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The exterior derivative is a linear map $d:{\Omega }^k(M)\to {\Omega }^{k+1}(M)$, agrees with the ordinary differential on functions, satisfies $d^2=0$, and obeys the graded Leibniz ruleIn local coordinates it differentiates the coefficient functions:Antisymmetry cancels the second partial derivatives, proving $d^2=0$ locally. These properties determine the coordinate-independent derivative.

For any smooth $\varphi :N\to M$, the pullback of a differential form is defined byThere is no requirement that $\varphi$ be invertible or an immersion. Pullback preserves wedge products and sends $df$ to $d(f\circ \varphi )$ by the chain rule. Locally every form is a sum of $fd{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}$; applying the preceding identities and $d^2{\varphi }^i=0$ proves

Describe the brane by an embedding $X:\Sigma \to M$, with nondegenerate induced worldvolume metric $h=X^{\ast }g$. Locally the Poincaré lemma gives $F=dA$ because $F$ is a closed differential form. For a timelike worldvolume, the geometric area and Wess-Zumino brane coupling give the action$T$ is the brane tension and $q$ the charge in this normalization. The geometric term is the higher-dimensional version of the Nambu–Goto action. Both integrals are invariant under changes of worldvolume coordinates.

The additive gauge transformation is $A\mapsto A+d\Lambda$. Its change of action isThe last step is the Generalized Stokes theorem. It vanishes for a closed worldvolume, or with suitable boundary restrictions. An open worldvolume can instead have a compensating boundary coupling, as in gauge invariance of an open-brane coupling.

More directly, for a variation vector $V$ of the embedding, Cartan's magic formula givesThus the interior force depends on $F$, not the gauge potential. Define the trace of the second fundamental form byThe area variation is $-\int \sqrt{|h|}g(V,K)$ plus a boundary term. For compactly supported interior variations the resulting equations areThey exhibit the field-strength force on a charged brane and are gauge invariant.

Two global qualifications are necessary. First, replacing $A$ by $d\Lambda$ alone, as printed, gives $d{A}^{\prime }=0$ and would erase a nonzero $F$; the intended transformation must be the additive shift above. Second, closedness of $F$ only guarantees a potential locally. If a global potential does not exist and the worldvolume bounds a filling $B$, its charged term can be defined as $q{\int }_{B}F$. Different fillings differ by a period of $F$; independence of the quantum phase requires $q{\int }_{C}F\in 2\pi \hslash \mathbb{Z}$ for the corresponding closed cycles. Without the required global potential or patching/filling data, the given closed form still defines the local variational equations, but not automatically a single globally defined real-valued action.

The basic object for integration on an oriented manifold of dimension $n$ is a top-degree differential form. In an orientation-preserving chart, write $\alpha =f(x)d{x}^1\wedge \cdots \wedge d{x}^n$. Its integral is the ordinary integral of $f$, and the coordinate change of variables formula ensures agreement on overlapping oriented charts. A partition of unity combines these local integrals into a global one, provided the form has compact support or satisfies appropriate convergence conditions. Reversing orientation reverses the integral. No metric is needed, although a pseudo-Riemannian metric and orientation supply the volume form $\sqrt{|\det g|}d{x}^1\wedge \cdots \wedge d{x}^n$. On a nonorientable manifold one instead integrates densities, or appropriately twisted top forms.

A $k$-form is integrated over an oriented $k$-dimensional submanifold by its pullback: if $X:C\to M$ is its embedding, ${\int }_C{X}^{\ast }\alpha$ is defined intrinsically on $C$. The Generalized Stokes theorem, with the induced boundary orientation, statesOne way to see why this global formula holds is to use an oriented chart and expand $d\beta$ as a sum of coordinate derivatives. The ordinary fundamental theorem of calculus gives the boundary contributions with their orientation signs. A partition of unity reduces the manifold to such charts; the artificial internal boundary contributions cancel, leaving only the actual boundary. Compactness or compact support controls the sums and the boundary at infinity.

This immediately explains how closed differential forms yield conservation laws. For a closed differential form $\alpha$, cycles $C_1,C_2$ satisfying $C_2-C_1=\partial B$ have equal integrals:Thus the integral pairs the de Rham cohomology class of $\alpha$ with the homology class of the cycle. An exact differential form integrates to zero on every closed cycle. A closed form need not be exact, and nonzero periods measure the global obstruction. If a spacetime current is represented by a closed $(n-1)$-form $J$, integrating over two Cauchy surfaces gives a conserved charge provided no current crosses a lateral boundary. A nonzero boundary flux must be retained in the Stokes calculation rather than silently discarded.

For a genuinely topological current, closure can follow identically from the field geometry, without the equations of motion. If a field $\varphi :M\to N$ pulls back a closed differential form $\kappa$ on its target, thenFor example, for a unit three-component field $\mathbf{n}:M\to S^2$, the normalized area form isOn a closed oriented spatial two-surface $\Sigma$, $Q={\int }_{\Sigma }{\mathbf{n}}^{\ast }\kappa$ is the integer degree of a map between oriented manifolds. Smooth evolution is a homotopy and conserves this topological charge. Singularities or flux through a spatial boundary can permit the charge to change. This is the pullback-volume representation of a topological current.

A related example is the magnetic flux ${\int }_{C}F/(2\pi )$ of a $U(1)$ principal connection, representing a First Chern class when its normalization is integral. Nonzero flux on a closed cycle requires nontrivial global bundle data or singular sources: if a single smooth global potential exists with $F=dA$, the flux on every closed cycle vanishes. This distinction is important when local differential equations are used to infer global conservation or quantization.

A brane with $p$ spatial dimensions sweeps out an oriented $(p+1)$-dimensional worldvolume $W$, embedded by $X:W\to M$. Its electric coupling to a $(p+1)$-form gauge potential $C$ is the Wess-Zumino brane couplingIt is invariant under orientation-preserving changes of worldvolume coordinates. Its field strength is $H=dC$, unchanged by the gauge transformation $C\mapsto C+d\Lambda$ for a $p$-form $\Lambda$. Since the pullback of a differential form commutes with the exterior derivative, its gauge variation isTherefore a closed brane has a gauge-invariant coupling under these transformations. For a noncompact worldvolume, support or decay conditions must also eliminate contributions at infinity.

For an open brane, the variation does not vanish automatically. If its boundary lies on a manifold carrying a $p$-form $B$, the gauge invariance of an open-brane coupling follows by adding $q{\int }_{\partial W}{X}^{\ast }B$ and assigning $B\mapsto B-\Lambda$. The two boundary variations cancel exactly with the induced orientation. Alternatively one may restrict the gauge parameter to have vanishing pullback at the boundary, if that is the chosen physical boundary condition. A particle has $p=0$ and couples by $q\int A$; a string has $p=1$ and couples to a two-form; a membrane has $p=2$ and couples to a three-form.

Finally, a gauge potential need not be globally a single differential form. Patchwise potentials and their overlap data define its global holonomy. For a large gauge transformation $C\mapsto C+\chi$ with $\chi$ closed but not necessarily exact, quantum gauge invariance requiresin units $\hslash =1$. This connects the coupling to integral periods and charge quantization. The local Stokes argument and these global period conditions are complementary: closure controls continuous conservation, while global topology controls which charges and gauge transformations are allowed.