Differential Forms Cheatsheet

A coordinate-free reference for forms, Hodge stars, and the Laplacian.

This post collects coordinate-free formulas for differential forms that I find myself constantly referencing. While these identities are scattered throughout textbooks, having them gathered in one place with consistent notation has proven invaluable for computation. The focus is on the interplay between Riemannian geometry and differential forms, particularly for working with the gradient, divergence, and Laplacian.

The Musical Isomorphism: $TM \leftrightarrow T^*M$

On a Riemannian manifold $(M,g)$, the metric provides a canonical isomorphism between tangent and cotangent bundles. This fundamental correspondence deserves careful statement.

The terms “flat” and “sharp” come from musical notation (♭ and ♯), emphasizing this is an isomorphism that can be traversed in both directions. In coordinate-based literature you’ll also see “raising and lowering indices” from index notation.

Differentials and the Gradient

The exterior derivative connects smooth functions to their directional derivatives through 1-forms, while the metric provides a geometric interpretation as a vector field.

The Exterior Derivative

Key properties:

• Closed forms: $\alpha$ is closed if $d\alpha = 0$
• Exact forms: $\alpha$ is exact if $\alpha = d\beta$ for some $\beta$
• Every exact form is closed (by nilpotency)
• Poincaré Lemma: On contractible domains, every closed form is exact

The exterior derivative is natural with respect to smooth maps, commuting with pullbacks: $d(\phi^*\omega) = \phi^*(d\omega)$.

Pullbacks

Smooth maps between manifolds induce pullback operations on differential forms, providing a powerful computational tool.

Properties of Pullbacks

The pullback satisfies several crucial identities:

1. Linearity: $\phi^*(\alpha + \beta) = \phi^*\alpha + \phi^*\beta$

2. Preserves wedge products: $\phi^*(\alpha \wedge \beta) = \phi^*\alpha \wedge \phi^*\beta$

3. Commutes with exterior derivative: $d(\phi^*\omega) = \phi^*(d\omega)$

4. Functoriality: $(\psi \circ \phi)^* = \phi^* \circ \psi^*$

5. Identity: $(\operatorname{id}_M)^* = \operatorname{id}_{\Omega^k(M)}$

Pullback on 0-Forms (Functions)

For $f \in C^\infty(N)$, the pullback is simply composition: $\phi^*f = f \circ \phi$

Computing with Pullbacks

The commutation with $d$ is particularly useful for computation. For instance, if $f: N \to \RR$ is a function, then: $\phi^*(df) = d(\phi^*f) = d(f \circ \phi)$

This allows us to compute differentials in one space by pulling back from another.

Relationship to Vector Fields

While there’s a pullback for forms, there’s no canonical pullback for vector fields (but there is a pushforward). However, if $\phi$ is a diffeomorphism, we can define: $\phi_*X := ((\phi^{-1})^*(X^\flat))^\sharp$

The interaction between pullbacks and the musical isomorphism on a Riemannian manifold involves the metric: $(\phi^*\alpha)^\sharp = g^{-1}(\phi^*\alpha, \cdot)$

Working with k-Forms

Differentiating Contractions: Cartan’s Magic Formula

When working with expressions like $\alpha(X)$ where $\alpha$ is a 1-form and $X$ is a vector field, we often need to differentiate. The resulting differential can be expressed coordinate-freely using Cartan’s Magic Formula.

Applying this to $\alpha(X) = \iota_X\alpha$, we obtain:

$$\begin{align} d(\alpha(X)) = d(\iota_X\alpha) &= \mathcal{L}_X\alpha - \iota_X d\alpha\\ &= X(\alpha(\cdot)) - \alpha([X,\cdot]) - (d\alpha)(X,\cdot) \end{align}$$

This formula eliminates the need to compute coordinate expressions, instead expressing everything through the Lie bracket, exterior derivative, and evaluation.

Interior Products and Contraction

Note that some authors place $X$ last rather than first. The interior product satisfies $\iota_X \circ \iota_X \equiv 0$ (inserting the same vector twice into an alternating form must vanish).

