Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by

$${\displaystyle {\begin{aligned}{\star }\,1&=dx\wedge dy\\{\star }\,dx&=dy\\{\star }\,dy&=-dx\\{\star }(dx\wedge dy)&=1.\end{aligned}}}$$

On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If z = x + iy is a holomorphic function of w = u + iv, then by the Cauchy–Riemann equations we have that ∂x/∂u = ∂y/∂v and ∂y/∂u = −∂x/∂v. In the new coordinates

$${\displaystyle \alpha \ =\ p\,dx+q\,dy\ =\ \left(p{\frac {\partial x}{\partial u}}+q{\frac {\partial y}{\partial u}}\right)\,du+\left(p{\frac {\partial x}{\partial v}}+q{\frac {\partial y}{\partial v}}\right)\,dv\ =\ p_{1}du+q_{1}\,dv,}$$

so that

$${\displaystyle {\begin{aligned}{\star }\alpha &=-q_{1}\,du+p_{1}\,dv\\[4pt]&=-\left(p{\frac {\partial x}{\partial v}}+q{\frac {\partial y}{\partial v}}\right)du+\left(p{\frac {\partial x}{\partial u}}+q{\frac {\partial y}{\partial u}}\right)dv\\[4pt]&=-q\left({\frac {\partial x}{\partial u}}du+{\frac {\partial x}{\partial v}}dv\right)+p\left({\frac {\partial y}{\partial u}}du+{\frac {\partial y}{\partial v}}dv\right)\\[4pt]&=-q\,dx+p\,dy,\end{aligned}}}$$

proving the claimed invariance.

Three dimensions

A common example of the Hodge star operator is the case n = 3, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R³ with the basis ${\displaystyle dx,dy,dz}$ of one-forms often used in vector calculus, one finds that

$${\displaystyle {\begin{aligned}{\star }\,dx&=dy\wedge dz\\{\star }\,dy&=dz\wedge dx\\{\star }\,dz&=dx\wedge dy.\end{aligned}}}$$

The Hodge star relates the exterior and cross product in three dimensions:^[2]

$${\displaystyle {\star }(\mathbf {u} \wedge \mathbf {v} )=\mathbf {u} \times \mathbf {v} \qquad {\star }(\mathbf {u} \times \mathbf {v} )=\mathbf {u} \wedge \mathbf {v} .}$$

Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector A and vice versa, that is:^[2] ${\displaystyle \mathbf {A} ={\star }\mathbf {a} ,\ \ \mathbf {a} ={\star }\mathbf {A} }$. The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. An inner product on a vector space ${\displaystyle V}$ gives an isomorphism ${\displaystyle V\cong V^{*}\!}$ identifying ${\displaystyle V}$ with its dual space, and the vector space ${\displaystyle L(V,V)}$ is naturally isomorphic to the tensor product ${\displaystyle V^{*}\!\!\otimes V\cong V\otimes V}$. Thus for ${\displaystyle V=\mathbb {R} ^{3}}$, the star mapping ${\textstyle \textstyle \star \colon V\to \bigwedge ^{\!2}\!V\subset V\otimes V}$ takes each vector ${\displaystyle \mathbf {v} }$ to a bivector ${\displaystyle \star \mathbf {v} \in V\otimes V}$, which corresponds to a linear operator ${\displaystyle L_{\mathbf {v} }\colon V\to V}$. Specifically, ${\displaystyle L_{\mathbf {v} }}$ is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis ${\displaystyle \mathbb {v} }$ are given by the matrix exponential ${\displaystyle \exp(tL_{\mathbf {v} })}$. With respect to the basis ${\displaystyle dx,dy,dz}$ of ${\displaystyle \mathbb {R} ^{3}}$, the tensor ${\displaystyle dx\otimes dy}$ corresponds to a coordinate matrix with 1 in the ${\displaystyle dx}$ row and ${\displaystyle dy}$ column, etc., and the wedge ${\displaystyle dx\wedge dy\,=\,dx\otimes dy-dy\otimes dx}$ is the skew-symmetric matrix ${\displaystyle \scriptscriptstyle \left[{\begin{array}{rrr}\,0\!\!&\!\!1&\!\!\!\!0\!\!\!\!\!\!\\[-.5em]\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!\\[-.5em]\,0\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!\end{array}}\!\!\!\right]}$, etc. That is, we may interpret the star operator as:

$${\displaystyle \mathbf {v} =a\,dx+b\,dy+c\,dz\quad \longrightarrow \quad \star {\mathbf {v} }\ \cong \ L_{\mathbf {v} }\ =\left[{\begin{array}{rrr}0&-c&b\\c&0&-a\\-b&a&0\end{array}}\right].}$$

Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: ${\displaystyle L_{\mathbf {u} \times \mathbf {v} }=L_{\mathbf {u} }L_{\mathbf {v} }-L_{\mathbf {v} }L_{\mathbf {u} }}$.

