3 ) and V is
an element of Vect(R3 ), then we can multiply V by f “pointwise” on
R3 to obtain another vector field, f V . That is, at each point x in R3 ,
(f V )|x = f (x) · V |x .
We can also “operate” on f with V to obtain a new smooth function, written V f , whose value at every point x in R3 is equal to the
directional derivative of f along the vector V |x based at x:
(V f )(x) = DV |x (f ).
∞
3
Since C (R ) and Vect(R3 ) are both vector spaces, it makes sense
to think about maps between them. A 1-form on R3 is a map
ω : Vect(R3 ) → C ∞ (R3 )
which is linear over C ∞ (R3 ). In other words, for all V, W ∈ Vect(R3 )
and f ∈ C ∞ (R3 ),
ω(f V ) = f ω(V )
and
DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS 7
ω(V + W ) = ω(V ) + ω(W ).
Perhaps this definition alone does not give us much of a sense for what
a 1-form really is, but the following important class of examples of 1forms ought to give us a good hint. Let f be any smooth function on
R3 , and define the differential of f to be the map from Vect(R3 ) to
C ∞ (R3 ) given by
df : V 7→ V f .
The linearity of this map over C ∞ (R3 ) is easily checked (try it), and
so we find that df is a 1-form on R3 . Now define a 1-form dx = d(f x ),
where f x denotes the first coordinate function on R3 . Define dy and
dz similarly. It can be shown that the set of 1-forms on R3 , denoted
Ω1 (R3 ), is a vector space over C ∞ (R3 ), with basis {dx, dy, dz}. That
is, every 1-form ω on R3 can be written uniquely as some linear combination
ω = f dx + gdy + hdz,
where f , g, and h are smooth functions on R3 . This is the vector
space over which we will build and use an external algebra to revisit
Maxwell’s Equations. We define the space of differential forms on
R3 to be
L k 3
Ω(R3 ) =
Ω (R ),
the exterior algebra generated over Ω1 (R3 ) by the wedge product we
introduced earlier, but this time with coefficients in C ∞ (R3 ) instead of
just R. In this case then, we have Ω0 (R3 ) ∼
= C ∞ (R3 ).
Notice that the differential operator defined above determines a map
d : Ω0 (R3 ) → Ω1 (R3 ). We now generalize this to define the exterior
derivative as the map
d : Ωk (R3 ) → Ωk+1 (R3 ),
determined uniquely for each integer k, which satisfies the following:
(1) d : Ω0 (R3 ) → Ω1 (R3 ) is the differential
(2) d is linear over R
(3) d(ω ∧µ) = dω ∧µ + (−1)k ω ∧dµ for ω ∈ Ωk (R3 ) and µ ∈ Ω(R3 )
(4) d(dω) = 0 for all ω ∈ Ω(R3 ).
These may seem like some strange conditions. Let us get our hands
dirty by explicitly calculating exterior derivatives of general k-forms
for k = 0, 1, and 2. We will find a nice surprise when we do this.
First let us consider the case d : Ω0 (R3 ) → Ω1 (R3 ). Let ω be an
element of Ω0 (R3 ). Then of course ω is just a smooth function on R3 ,
and dω will be a 1-form on R3 , so for all V ∈ Vect (R3 ), we have
dω (V ) = V ω ∈ C ∞ (R3 ).
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FORREST GORDON
Now for all p ∈ R3 ,
d
(ω(p + tV |p ))
(dω(V ))(p) =
dt
d
=
( ωhpx + tVx (p), py + tVy (p), pz + tVz (p)i )
dt
d
d
= (px + tVx (p))∂x ω(p) + (py + tVy (p))∂y ω(p)+
dt
dt
d
(pz + tVz (p))∂z ω(p)
dt
= Vx (p)(∂x ω(p)) + Vy (p)(∂y ω(p)) + Vz (p)(∂z ω(p))
= ∇ω|p · V |p .
Thus globally we have dω(V ) = ∇ω · V . Since V is arbitrary in this
equality, we may write simply
dω ≡ ∇ω,
(3.1)
and so we see that taking the exterior derivative of a 0-form on R3 is
“the same as” taking the gradient of a smooth function on R3 !
Let us move on to the case d : Ω1 (R3 ) → Ω2 (R3 ). Let ω be an
element of Ω1 (R3 ). Since {dx, dy, dz} is a basis for Ω1 (R3 ), we may
write
ω = ωx dx + ωy dy + ωz dz,
∞
where ωx , ωy , ωz ∈ C (R3 ). By the linearity of d over R, we have
dω = d(ωx dx + ωy dy + ωz dz)
= d(ωx dx) + d(ωy dy) + d(ωz dz).
Now by the definitions of the wedge product and the exterior derivative
and by (3.1), we can manipulate the first term in the right-hand side
of this last equation:
d(ωx dx) = d(ωx ∧ dx)
= dωx ∧ dx + (−1)0 ω ∧ d(dx)
= dωx ∧ dx + 0
= ((∂x ωx )dx + (∂y ωx )dy + (∂z ωx )dz) ∧ dx
= (∂x ωx )dx ∧ dx + (∂y ωx )dy ∧ dx + (∂z ωx )dz ∧ dx
= 0 + (−∂y ωx )dx ∧ dy + (∂z ωx )dz ∧ dx.