For wedge products $\alpha \wedge \beta$ where $\alpha$ is a $p$-form, we have the derivation rule: $\iota_X(\alpha\wedge \beta) = (\iota_X\alpha)\wedge\beta + (-1)^p\alpha\wedge(\iota_X\beta)$

The Riemannian Volume Form

Pairing Vectors and $(n-1)$-Forms

Contracting the volume form establishes a natural pairing between vector fields and $(n-1)$-forms. Given a vector field $X$, the $(n-1)$-form $\iota_X\omega_g$ captures the oriented $(n-1)$-volume perpendicular to $X$. This perspective is fundamental to the classical relationship between vector fields and forms in vector calculus.

Extending the Metric to Forms

The Riemannian metric naturally extends to differential forms through the musical isomorphism and tensor products.

Metric on 1-Forms

Given 1-forms $\alpha, \beta$, we define: $g(\alpha,\beta) := g(\alpha^\sharp,\beta^\sharp)$

This gives us norms: $\|\alpha\|^2 := g(\alpha,\alpha) = g(\alpha^\sharp,\alpha^\sharp)$.

Metric on Tensor Powers

The metric $g$ naturally extends to tensor powers of the tangent bundle: elements of $\bigotimes^k TM$ are linear combinations of elementary tensor products $v_1\otimes v_2\otimes\cdots\otimes v_k$, and by bilinearity it suffices to define the extension on simple tensors:

$g(v_1\otimes\cdots\otimes v_k, w_1\otimes\cdots\otimes w_k):=g(v_1,w_1)g(v_2,w_2)\cdots g(v_k,w_k)$

Note: It’s often convenient to introduce a factor of $\frac{1}{k!}$ into this definition depending on conventions.

Metric on k-Forms

The extension to $k$-forms uses the tensor product structure. For wedge products $\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k, \quad \beta = \beta_1 \wedge \cdots \wedge \beta_k$

the metric is given by the Gram determinant:

$$g(\alpha,\beta) := \det\begin{pmatrix} g(\alpha_1,\beta_1) & \cdots & g(\alpha_1,\beta_k)\\ \vdots & \ddots & \vdots\\ g(\alpha_k,\beta_1) & \cdots & g(\alpha_k,\beta_k) \end{pmatrix}$$

where each entry uses the 1-form metric $g(\alpha_i,\beta_j) = g(\alpha_i^\sharp,\beta_j^\sharp)$.

This determinant formula arises naturally: $k$-forms are the alternating part of $k$-tensors, and alternation converts products into determinants.

The Lie Derivative

The Lie derivative measures how tensors change along the flow of a vector field, allowing the measuring apparatus itself to be transported by the flow.

Computing Lie Derivatives

For the basic objects:

Functions: $\mathcal{L}_X f = X(f)$ (directional derivative)

Vector fields: $\mathcal{L}_X Y = [X,Y]$ (Lie bracket)

1-forms: Using the Leibniz rule on $\alpha(Y)$:

$$\begin{align} \mathcal{L}_X(\alpha(Y)) &= (\mathcal{L}_X\alpha)(Y) + \alpha(\mathcal{L}_X Y)\\ X(\alpha(Y)) &= (\mathcal{L}_X\alpha)(Y) + \alpha([X,Y]) \end{align}$$

Therefore: $(\mathcal{L}_X\alpha)(Y) = X(\alpha(Y)) - \alpha([X,Y])$.

General Tensors

For a tensor $T$ taking vector fields $U, V$ and 1-form $\alpha$ as input, the Leibniz rule gives: $(\mathcal{L}_X T)(U,V,\alpha) = X(T(U,V,\alpha)) - T([X,U],V,\alpha) - T(U,[X,V],\alpha) - T(U,V,\mathcal{L}_X\alpha)$

Each argument contributes a term involving the Lie derivative or bracket of that argument.

Cartan’s Magic Formula

For differential forms, there’s a much more efficient formula relating the Lie derivative to the exterior derivative and interior product:

$\mathcal{L}_X\omega = d(\iota_X\omega) + \iota_X(d\omega)$

This is invaluable for computing Lie derivatives of $k$-forms, replacing the $(k+1)$-term Leibniz expansion with just two terms.

