In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
Hodge star operator or
Hodge dual is a significant linear map introduced in general by
W. V. D. HodgeWilliam Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
. It is defined on the
exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of a finite-dimensional
orientedIn mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
inner product spaceIn mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
.
Dimensions and algebra
The Hodge star operator establishes a one-to-one correspondence between the space of
k-vectors and the space of (
n −
k)-vectors. The image of a
k-vector under this correspondence is called the
Hodge dual of the
k-vector. The former space, of
k-vectors, has dimension
while the latter has dimension
and by the symmetry of the
binomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
s, these two dimensions are in fact equal.
Two
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s with the same dimension are always isomorphic; but not necessarily in a natural or canonical way. The Hodge duality, however, in this case exploits the inner product and orientation of the vector space. It singles out a unique isomorphism, that reflects therefore the pattern of the binomial coefficients in algebra. This in turn induces an inner product on the space of
k-vectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.
The first interesting case is on three-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
V. In this context the relevant row of
Pascal's triangleIn mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
reads
- 1, 3, 3, 1
and the Hodge dual sets up an isomorphism between the two spaces of dimension 3, which are
V itself and the space of wedge products of two vectors from
V. See the Examples section for details. In this case the content is just that of the
cross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of traditional
vector calculus. While the properties of the cross product are special to three dimensions, the Hodge dual is available in all dimensions.
Extensions
Since the space of alternating linear forms in
k arguments on a vector space is naturally isomorphic to the dual of the space of
k-vectors over that vector space, the Hodge dual can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge dual can then be extended to a
vector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
. Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle (i.e. the space of differential forms on a manifold) where it can be used to construct the
codifferential from the
exterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
, and thus the Laplace-de Rham operator, which leads to the Hodge decomposition of differential forms in the case of
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s.
Formal definition of the Hodge star of k-vectors
The
Hodge star operator on an
orientedIn mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
inner product
spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V is a linear operator on the
exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of
V, interchanging the subspaces of
k-vectors and
n−k-vectors where
n = dim
V, for 0 ≤
k ≤
n. It has the following property, which defines it completely: given an oriented orthonormal basis
we have
where
is an even permutation of
.
Of these
relations, of course only
are independent. The first one in the usual lexicographical order reads
Index notation for the star operator
Using index notation, the Hodge dual is obtained by contracting the indices of a
k-form with the
n-dimensional completely antisymmetric
Levi-Civita tensor. This differs from the
Levi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
by an extra-factor of (det
g)
½, where
g is an inner product. (One uses an absolute value around the determinant if
g is not positive-definite, e.g. for tangent spaces to Lorentzian manifolds.)
Thus one writes
where η is an arbitrary antisymmetric
tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
in
k indices. It is understood that
indices are raised and loweredIn mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a type tensor to a tensor or to a tensor...
using the same inner product
g as in the definition of the Levi-Civita tensor. Although one can take the star of any tensor, the result is antisymmetric, since the symmetric components of the tensor completely cancel out when contracted with the completely anti-symmetric Levi-Civita symbol.
Examples
A common example of the star operator is the case
n = 3, when it can be taken as the correspondence between the vectors and the
skew-symmetric matricesIn mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
of that size. This is used implicitly in
vector calculus, for example to create the
cross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
vector from the wedge product of two vectors. Specifically, for
EuclideanIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
R3, one easily finds that
and
and
where
dx,
dy and
dz are the standard orthonormal differential
one-formIn linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s on
R3. The Hodge dual in this case clearly corresponds to the cross-product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.
Three-dimensional example
Applied to three dimensions, the Hodge dual provides an
isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
between axial vectors and
bivectorIn mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...
s, so each axial vector
a is associated with a bivector
A and vice-versa, that is:
-
where * indicates the dual operation. These dual relations can be implemented using multiplication by the unit pseudoscalar in
Cl
3(R),
i =
e1e2e3 (the vectors are an orthonormal basis in three dimensional Euclidean space) according to the relations:
-
The dual of a vector is obtained by multiplication by
i, as established using the properties of the geometric product of the algebra as follows:
-
and also, in the dual space spanned by }:
-
In establishing these results, the identities are used:
and:
These relations between the dual
and
i apply to any vectors. Here they are applied to relate the axial vector created as the
cross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
a = to the bivector-valued exterior product
A = of two polar (that is, not axial) vectors
u and
v; the two products can be written as
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
s expressed in the same way:
-
using the notation
elm =
elem. These expressions show these two types of vector are Hodge duals:
as a result of the relations:
with
l, m, n cyclic,
and:
also with
l, m, n cyclic.