By similar arguments, we obtain
d(ωy dy) = (−∂z ωy )dy ∧ dz + (∂x ωy )dx ∧ dy
and
DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS 9
d(ωz dz) = (−∂x ωz )dz ∧ dx + (∂y ωz )dy ∧ dz,
and so we add these three summands together to find that
dω = d(ωx dx) + d(ωy dy) + d(ωz dz)
= (∂x ωy − ∂y ωx )dx ∧ dy + (∂y ωz − ∂z ωy )dy ∧ dz
+ (∂z ωx − ∂x ωz )dz ∧ dx.
Thus the exterior derivative of a 1-form on R3 is “the same as” the curl
of a vector field on R3 !
Finally, let us consider the case d : Ω2 (R3 ) → Ω3 (R3 ). Let ω be an
element of Ω2 (R3 ); we may write
ω = ωxy dx ∧ dy + ωyz dy ∧ dz + ωzx dz ∧ dx,
where again ωxy , ωyz , ωzx ∈ C ∞ (R3 ). Then
dω = d(ωxy dx ∧ dy + ωyz dy ∧ dz + ωzx dz ∧ dx)
= d(ωxy dx ∧ dy) + d(ωyz dy ∧ dz) + d(ωzx dz ∧ dx).
Again, we use the definitions above and (3.1) to manipulate the first
term in the right hand side:
d(ωxy dx ∧ dy) = d(ωxy ∧ (dx ∧ dy))
= (dωxy ) ∧ (dx ∧ dy) + (−1)0 ωxy ∧ d(dx ∧ dy)
= [(∂x ωxy )dx + (∂y ωxy )dy + (∂z ωxy )dz] ∧ dx ∧ dy
= (∂z ωxy ) dz ∧ dx ∧ dy
= ∂z ωxy dx ∧ dy ∧ dz.
Similarly, we obtain
d(ωyz ∧ dy ∧ dz) = (∂x ωyz ) ∧ dx ∧ dy ∧ dz
and
d(ωzx ∧ dz ∧ dx) = (∂y ωzx ) ∧ dx ∧ dy ∧ dz,
and so by adding these three terms together we find that
dω = (∂x ωyz + ∂y ωzx + ∂z ωxy ) dx ∧ dy ∧ dz.
That is, the exterior derivative of a 2-form on R3 is “the same as” the
divergence of a vector field on R3 ! So we have seen that the gradient
of a function, the curl of a vector field, and the divergence of a vector
field are all special cases of this exterior derivative with which we are
dealing. With this new revelation in mind, we return to Maxwell’s
Equations.
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FORREST GORDON
4. Maxwell’s Equations Revisited
Let us consider again the first pair of Maxwell’s Equations:
~ =0
∇·B
~
~ + dB = 0
∇×E
dt
~ and the curl of
These involve the divergence of the magnetic field B
~ Thus, in this pair of equations, we see B
~ acting
the electric field E.
~ acting like a 1-form. Instead of writing these as
like a 2-form and E
~
~ = hEx , Ey , Ez i, let us write the
vector fields B = hBx , By , Bz i and E
magnetic field as a 2-form:
B = Bxy dx ∧ dy + Byz dy ∧ dz + Bzx dz ∧ dx ∈ Ω2 (R3 )
and the electric field as a 1-form:
E = Ex dx + Ey dy + Ez dz ∈ Ω1 (R3 ).
We are very close to unifying these two fields, but to do so we need
move out into R4 . The extra dimension will of course represent time,
and we will denote by dt the 1-form on R4 corresponding to this new
coordinate. The wedge product of the 1-form E with this new 1-form
will be a 2-form, which we can combine with the 2-form B via simple
addition. Thus we define the unified electromagnetic field F as a 2-form
on Minkowski spacetime R4 :
F = B + E ∧ dt.
Written in terms of this new 2-form, Equations (1.1) and (1.2) together
become the simple conservation law dF = 0. The electromagnetic
field is a 2-form on spacetime whose exterior derivative is zero. The
verification of this equality involves splitting the exterior derivative
d into so-called “space-like” and “time-like” parts, and we will not
describe this process here.
Equations (1.3) and (1.4) are trickier to rewrite. Unlike the first pair,
these lack a property called general covariance; what this means is that
re-writing them in terms of F involves the use of a Riemannian metric
on Minkowski spacetime. Their re-written form also involves the Hodge
star operator, which swaps k-forms with (n − k)-forms in a particularly
nice way. This machinery is above the level of this survey, but as
promised, we conclude by displaying Maxwell’s Equations rewritten in
terms of differential forms:
dF = 0
?d ? F = J.
DIFFERENTIAL FORMS ON R3 :
REWRITING MAXWELL’S EQUATIONS 11
References
[1] Baez, John and Javier P. Muniain, Gauge Fields, Knots and Gravity, Series on
Knots and Everything, Volume 4. World Scientific, Singapore, 1994.
[2] Lee, John M. Introduction to Smooth Manifolds Springer, New York, 2003.
Mathematics Department, Louisiana State University, Baton Rouge,
Louisiana
E-mail address: [email protected]