Differential Forms
The alternating tensors built in the preceding development live on a single vector space.
To do calculus we let them vary from point to point across a manifold, exactly as covectors became covector fields.
The result is a differential form: a smoothly varying choice of alternating tensor on each tangent space.
These are the objects that integrate over curved spaces and whose derivative encodes the classical operators of vector calculus.
Recall that a
covector field
assigns to each point of a manifold a linear functional on the tangent space there. A differential
form generalizes this by replacing the single covector with an alternating tensor of arbitrary degree.
The extreme degrees recover familiar objects. A \(0\)-form is a smooth real-valued function, since
\(\Lambda^0(T^*_p M) = \mathbb{R}\); thus \(\Omega^0(M) = C^\infty(M)\). A \(1\)-form is exactly a covector field,
so \(\Omega^1(M)\) is the space of smooth covector fields already studied. For \(k \geq 2\),
differential forms are the genuinely new objects, and they vanish identically once \(k\) exceeds the dimension of \(M\),
because the fiber \(\Lambda^k(T^*_p M)\) is then the zero space.
In any smooth chart with coordinates \((x^1, \dots, x^n)\), the coordinate differentials \(dx^1, \dots, dx^n\) form a coframe
— a basis of \(T^*_p M\) at each point of the chart. The
elementary alternating tensors
built from this coframe, indexed by increasing multi-indices \(I = (i_1, \dots, i_k)\), give a pointwise basis for \(\Lambda^k(T^*_p M)\):
\[
dx^I = dx^{i_1} \wedge \cdots \wedge dx^{i_k}.
\]
Every \(k\)-form therefore has a unique local expression
\[
\omega = {\sum_I}'\, \omega_I\, dx^I,
\]
the primed sum running over increasing multi-indices of length \(k\), with smooth coefficient functions
\(\omega_I = \omega\bigl(\tfrac{\partial}{\partial x^{i_1}}, \dots, \tfrac{\partial}{\partial x^{i_k}}\bigr)\).
The form is smooth precisely when all coefficient functions \(\omega_I\) are smooth.
The wedge product extends to differential forms by performing it pointwise: \((\omega \wedge \eta)_p = \omega_p \wedge \eta_p\). All the
algebraic properties of the wedge product
— bilinearity, associativity, and the anticommutativity \(\omega \wedge \eta = (-1)^{kl}\eta \wedge \omega\)
— hold verbatim for differential forms, since they hold at every point. The forms of all degrees on \(M\)
thus assemble into an algebra under the wedge product, the smoothly varying counterpart of the exterior algebra of a single tangent space.
Example: Forms in Low Degree on \(\mathbb{R}^3\)
A \(0\)-form is just a smooth real-valued function, and a \(1\)-form is a covector field. On \(\mathbb{R}^3\)
with coordinates \((x, y, z)\), some smooth \(2\)-forms are
\[
\begin{align*}
\omega &= (\sin xy)\, dy \wedge dz, \\\\
\eta &= dx \wedge dy + dx \wedge dz + dy \wedge dz,
\end{align*}
\]
and every \(3\)-form on \(\mathbb{R}^3\) is a smooth function times the single top-degree basis element \(dx \wedge dy \wedge dz\).
The pattern reflects the dimension count from the algebra of alternating tensors: on a \(3\)-dimensional space there are
three independent \(2\)-forms and exactly one independent \(3\)-form.
Pullbacks of Differential Forms
A smooth map between manifolds transports differential forms backward, from the target to the source.
This pullback operation is the mechanism through which forms change coordinates, restrict to submanifolds,
and ultimately acquire the change-of-variables factor in integration. Because a differential
form is a covariant tensor field that happens to be alternating, its pullback is a special case of the pullback already defined
for tensor fields.
Proof:
That \(F^*\omega\) is alternating, smooth, and depends \(\mathbb{R}\)-linearly on \(\omega\) follows
from the corresponding facts for the pullback of covariant tensor fields, since inserting the same vector
into two slots of \(F^*\omega\) inserts the same image vector into two slots of \(\omega\), which vanishes.
This gives Part (1).
Part (2).
The wedge product is built from the tensor product by alternation, and the pullback commutes
with the tensor product of covariant tensors, \(F^*(\alpha \otimes \beta) = (F^*\alpha) \otimes (F^*\beta)\),
because \(dF_p\) is applied slot by slot in both factors. Alternation is defined by averaging over permutations
with signs, an operation that commutes with \(F^*\) since \(F^*\) acts identically on each permuted term.
The normalizing coefficient \((k+l)!/(k!\,l!)\) is the same on both sides.
