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 we integrate over curved spaces, and their
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
smooth 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 \((x^1, \dots, x^n)\), where \(n = \dim M\), 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 coefficient functions
\(\omega_I = \omega\bigl(\tfrac{\partial}{\partial x^{i_1}}, \dots, \tfrac{\partial}{\partial x^{i_k}}\bigr)\).
The proposition below shows that, for \(1 \le k \le n\), a section \(\omega\)
of \(\Lambda^k T^*M\) is smooth precisely when the functions \(\omega_I\) given by the same formula
are smooth in every chart.
The definition of a differential form presupposes that \(\Lambda^k T^*M\) carries the structure of
a smooth vector bundle, so that smoothness of a section has a meaning. For \(1 \le k \le n\) that
structure comes from the bundle \(T^k T^*M\) of covariant \(k\)-tensors, a
smooth vector bundle
which contains \(\Lambda^k T^*M\) fiber by fiber.
Proposition: The Bundle of Alternating Tensors Is a Smooth Vector Bundle
Let \(M\) be a smooth \(n\)-manifold with or without boundary and let \(1 \le k \le n\). Then
\(\Lambda^k T^*M\) is a smooth subbundle of \(T^k T^*M\) of rank \(\binom{n}{k}\), with a
smooth structure in which, for every smooth chart \((x^i)\), the sections \(dx^I\) with \(I\)
increasing of length \(k\) form a smooth local frame over the chart domain. A section
\(\omega\) of \(\Lambda^k T^*M\) is smooth if and only if its coefficient functions
\(\omega_I\) are smooth in every smooth chart, and if and only if it is smooth as a section of
\(T^k T^*M\).
Proof:
Fix a smooth chart \((U, (x^i))\) and write \(\partial_i\) for \(\partial/\partial x^i\). By
the determinant formula for wedges of covectors,
for an increasing multi-index \(I\) of length \(k\) and any indices \(j_1, \dots, j_k\), the
tensor component
\(dx^I(\partial_{j_1}, \dots, \partial_{j_k}) = \det\bigl(dx^{i_b}(\partial_{j_a})\bigr)\) is
the determinant of a matrix of zeros and ones, and for increasing \(J = (j_1, \dots, j_k)\) it
equals \(1\) when \(J = I\) and \(0\) otherwise. The tensor components of \(dx^I\) are thus
constant on \(U\), so the smoothness criteria for tensor fields,
applied on the open submanifold \(U\), show that each \(dx^I\) is a smooth section of
\(T^k T^*M\) over \(U\). For \(p \in U\), the basis theorem for alternating tensors
shows that the alternating tensors \(dx^I|_p\), \(I\) increasing of length \(k\), form a basis
of \(\Lambda^k(T^*_p M)\). The local frame criterion for subbundles
therefore makes \(\Lambda^k T^*M\) a smooth subbundle of \(T^k T^*M\) of rank \(\binom{n}{k}\),
with a smooth structure in which each coordinate family \((dx^I)\) is a smooth local frame over
its chart domain. If \(\omega\) is a section of \(\Lambda^k T^*M\) and
\(\omega_p = {\sum_I}'\, c_I\, dx^I|_p\), evaluating both sides on
\((\partial_{j_1}|_p, \dots, \partial_{j_k}|_p)\) with \(J\) increasing gives
\(c_J = \omega_p(\partial_{j_1}|_p, \dots, \partial_{j_k}|_p)\), so the components of
\(\omega\) in this frame are the coefficient functions \(\omega_I\). The
local frame criterion for smoothness
then shows that \(\omega\) is smooth on \(U\) exactly when its coefficients \(\omega_I\) are
smooth there.
For the comparison with \(T^k T^*M\), let \(\omega\) be a section of \(\Lambda^k T^*M\). Its
tensor components \(\omega(\partial_{j_1}, \dots, \partial_{j_k})\) on \(U\) vanish when two
indices coincide, and otherwise equal \(\pm\,\omega_I\), where \(I\) is the increasing
rearrangement of \((j_1, \dots, j_k)\) and the sign is that of the rearranging permutation,
because \(\omega\) is alternating at each point. The tensor components are therefore smooth
exactly when the coefficients \(\omega_I\) are, and the smoothness criteria for tensor fields
complete the proof.
The degrees outside the proposition need no construction. For \(k = 0\) the bundle is the product
\(M \times \mathbb{R}\), so \(0\)-forms are smooth functions, and for \(k \gt n\) every fiber is
the zero space, so \(\Lambda^k T^*M\) is the zero bundle \(M \times \{0\}\). Degree \(1\), which
the proposition covers, adds nothing new: every covector is alternating, so \(\Lambda^1 T^*M\) is
all of \(T^1 T^*M = T^*M\). In every degree a differential form is therefore, equivalently, a
smooth covariant tensor field whose value at each point is alternating (for \(k = 0\), a smooth
function), which is the reading used for pullbacks below.
The wedge product extends to differential forms by performing it pointwise: for \(\omega \in \Omega^k(M)\) and
\(\eta \in \Omega^l(M)\), \((\omega \wedge \eta)_p = \omega_p \wedge \eta_p\). The result is again smooth. In a chart,
bilinearity gives \(\omega \wedge \eta = {\sum_I}' {\sum_J}'\, \omega_I\, \eta_J\, dx^I \wedge dx^J\), and each
\(dx^I \wedge dx^J\) is either zero or \(\pm dx^K\) for an increasing multi-index \(K\), so the coefficients of
\(\omega \wedge \eta\) are sums of products of smooth functions, and the proposition above shows that \(\omega \wedge \eta\)
is smooth (for \(k + l \gt n\) it is zero). All the
algebraic properties of the wedge product
hold verbatim for differential forms, since they hold at every point. They include bilinearity, associativity, and the
anticommutativity \(\omega \wedge \eta = (-1)^{kl}\eta \wedge \omega\). 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 smooth 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 each tangent space of \(\mathbb{R}^3\) the alternating \(2\)-tensors
form a \(3\)-dimensional space and the alternating \(3\)-tensors a \(1\)-dimensional one.
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:
By the proposition on the bundle of alternating tensors
and the paragraph following it, a section of \(\Lambda^k T^*M\) in any degree is smooth exactly
when it is smooth as a covariant tensor field, and the same holds on \(N\). So \(\omega\) is a
smooth covariant tensor field on \(N\), and the \(\mathbb{R}\)-linear dependence of
\(F^*\omega\) on \(\omega\) and the smoothness of \(F^*\omega\) as a covariant tensor field
follow from the corresponding facts for the pullback of covariant tensor fields. Once
\(F^*\omega\) is known to be alternating, it is therefore smooth as a form. The alternating
property needs a separate observation, based on the
characterizations of alternating tensors.
A covariant tensor is alternating exactly when it gives the value zero whenever two of its
arguments are equal. Inserting the same vector into two slots of
\(F^*\omega\) inserts the same image vector into two slots of \(\omega\), where the value
vanishes. So \(F^*\omega\) is again alternating and lies in \(\Omega^k(M)\), while Part (1) is
the linearity noted above.
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 facts, we obtain
\[
F^*(\omega \wedge \eta) = \tfrac{(k+l)!}{k!\,l!}\operatorname{Alt}\bigl((F^*\omega) \otimes (F^*\eta)\bigr) = (F^*\omega) \wedge (F^*\eta).
\]
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, fix
\(p \in F^{-1}(V)\) and a smooth chart \((x^i)\) on \(M\) around \(p\), and evaluate both sides on a coordinate vector
\(\partial/\partial x^i\) at \(p\). Writing \(F^m = y^m \circ F\) for the component functions, the differential
is given by
\[
dF_p(\partial/\partial x^i) = \sum_m (\partial F^m/\partial x^i)\, \partial/\partial y^m,
\]
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 identity is exactly the chain rule. Applying Part (2) to a
wedge of coordinate differentials, Part (2) with a \(0\)-form factor to the coefficients, which gives
\(F^*(\omega_J\, dy^J) = (\omega_J \circ F)\, F^*(dy^J)\), 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 \(F(u, v) = (u,\, v,\, u^2 - v^2)\). With coordinates \((x, y, z)\) on the
target, let \(\omega = y\, dx \wedge dz + x\, dy \wedge dz\), a \(2\)-form on \(\mathbb{R}^3\). The chart formula
replaces each coefficient by its composition with \(F\) and each differential by the
differential of the corresponding component:
\[
F^*(dx) = du, \quad F^*(dy) = dv, \quad F^*(dz) = d(u^2 - v^2) = 2u\, du - 2v\, dv.
\]
Substituting these 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 \(du \wedge dv\), we obtain
\[
\begin{align*}
F^*\omega
&= -2v^2\, du \wedge dv - 2u^2\, du \wedge dv \\\\
&= -2\bigl(u^2 + v^2\bigr)\, du \wedge dv.
\end{align*}
\]
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. Here the determinant from the algebra of alternating tensors reappears as the change-of-variables factor. Integration of
forms rests on this result, and so does the construction of invariant integrals on groups.
Proof:
By the coordinate formula for pullbacks and the fact that \(F^*u = u \circ F\),
\[
F^*\bigl(u\, dy^1 \wedge \cdots \wedge dy^n\bigr) = (u \circ F)\, dF^1 \wedge \cdots \wedge dF^n,
\]
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 \((dx^i)\). Both
\(dF^1 \wedge \cdots \wedge dF^n\) and \(dx^1 \wedge \cdots \wedge dx^n\) are alternating \(n\)-tensors on an
\(n\)-dimensional space, the top degree, where the space of such tensors is one-dimensional, so the first is a scalar
multiple of the second. To identify the scalar, evaluate both on the coordinate frame
\((\partial/\partial x^1, \dots, \partial/\partial x^n)\). A wedge of \(n\) covectors evaluates to the
determinant of the matrix of evaluations,
and \(dF^j(\partial/\partial x^i) = \partial F^j/\partial x^i\), so the left side gives \(\det DF\) while the right side
gives \(1\):
\[
dF^1 \wedge \cdots \wedge dF^n = (\det DF)\, dx^1 \wedge \cdots \wedge dx^n.
\]
Substituting into the previous display yields the stated formula.
The factor \(\det DF\) is the Jacobian determinant of \(F\). The classical substitution rule for
multiple integrals uses its absolute value \(|\det DF|\) instead. For a diffeomorphism, \(\det DF\)
never vanishes, so on any connected open set where \(DF\) is defined it has a constant sign and
equals \(\pm|\det DF|\). Up to that sign, the Jacobian factor of the classical rule appears
automatically when a top-degree form is pulled back, 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
\[
dx = \cos\theta\, dr - r\sin\theta\, d\theta, \quad dy = \sin\theta\, dr + r\cos\theta\, d\theta.
\]
Wedging the two and distributing, we find that 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
\(dr \wedge d\theta\), we obtain
\[
\begin{align*}
dx \wedge dy &= \cos\theta\,(r\cos\theta)\, dr \wedge d\theta + (-r\sin\theta)(\sin\theta)\, d\theta \wedge dr\\\\
&= \bigl(r\cos^2\theta + r\sin^2\theta\bigr)\, dr \wedge d\theta\\\\
&= r\, dr \wedge d\theta.
\end{align*}
\]
The coefficient \(r\) is the Jacobian determinant of the polar map, and it recovers the factor that appears when a double
integral is rewritten in polar coordinates. The wedge product produces this factor 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 \(\tilde{x}^j = \tilde{x}^j(x^1, \dots, x^n)\). Its Jacobian is therefore 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, an object whose single coefficient is multiplied by the absolute value instead, 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.