Commutator Relations

The various operators on differential forms satisfy important commutation relations that are essential for computation and understanding their algebraic structure.

Understanding the Relations

These commutator relations reveal deep structural properties:

• Relation (1) is the foundation of de Rham cohomology: closed forms ($d\omega = 0$) modulo exact forms ($\omega = d\eta$).

• Relation (2) reflects the antisymmetry of differential forms: contracting with two vectors in either order gives the same result.

• Relation (3) shows that the Lie derivative is a “Lie algebra action” of vector fields on forms.

• Relation (4) expresses that $d$ is natural: it doesn’t depend on any additional structure and commutes with all flows.

• Relation (5) shows how interior products transform under flows.

• Relation (6) is Cartan’s Magic Formula written as a commutator. The fact that the anti-commutator (not commutator) gives the Lie derivative is the key computational tool.

Computational Applications

These relations allow you to reorder operations. For example, to compute $\mathcal{L}_X(d\alpha)$: $\mathcal{L}_X(d\alpha) = d(\mathcal{L}_X\alpha)$

Or to express $\mathcal{L}_X(\iota_Y\omega)$: $\mathcal{L}_X(\iota_Y\omega) = \iota_Y(\mathcal{L}_X\omega) + \iota_{[X,Y]}\omega$

The Hodge Star

The Hodge star is an isometry between $k$-forms and $(n-k)$-forms. Key properties:

• $\star 1 = \omega_g$ and $\star\omega_g = 1$
• $\star \circ \star = (-1)^{k(n-k)} \operatorname{id}$ on $k$-forms
• $g(\eta,\zeta) = \star(\eta \wedge \star\zeta) = \star(\zeta \wedge \star\eta)$

Hodge Star on 1-Forms

For 1-forms $\alpha, \beta$: $\star g(\alpha,\beta) = g(\alpha,\beta)\omega_g = \alpha \wedge \star\beta = \beta \wedge \star\alpha$

This identity is particularly useful when $\alpha = df$ is the differential of a function.

Contracting the Volume Form

There’s a beautiful relationship between interior contraction of the volume form and the musical isomorphism:

Applying this to the volume form with $\star\omega_g = 1$: $\star(\iota_X\omega_g) = X^\flat \wedge 1 = X^\flat$

This shows that contracting the volume form and applying the Hodge star recovers the musical isomorphism: the two ways of converting a vector to a 1-form are identical.

Divergence

Using Cartan’s Magic Formula:

$$\begin{align} (\div X)\omega_g &= \mathcal{L}_X\omega_g\\ &= d(\iota_X\omega_g) + \iota_X(d\omega_g)\\ &= d(\iota_X\omega_g) \end{align}$$

The second term vanishes because $\omega_g$ is top-dimensional, so $d\omega_g = 0$.

Using the relationship between contraction and the Hodge star: $(\div X)\omega_g = d(\iota_X\omega_g) = d(\star\star\iota_X\omega_g) = d(\star X^\flat)$

Therefore: $\div X = \star d \star X^\flat$

This formula elegantly packages all metric dependence into the Hodge star operators.

The Laplacian

This coordinate-free formula reveals the Laplacian as a composition of metric-dependent operations (the two Hodge stars) with metric-independent ones (exterior derivatives).

The Laplacian equals the divergence of the gradient, which we can verify:

$$\begin{align} \Delta f &= \div(\grad f)\\ &= \star d \star (\grad f)^\flat\\ &= \star d \star df \end{align}$$

using $(\grad f)^\flat = df$ by definition of the gradient.