Combining these, \(F^*\) carries
\[
\omega \wedge \eta = \tfrac{(k+l)!}{k!\,l!}\operatorname{Alt}(\omega \otimes \eta)
\]
to
\[
(F^*\omega) \wedge (F^*\eta).
\]
Part (3).
For a \(0\)-form, that is a function \(u\), the pullback is composition: \(F^*u = u \circ F\).
For a coordinate differential \(dy^j\), the pullback is the differential of the composite,
\(F^*(dy^j) = d(y^j \circ F)\). To see this, evaluate both sides on a coordinate vector \(\partial/\partial x^i\).
Writing \(F^k = y^k \circ F\) for the component functions, the differential sends
\[
dF_p(\partial/\partial x^i) = \sum_k (\partial F^k/\partial x^i)\, \partial/\partial y^k,
\]
so
\[
\begin{align*}
\bigl(F^*(dy^j)\bigr)\!\left(\frac{\partial}{\partial x^i}\right)
&= dy^j\!\left(dF_p\!\left(\frac{\partial}{\partial x^i}\right)\right) \\\\
&= \frac{\partial F^j}{\partial x^i} = \frac{\partial (y^j \circ F)}{\partial x^i} \\\\
&= \bigl(d(y^j \circ F)\bigr)\!\left(\frac{\partial}{\partial x^i}\right).
\end{align*}
\]
Both sides agree on every coordinate vector, hence are equal; this is exactly the chain rule.
Applying Part (2) to a wedge of coordinate differentials and linearity from Part (1) to the sum yields
\[
F^*\Bigl({\sum_J}'\, \omega_J\, dy^J\Bigr) = {\sum_J}'\, (\omega_J \circ F)\, d(y^{j_1} \circ F) \wedge \cdots \wedge d(y^{j_k} \circ F),
\]
the claimed coordinate formula.
The third property is what makes pullbacks computable: one substitutes the component functions of \(F\) for the target coordinates
and replaces each \(dy^j\) by the differential of the corresponding component.
Example: Computing a Pullback
Let \(F : \mathbb{R}^2 \to \mathbb{R}^3\) be \(F(u, v) = (u,\, v,\, u^2 - v^2)\),
and let \(\omega = y\, dx \wedge dz + x\, dy \wedge dz\) be a \(2\)-form on \(\mathbb{R}^3\).
Writing the target coordinates as \((x, y, z)\), the chart formula replaces each coefficient
by its composition with \(F\) and each differential by the differential of the corresponding component:
\[
dx = du, \qquad dy = dv, \qquad dz = d(u^2 - v^2) = 2u\, du - 2v\, dv.
\]
Substituting \(dx = du\), \(dy = dv\), and \(dz = 2u\, du - 2v\, dv\) into \(\omega\)
gives
\[
F^*\omega = v\, du \wedge (2u\, du - 2v\, dv) + u\, dv \wedge (2u\, du - 2v\, dv).
\]
We expand each wedge by distributing over the sum and pulling the scalar coefficients out front:
\[
\begin{align*}
v\, du \wedge (2u\, du - 2v\, dv) &= 2uv\, (du \wedge du) - 2v^2\, (du \wedge dv),\\\\
u\, dv \wedge (2u\, du - 2v\, dv) &= 2u^2\, (dv \wedge du) - 2uv\, (dv \wedge dv).
\end{align*}
\]
The terms \(du \wedge du\) and \(dv \wedge dv\) vanish, because the wedge of a \(1\)-form with itself is zero.
In the surviving term \(dv \wedge du\), anticommutativity of \(1\)-forms gives \(dv \wedge du = -\,du \wedge dv\).
Collecting the two nonzero contributions over the common basis element \(du \wedge dv\),
\[
\begin{align*}
F^*\omega
&= -2v^2\, du \wedge dv - 2u^2\, du \wedge dv \\\\
&= -2\bigl(u^2 + v^2\bigr)\, du \wedge dv.
\end{align*}
\]
The same technique computes the expression for a form in any second chart, by reading the change
of coordinates as the identity map written with different coordinates on its domain and codomain.
The next section pushes this computation to its sharpest form, where the source and target have the same dimension
and the wedge of all the differentials collapses into a single determinant.
The Pullback Formula for Top-Degree Forms
When a smooth map relates two manifolds of the same dimension and we pull back a form of top degree,
the chart formula simplifies sharply: the entire wedge of pulled-back differentials reduces to multiplication
by a single scalar, the Jacobian determinant of the map. This is the precise sense in which the determinant
of the earlier algebra reappears as the change-of-variables factor, and it is the result on which the integration of forms
— and the construction of invariant integrals on groups — ultimately rests.