Using the implementation of ? based upon
i, the commonly used relations are:
Four dimensions
In case
n = 4 the Hodge dual acts as an
endomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
of the second exterior power, of dimension 6. It is an involution, so it splits it into
self-dual and
anti-self-dual subspaces, on which it acts respectively as +1 and −1.
Another useful example is
n=4 Minkowski spacetime with metric signature (+,-,-,-) and coordinates
(t,x,y,z) where (using
)
for
one-formIn linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s while
for
two-formIn linear algebra, a two-form is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skew-symmetric....
s.
Inner product of k-vectors
The Hodge dual induces an inner product on the space of
k-vectors, that is, on the
exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of
V. Given two
k-vectors
and
, one has
where
ω is the normalised
n-form (i.e.
ω ∧
*ω=
ω). In the calculus of exterior
differential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s on a
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
of dimension
n, the normalised n-form is called the
volume formIn mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
and can be written as
where
is the metric on the manifold.
It can be shown that
is an inner product, in that it is sesquilinear and defines a
normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
. Conversely, if an inner product is given on
, then this equation can be regarded as an alternative definition of the Hodge dual.
The wedge products of elements of an orthonormal basis in
V form an orthonormal basis of the exterior algebra of
V.
Duality
The Hodge star defines a dual in that when it is applied twice, the result is an identity on the exterior algebra, up to sign. Given a
k-vector
in an
n-dimensional space
V, one has
where
s is related to the
signatureThe signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
of the inner product on
V. Specifically,
s is the sign of the
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the inner product tensor. Thus, for example, if
n=4 and the signature of the inner product is either
(+,−,−,−) or (−,+,+,+) then
s=-1. For ordinary Euclidean spaces, the signature is always positive, and so
s=+1. When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.
Note that the above identity implies that the inverse of
can be given as
Note that if n is odd k(n-k) is even for any k whereas if n is even k(n-k) has the parity of k.
Therefore, if n is odd, it holds for any k
whereas, if n is even, it holds
where k is the degree of the forms operated on.
Hodge star on manifolds
One can repeat the construction above for each
cotangent spaceIn differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
of an
n-dimensional oriented
RiemannianIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
or pseudo-Riemannian manifold, and get the
Hodge dual n− k-form, of a
k-formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
. The Hodge star then induces an
L2-normIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
inner product on the differential forms on the manifold. One writes
for the inner product of sections
and
of
. (The set of sections is frequently denoted as
. Elements of
are called exterior k-forms).
More generally, in the non-oriented case, one can define the hodge star of a
k-form as a (
n− k)-
pseudo differential formIn physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation preserving coordinate transformation , but gains an additional sign flip under an orientation reversing coordinate transformation In physics and mathematics, a pseudotensor is usually...
; that is, a differential forms with values in the
canonical line bundleThe canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes...
.
The codifferential
The most important application of the Hodge dual on manifolds to is to define the
codifferential δ. Let
where
d is the
exterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
, or differential.
s=+1 for Riemannian manifolds.
while
The codifferential is not an antiderivation on the exterior algebra,
in contrast to the exterior derivative.
The codifferential is the adjoint of the exterior derivative, in that
where ζ is a (k+1)-form and η a k-form. This identity follows from Stokes' theorem for smooth forms, when
i.e. when M has empty boundary or when η or *ζ has zero boundary values (Of course, true adjointness follows after continuous continuation to the appropriate topological vector spaces as closures of the spaces of smooth forms).
Notice that since the differential satisfies
, the codifferential has the corresponding property
The
Laplace-deRhamIn differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami...
operator is given by
and lies at the heart of
Hodge theoryIn mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
. It is symmetric:
and non-negative:
The Hodge dual sends harmonic forms to harmonic forms. As a consequence of the
Hodge theorem, the
de Rham cohomologyIn mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
is naturally isomorphic to the space of harmonic
k-forms, and so the Hodge star induces an isomorphism of cohomology groups
which in turn gives canonical identifications via
Poincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
of
Hk(
M) with its
dual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
.
Derivatives in three dimensions
The combination of the
operator and the
exterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
generates the classical operators
gradIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
, curl, and
divIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
, in three dimensions. This works out as follows:
can take a 0-form (function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (applied to a 3-form it just gives zero).
For a 0-form,
, the first case written out in components is identifiable as the grad operator:
The second case followed by
is an operator on 1-forms (
) that in components is the
operator:
Applying the Hodge star gives:
The final case prefaced and followed by
, takes a 1-form (
) to a 0-form (function); written out in components it is the divergence operator:
One advantage of this expression is that the identity
, which is true in all cases, sums up two others, namely that
and
. In particular,
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star.
One can also obtain the Laplacian. Using the information above and the fact that
then for a 0-form,
:
