Differential Forms 03: Exterior Derivative
Differential forms / 03
Exterior derivative
The exterior derivative is the unique calculus operation that extends ordinary differentiation and makes boundary integrals equal interior integrals.
Calculus root: derivative, boundary, and mixed partials
The exterior derivative extends the one-variable derivative. For a function on the line,
\[d(f)=f'(x)\,dx.\]The fundamental theorem of calculus then says
\[\int_{[a,b]}df=f(b)-f(a)=\int_{\partial[a,b]}f.\]Exterior differentiation is designed so the same boundary principle remains true in every dimension.
Example: curl from failure of a 1-form to be a derivative
For \(\eta=P\,dx+Q\,dy\),
\[d\eta=(Q_x-P_y)\,dx\wedge dy.\]If \(\eta=df\), then \(P=f_x\) and \(Q=f_y\), so \(Q_x-P_y=f_{yx}-f_{xy}=0\). The identity \(d^2=0\) is the equality of mixed partials in invariant form.
Example: divergence from the same operation
In \(\mathbb R^3\), associate \(F=(P,Q,R)\) to
\[\alpha=P\,dy\wedge dz+Q\,dz\wedge dx+R\,dx\wedge dy.\]Then
\[d\alpha=(P_x+Q_y+R_z)\,dx\wedge dy\wedge dz.\]Gradient, curl, and divergence are not separate miracles; they are different degrees of one derivative.
For a local expression
\[\omega=\sum_I a_I\,dx_I,\]the exterior derivative is
\[d\omega=\sum_I da_I\wedge dx_I, \qquad da_I=\sum_j {\partial a_I\over \partial x_j}\,dx_j.\]This is the computational rule used in Section 4.1 of the PDF. The definitions are arranged so that two structural identities hold:
\[d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^{\deg\alpha}\alpha\wedge d\beta, \qquad d^2=0.\]Functions and 1-forms
For a function \(f\),
\[df=\sum_i f_{x_i}\,dx_i.\]For a planar 1-form \(\eta=P\,dx+Q\,dy\),
\[d\eta=dP\wedge dx+dQ\wedge dy=(Q_x-P_y)\,dx\wedge dy.\]Example 1: curl as exterior derivative
Let
\[\eta=(x^2y)\,dx+(\sin x+y^3)\,dy.\]Then
\[d\eta=(\cos x- x^2)\,dx\wedge dy.\]The coefficient is the scalar curl \(Q_x-P_y\).
Example 2: divergence as derivative of a 2-form
In \(\mathbb R^3\) associate a vector field \(F=(P,Q,R)\) to
\[\alpha=P\,dy\wedge dz+Q\,dz\wedge dx+R\,dx\wedge dy.\]Then
\[d\alpha=(P_x+Q_y+R_z)\,dx\wedge dy\wedge dz.\]Thus the divergence theorem is a Stokes theorem for this 2-form.
Why the sign in Leibniz appears
If \(\alpha=a_I dx_I\) is degree \(k\) and \(\beta=b_Jdx_J\), then
\[d(\alpha\wedge\beta)= d(a_Ib_J)\wedge dx_I\wedge dx_J.\]The term \(a_I\,db_J\) must move past the \(k\) one-forms in \(dx_I\), producing the factor \((-1)^k\). Therefore
\[d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^k\alpha\wedge d\beta.\]Proof sketch of \(d^2=0\)
It is enough to test \(\omega=a\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\). Then
\[d\omega=\sum_p a_{x_p}\,dx_p\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}.\]Applying \(d\) again gives
\[d^2\omega= \sum_{p,q} a_{x_qx_p}\, dx_q\wedge dx_p\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}.\]The term \((p,q)\) cancels the term \((q,p)\) because mixed partials agree while \(dx_q\wedge dx_p=-dx_p\wedge dx_q\). Terms with \(p=q\) vanish because \(dx_p\wedge dx_p=0\).
Closed and exact
A form \(\omega\) is closed if \(d\omega=0\). It is exact if \(\omega=d\eta\) for some form \(\eta\). Since \(d^2=0\),
\[\text{exact}\quad\Longrightarrow\quad\text{closed}.\]For positive-degree forms, the converse is local on small balls but not global on arbitrary domains. In degree zero, closed means locally constant; there are no \((-1)\)-forms whose exterior derivative could produce an exact 0-form.
Example 3: exact form
Let \(f=x^2y+\cos y\). Then
\[df=2xy\,dx+(x^2-\sin y)\,dy.\]The form is closed because
\(d(df)=((2x)-(2x))\,dx\wedge dy=0.\)
Example 4: a closed 2-form for degree reasons
On \(\mathbb R^2\), every 2-form \(f\,dx\wedge dy\) is closed. But it is not meaningful to infer topology from this alone; \(d\) would land in \(\Omega^3(\mathbb R^2)=0\). Degree can force closedness without saying anything about primitives.
Reading Stokes in advance
The exterior derivative is constructed so that
\[\int_{\partial M}\omega=\int_M d\omega\]has a uniform meaning. In dimension one this is the fundamental theorem of calculus. In dimension two it becomes Green’s theorem. In dimension three it becomes classical Stokes or divergence, depending on the degree of \(\omega\).