Proof:
By the coordinate formula for pullbacks and the fact that \(F^*u = u \circ F\),
\[
F^*\bigl(u\, dy^1 \wedge \cdots \wedge dy^n\bigr) = (u \circ F)\; d F^1 \wedge \cdots \wedge dF^n,
\]
where \(dF^j = d(y^j \circ F) = \sum_i \dfrac{\partial F^j}{\partial x^i}\, dx^i\) is the differential
of the \(j\)th component. It remains to evaluate the wedge \(dF^1 \wedge \cdots \wedge dF^n\).
Each \(dF^j\) is a covector with components \(\partial F^j/\partial x^i\) in the coframe \((dx^i)\);
equivalently, the \(n\) covectors \(dF^1, \dots, dF^n\) are obtained from the basis covectors \(dx^1, \dots, dx^n\)
by the linear map whose matrix is \(DF\). Now \(dx^1 \wedge \cdots \wedge dx^n\) is an alternating \(n\)-tensor
on an \(n\)-dimensional space, the top degree, where the space of such tensors is one-dimensional. On that one-dimensional space,
applying a linear map to all \(n\) arguments simply scales the tensor by the
determinant of the map.
Feeding the \(dF^j\), which are the \(dx^i\) transformed by \(DF\), into the top-degree wedge
therefore reproduces \(dx^1 \wedge \cdots \wedge dx^n\) scaled by \(\det DF\):
\[
dF^1 \wedge \cdots \wedge dF^n = (\det DF)\; dx^1 \wedge \cdots \wedge dx^n.
\]
Substituting into the previous display yields the stated formula.
The factor \(\det DF\) is exactly the Jacobian determinant that governs the change of variables in multiple integrals.
Pulling back a top-degree form and integrating reproduces the classical substitution rule with the Jacobian appearing
automatically — no separate correction term is needed, because the determinant is already built into how top-degree forms transform.
The following example shows this mechanism in a standard change of coordinates.
Example: Polar Coordinates
Let \(F(r, \theta) = (r\cos\theta,\, r\sin\theta)\) be the polar-coordinate map,
with \(x = r\cos\theta\) and \(y = r\sin\theta\). Its differentials are
\[
dx = \cos\theta\, dr - r\sin\theta\, d\theta, \qquad dy = \sin\theta\, dr + r\cos\theta\, d\theta.
\]
Wedging the two and distributing, the terms \(dr \wedge dr\) and \(d\theta \wedge d\theta\) vanish,
leaving only the mixed products. Using \(d\theta \wedge dr = -\,dr \wedge d\theta\) to bring both
onto the basis element \(dr \wedge d\theta\),
\[
\begin{align*}
dx \wedge dy &= \cos\theta\,(r\cos\theta)\, dr \wedge d\theta + (-r\sin\theta)(\sin\theta)\, d\theta \wedge dr\\\\
&= \bigl(r\cos^2\theta + r\sin^2\theta\bigr)\, dr \wedge d\theta\\\\
&= r\, dr \wedge d\theta.
\end{align*}
\]
The coefficient \(r\) is the Jacobian determinant of the polar map, recovering the factor that appears
when a double integral is rewritten in polar coordinates. The wedge product produces it without any separate computation.
The same reasoning applies in any dimension and for any change of coordinates.
A particularly important instance arises when the map is the identity, expressed in two different coordinate systems
on an overlap.
Proof:
Apply the top-degree pullback formula to the identity map, written with coordinates \((x^i)\)
on its domain and \((\tilde{x}^j)\) on its codomain. Although the map is the identity on points,
its coordinate expression is the genuine coordinate-change function \(\tilde{x}^j = \tilde{x}^j(x^1, \dots, x^n)\),
so its Jacobian is not the identity matrix but the matrix \((\partial \tilde{x}^j / \partial x^i)\) of partial derivatives
of the new coordinates with respect to the old. The function \(u\) is \(1\), its composition with the identity is again \(1\),
and substituting this Jacobian into the pullback formula gives the stated identity.
This is the precise sense in which a top-degree form carries an intrinsic transformation law: its single coefficient
is multiplied by the Jacobian determinant of the coordinate change. That determinant — rather than its absolute value — is
what distinguishes a top-degree form from a density, and it is the algebraic seed of orientation. This identity is the bridge
between the linear-algebraic determinant and the analytic change-of-variables formula, and it is the form
in which top-degree forms feed into integration over manifolds and the averaging constructions used to build invariant structures
on Lie groups.