Four dimensions

In case ${\displaystyle n=4}$, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since 4 − 2 = 2). If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see § Duality below. This particular endomorphism property of 2-forms in four dimensions makes self-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues ${\displaystyle \pm 1}$ (or ${\displaystyle \pm i}$, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where ${\displaystyle n=4}$ with metric signature (− + + +) and coordinates ${\displaystyle (t,x,y,z)}$. The volume form is oriented as ${\displaystyle \varepsilon _{0123}=1}$. For one-forms,

$${\displaystyle {\begin{aligned}\star dt&=-dx\wedge dy\wedge dz\,,\\\star dx&=-dt\wedge dy\wedge dz\,,\\\star dy&=dt\wedge dx\wedge dz\,,\\\star dz&=-dt\wedge dx\wedge dy\,,\end{aligned}}}$$

while for 2-forms,

$${\displaystyle {\begin{aligned}\star (dt\wedge dx)&=-dy\wedge dz\,,\\\star (dt\wedge dy)&=-dz\wedge dx\,,\\\star (dt\wedge dz)&=-dx\wedge dy\,,\\\star (dx\wedge dy)&=dt\wedge dz\,,\\\star (dx\wedge dz)&=-dt\wedge dy\,,\\\star (dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}}$$

These are summarized in the index notation as

$${\displaystyle {\begin{aligned}\star (dx^{\mu })&=\eta ^{\mu \lambda }\varepsilon _{\lambda \nu \rho \sigma }{\frac {1}{3!}}dx^{\nu }\wedge dx^{\rho }\wedge dx^{\sigma }\,,\\\star (dx^{\mu }\wedge dx^{\nu })&=\eta ^{\mu \kappa }\eta ^{\nu \lambda }\varepsilon _{\kappa \lambda \rho \sigma }{\frac {1}{2!}}dx^{\rho }\wedge dx^{\sigma }\,.\end{aligned}}}$$

Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, ${\displaystyle (\star )^{2}=1}$ for odd-rank forms and ${\displaystyle (\star )^{2}=-1}$ for even-rank forms. An easy rule to remember for these Hodge operations is that given a form ${\displaystyle \alpha }$, its Hodge dual ${\displaystyle {\star }\alpha }$ may be obtained by writing the components not involved in ${\displaystyle \alpha }$ in an order such that ${\displaystyle \alpha \wedge (\star \alpha )=dt\wedge dx\wedge dy\wedge dz}$. An extra minus sign will enter only if ${\displaystyle \alpha }$ contains ${\displaystyle dt}$. (For (+ − − −), one puts in a minus sign only if ${\displaystyle \alpha }$ involves an odd number of the space-associated forms ${\displaystyle dx}$, ${\displaystyle dy}$ and ${\displaystyle dz}$.)

Note that the combinations

$${\displaystyle (dx^{\mu }\wedge dx^{\nu })^{\pm }:={\frac {1}{2}}{\big (}dx^{\mu }\wedge dx^{\nu }\mp i\star (dx^{\mu }\wedge dx^{\nu }){\big )}}$$

take ${\displaystyle \pm i}$ as the eigenvalue for Hodge star operator, i.e.,

$${\displaystyle \star (dx^{\mu }\wedge dx^{\nu })^{\pm }=\pm i(dx^{\mu }\wedge dx^{\nu })^{\pm },}$$

and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

Conformal invariance

The Hodge star is conformally invariant on n forms on a 2n dimensional vector space V, i.e. if ${\displaystyle g}$ is a metric on ${\displaystyle V}$ and ${\displaystyle \lambda >0}$, then the induced Hodge stars

$${\displaystyle \star _{g},\star _{\lambda g}\colon \Lambda ^{n}V\to \Lambda ^{n}V}$$

are the same.

Example: Derivatives in three dimensions

The combination of the ${\displaystyle \star }$ operator and the exterior derivative d generates the classical operators grad, curl, and div on vector fields in three-dimensional Euclidean space. This works out as follows: d takes a 0-form (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). For a 0-form ${\displaystyle f=f(x,y,z)}$, the first case written out in components gives:

$${\displaystyle df={\frac {\partial f}{\partial x}}\,dx+{\frac {\partial f}{\partial y}}\,dy+{\frac {\partial f}{\partial z}}\,dz.}$$

The inner product identifies 1-forms with vector fields as ${\displaystyle dx\mapsto (1,0,0)}$, etc., so that ${\displaystyle df}$ becomes ${\textstyle \operatorname {grad} f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\right)}$.