Connection to Vector Calculus

In three-dimensional Euclidean space $\RR^3$ with the standard metric, differential forms provide a coordinate-free formulation of classical vector calculus. Here’s the dictionary:

The Correspondence

Vector Calculus	Differential Forms	Dimension
Scalar field $f$	0-form $f$	$n=3$
Vector field $\mathbf{F}$	1-form $\mathbf{F}^\flat$ or 2-form $\star\mathbf{F}^\flat$	$n=3$
$\grad f$	$(df)^\sharp$	0→1
$\div \mathbf{F}$	$\star d \star \mathbf{F}^\flat$	1→0
$\curl \mathbf{F}$	$(\star d \mathbf{F}^\flat)^\sharp$	1→1
$\Delta f = \div \grad f$	$\star d \star df$	0→0

Detailed Formulas

Gradient: For a function $f: \RR^3 \to \RR$, $\grad f = (df)^\sharp$

In coordinates: if $df = f_x\,dx + f_y\,dy + f_z\,dz$, then $\grad f = f_x\partial_x + f_y\partial_y + f_z\partial_z$.

Divergence: For a vector field $\mathbf{F}$, $\div \mathbf{F} = \star d \star \mathbf{F}^\flat$

The form $\mathbf{F}^\flat$ is a 1-form, $\star\mathbf{F}^\flat$ is a 2-form, $d(\star\mathbf{F}^\flat)$ is a 3-form, and $\star$ converts it back to a function (0-form).

Curl: For a vector field $\mathbf{F}$ in $\RR^3$, $\curl \mathbf{F} = (\star d \mathbf{F}^\flat)^\sharp$

Here $\mathbf{F}^\flat$ is a 1-form, $d\mathbf{F}^\flat$ is a 2-form, $\star d\mathbf{F}^\flat$ is a 1-form, and raising the index gives back a vector field.

Laplacian: $\Delta f = \div(\grad f) = \star d \star df$

Classical Identities via Differential Forms

The fundamental identities of vector calculus follow immediately from $d^2 = 0$:

Generalization to Arbitrary Dimensions

While curl is specific to 3D (because only in 3D are 1-forms and 2-forms paired by the Hodge star), gradient and divergence make sense in any dimension:

In dimension $n$:

• $\grad f = (df)^\sharp$ takes 0-forms to vector fields
• $\div X = \star d \star X^\flat$ takes vector fields to 0-forms
• $\Delta f = \div \grad f = \star d \star df$ is the Laplacian on functions

The curl generalizes to the “vorticity 2-form” $d\mathbf{F}^\flat$ in any dimension, but it remains a 2-form rather than a vector field (except in 3D where the Hodge star provides the isomorphism).

Stokes’ Theorem

The fundamental theorem of calculus generalizes to manifolds:

In classical vector calculus, this subsumes:

• Fundamental Theorem of Calculus: $\int_a^b f'(x)\,dx = f(b) - f(a)$
• Green’s Theorem: $\oint_{\partial R} (L\,dx + M\,dy) = \iint_R \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx\,dy$
• Kelvin-Stokes Theorem: $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\curl \mathbf{F}) \cdot d\mathbf{S}$
• Divergence Theorem: $\oint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V \div \mathbf{F}\, dV$

Each is simply Stokes’ theorem applied to forms of different degrees in different dimensions.

Working in Coordinates

Coordinates provide concrete computational tools, though at the cost of manifest coordinate independence.

If $(u,v)$ are local coordinates with coordinate functions $u,v: M \to \RR$, they induce:

• Coordinate vector fields: $\partial_u, \partial_v$ defined by $\partial_u(f) := \frac{\partial(f \circ \Psi)}{\partial u} \circ \Phi$ where $\Phi$ is the coordinate map and $\Psi$ its inverse

• Coordinate 1-forms: $du, dv$ defined by $du(X) := X(u)$

These form dual bases satisfying: $du(\partial_u) = dv(\partial_v) = 1, \quad du(\partial_v) = dv(\partial_u) = 0$

To verify: $du(\partial_u) = \partial_u(u) = \frac{\partial(u \circ \Psi)}{\partial u} \circ \Phi$. Since $u \circ \Psi$ is just the projection $(u,v) \mapsto u$ in $\mathbb{R}^2$, its $u$-derivative is constantly $1$. Similarly, $dv(\partial_u) = 0$ since we’re differentiating the $v$-projection with respect to $u$.

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Note: For rigorous statements, “coordinates $(u,v)$” should be “a coordinate chart $\Phi: U \to \mathbb{R}^2$ on an open set $U \subset M$.” I’ve omitted these technicalities for readability.