In the second case, a vector field ${\displaystyle \mathbf {F} =(A,B,C)}$ corresponds to the 1-form ${\displaystyle \varphi =A\,dx+B\,dy+C\,dz}$, which has exterior derivative:

$${\displaystyle d\varphi =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}}\right)dy\wedge dz+\left({\frac {\partial C}{\partial x}}-{\frac {\partial A}{\partial z}}\right)dx\wedge dz+\left({\partial B \over \partial x}-{\frac {\partial A}{\partial y}}\right)dx\wedge dy.}$$

Applying the Hodge star gives the 1-form:

$${\displaystyle \star d\varphi =\left({\partial C \over \partial y}-{\partial B \over \partial z}\right)\,dx-\left({\partial C \over \partial x}-{\partial A \over \partial z}\right)\,dy+\left({\partial B \over \partial x}-{\partial A \over \partial y}\right)\,dz,}$$

which becomes the vector field ${\textstyle \operatorname {curl} \mathbf {F} =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}},\,-{\frac {\partial C}{\partial x}}+{\frac {\partial A}{\partial z}},\,{\frac {\partial B}{\partial x}}-{\frac {\partial A}{\partial y}}\right)}$.

In the third case, ${\displaystyle \mathbf {F} =(A,B,C)}$ again corresponds to ${\displaystyle \varphi =A\,dx+B\,dy+C\,dz}$. Applying Hodge star, exterior derivative, and Hodge star again:

$${\displaystyle {\begin{aligned}\star \varphi &=A\,dy\wedge dz-B\,dx\wedge dz+C\,dx\wedge dy,\\d{\star \varphi }&=\left({\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}\right)dx\wedge dy\wedge dz,\\\star d{\star \varphi }&={\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}=\operatorname {div} \mathbf {F} .\end{aligned}}}$$

One advantage of this expression is that the identity d² = 0, which is true in all cases, has as special cases two other identities: 1) curl grad f = 0, and 2) div curl F = 0. In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression ${\displaystyle \star d\star }$ (multiplied by an appropriate power of -1) is called the codifferential; it is defined in full generality, for any dimension, further in the article below.

One can also obtain the Laplacian Δf = div grad f in terms of the above operations:

$${\displaystyle \Delta f=\star d{\star df}={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}$$

The Laplacian can also be seen as a special case of the more general Laplace–deRham operator ${\displaystyle \Delta =d\delta +\delta d}$ where ${\displaystyle \delta =(-1)^{k}\star d\star }$ is the codifferential for ${\displaystyle k}$-forms. Any function ${\displaystyle f}$ is a 0-form, and ${\displaystyle \delta f=0}$ and so this reduces to the ordinary Laplacian. For the 1-form ${\displaystyle \varphi }$ above, the codifferential is ${\displaystyle \delta =-\star d\star }$ and after some straightforward calculations one obtains the Laplacian acting on ${\displaystyle \varphi }$.

Computation in index notation

We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis ${\textstyle \left\{{\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right\}}$ in a tangent space ${\displaystyle V=T_{p}M}$ and its dual basis ${\displaystyle \{dx_{1},\ldots ,dx_{n}\}}$ in ${\displaystyle V^{*}=T_{p}^{*}M}$, having the metric matrix ${\textstyle (g_{ij})=\left(\left\langle {\frac {\partial }{\partial x_{i}}},{\frac {\partial }{\partial x_{j}}}\right\rangle \right)}$ and its inverse matrix ${\displaystyle (g^{ij})=(\langle dx^{i},dx^{j}\rangle )}$. The Hodge dual of a decomposable k-form is:

$${\displaystyle \star \left(dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}\right)\ =\ {\frac {\sqrt {\left|\det[g_{ij}]\right|}}{(n-k)!}}g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}\varepsilon _{j_{1}\dots j_{n}}dx^{j_{k+1}}\wedge \dots \wedge dx^{j_{n}}.}$$

Here ${\displaystyle \varepsilon _{j_{1}\dots j_{n}}}$ is the Levi-Civita symbol with ${\displaystyle \varepsilon _{1\dots n}=1}$, and we implicitly take the sum over all values of the repeated indices ${\displaystyle j_{1},\ldots ,j_{n}}$. The factorial ${\displaystyle (n-k)!}$ accounts for double counting, and is not present if the summation indices are restricted so that ${\displaystyle j_{k+1}<\dots <j_{n}}$. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds.

An arbitrary differential form can be written as follows:

$${\displaystyle \alpha \ =\ {\frac {1}{k!}}\alpha _{i_{1},\dots ,i_{k}}dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}\ =\ \sum _{i_{1}<\dots <i_{k}}\alpha _{i_{1},\dots ,i_{k}}dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}.}$$

The factorial ${\displaystyle k!}$ is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component ${\displaystyle \alpha _{i_{1},\dots ,i_{k}}}$ so that the Hodge dual of the form is given by

$${\displaystyle \star \alpha ={\frac {1}{(n-k)!}}(\star \alpha )_{i_{k+1},\dots ,i_{n}}dx^{i_{k+1}}\wedge \dots \wedge dx^{i_{n}}.}$$

Using the above expression for the Hodge dual of ${\displaystyle dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}}$, we find:^[3]

$${\displaystyle (\star \alpha )_{i_{k+1},\dots ,i_{n}}={\frac {\sqrt {\left|\det[g_{ab}]\right|}}{k!}}\alpha ^{i_{1},\dots ,i_{k}}\,\,\varepsilon _{i_{1},\dots ,i_{n}}.}$$

Although one can apply this expression to any tensor ${\displaystyle \alpha }$, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor. It is thus equivalent to antisymmetrization followed by applying the Hodge star.

The unit volume form ${\textstyle \omega =\star 1\in \bigwedge ^{n}V^{*}}$ is given by:

$${\displaystyle \omega ={\sqrt {\left|\det[g_{ij}]\right|}}\;dx^{1}\wedge \cdots \wedge dx^{n}.}$$

Codifferential

The most important application of the Hodge star on manifolds is to define the codifferential ${\displaystyle \delta }$ on k-forms. Let

$${\displaystyle \delta =(-1)^{n(k-1)+1}s\ {\star }d{\star }=(-1)^{k}\,{\star }^{-1}d{\star }}$$

where ${\displaystyle d}$ is the exterior derivative or differential, and ${\displaystyle s=1}$ for Riemannian manifolds. Then

$${\displaystyle d:\Omega ^{k}(M)\to \Omega ^{k+1}(M)}$$

while

$${\displaystyle \delta$$ :\Omega ^{k}(M)\to \Omega ^{k-1}(M).}

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 with respect to the square-integrable inner product:

$${\displaystyle \langle \!\langle \eta ,\delta \zeta \rangle \!\rangle \ =\ \langle \!\langle d\eta ,\zeta \rangle \!\rangle ,}$$

where ${\displaystyle \zeta }$ is a (k + 1)-form and ${\displaystyle \eta }$ a k-form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved from Stokes' theorem for smooth forms:

$${\displaystyle 0\ =\ \int _{M}d(\eta \wedge {\star }\zeta )\ =\ \int _{M}\left(d\eta \wedge {\star }\zeta -\eta \wedge {\star }(-1)^{k+1}\,{\star }^{-1}d{\star }\zeta \right)\ =\ \langle \!\langle d\eta ,\zeta \rangle \!\rangle -\langle \!\langle \eta ,\delta \zeta \rangle \!\rangle ,}$$

provided M has empty boundary, or ${\displaystyle \eta }$ or ${\displaystyle \star \zeta }$ has zero boundary values. (The proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms. The Sobolev space is conventionally used; it allows the convergent sequence of forms ${\displaystyle \zeta _{i}\to \zeta }$ (as ${\displaystyle i\to \infty }$) to be interchanged with the combined differential and integral operations, so that ${\displaystyle \langle \!\langle \eta ,\delta \zeta _{i}\rangle \!\rangle \to \langle \!\langle \eta ,\delta \zeta \rangle \!\rangle }$ and likewise for sequences converging to ${\displaystyle \eta }$.)

Since the differential satisfies ${\displaystyle d^{2}=0}$, the codifferential has the corresponding property

$${\displaystyle \delta ^{2}=s^{2}{\star }d{\star }{\star }d{\star }=(-1)^{k(n-k)}s^{3}{\star }d^{2}{\star }=0.}$$

The Laplace–deRham operator is given by

$${\displaystyle \Delta =(\delta +d)^{2}=\delta d+d\delta }$$

and lies at the heart of Hodge theory. It is symmetric:

$${\displaystyle \langle \!\langle \Delta \zeta ,\eta \rangle \!\rangle =\langle \!\langle \zeta ,\Delta \eta \rangle \!\rangle }$$

and non-negative:

$${\displaystyle \langle \!\langle \Delta \eta ,\eta \rangle \!\rangle \geq 0.}$$

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups

$${\displaystyle {\star }:H_{\Delta }^{k}(M)\to H_{\Delta }^{n-k}(M),}$$

which in turn gives canonical identifications via Poincaré duality of H^k(M) with its dual space.

In coordinates, with notation as above, the codifferential of the form ${\displaystyle \alpha }$ may be written as

$${\displaystyle \delta \alpha =\ -{\frac {1}{k!}}g^{ml}\left({\frac {\partial }{\partial x_{l}}}\alpha _{m,i_{1},\dots ,i_{k-1}}-\Gamma _{ml}^{j}\alpha _{j,i_{1},\dots ,i_{k-1}}\right)dx^{i_{1}}\wedge \dots \wedge dx^{i_{k-1}},}$$

where here ${\displaystyle \Gamma _{ml}^{j}}$ denotes the Christoffel symbols of ${\textstyle \left\{{\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right\}}$.