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The pure Abelian Chern--Simons theory has zero metric stress. For a source written as $-c\int \star J\wedge A$, holding the covariant components of the one-forms $J,A$ fixed makes the source equal to $-c\int \sqrt{-g}{g}^{\mu \nu }{J}_{\mu }{A}_{\nu }{d}^{3}x$. Variation using $\delta \sqrt{-g}=-\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }/2$ gives the displayed stress-energy tensor. If the independent source is instead the fixed two-form $\mathcal{J}=\star J$, the source action is metric-independent and its stress vanishes. The variational data must therefore be stated.

For a codimension-one tangent distribution, a coorientation is an orientation of its normal quotient line bundle. An oriented real line bundle is trivial: local positive sections can be combined with a partition of unity without cancellation. Hence a cooriented distribution is the kernel of a global nowhere-vanishing one-form. This choice is independent of the orientation of the ambient manifold, and reversing it does not change the Godbillon-Vey invariant.

A differential form is a smooth field of alternating multilinear maps that can be integrated over an appropriately oriented manifold of matching dimension. A one-form measures a directed displacement; a two-form measures an oriented area element. The point is to express what is being accumulated along a curve or over a surface without tying the answer to one set of coordinates. This makes line integrals, surface integrals, and the integral theorems of vector calculus parts of one construction.

For a concrete one-form, take $\alpha =xdy$ in the plane. Travel from $(0,0)$ to $(1,1)$ along the bottom and then the right edge of the unit square. The horizontal leg contributes zero because $dy=0$; the vertical leg has $x=1$ and contributes ${\int }_0^{1}dy=1$. Taking the left and then the top edge instead gives zero: the vertical leg has $x=0$, and the horizontal leg again has $dy=0$. The two paths have the same endpoints but different integrals.

Now replace $\alpha$ with the exact differential $\beta =d(xy)=ydx+xdy$. Both paths give $1$, because integrating an exact differential along a curve gives the endpoint difference of its potential. Here that difference is $xy{|}_{(1,1)}-xy{|}_{(0,0)}=1$. This is the fundamental theorem of calculus applied to a function along a parametric curve, and is the basic reason conservative vector fields have path-independent work.

The exterior derivative records the local obstruction in the first example: $d(xdy)=dx\wedge dy$. The Generalized Stokes theorem turns the difference between the two path integrals into the integral of this two-form over the square, whose oriented area is $1$. Interior edges cancel when adjoining pieces are added; only the outer boundary survives. That cancellation connects a local calculation with a global measurement.

More precisely, a degree-$k$ differential form assigns to each point $p$ a multilinear scalar-valued function of $k$ tangent vectors at $p$. Interchanging two arguments changes its sign. A one-form is therefore a covector field, while a two-form vanishes on a pair of parallel vectors. Degree zero simply means a smooth function. The space ${\Omega }^k(M)$ is a vector space; the wedge product of differential forms combines degrees, and the exterior derivative raises degree by one.

The usefulness goes beyond bookkeeping. Pullbacks of a differential form supply the right transformation law automatically under a coordinate change. The Generalized Stokes theorem unifies the fundamental theorem of calculus, Green theorem, Stokes theorem, and divergence theorem. Meanwhile, the distinction between closed differential forms and exact differential forms detects holes through de Rham cohomology. Reyer Sjamaar develops these connections in sections 5.4 and 9.3 of his university notes. ^[ref].

For the Levi-Civita connection of a Riemannian metric, the divergence of $X$ is ${\mathrm{div}}_{g}X=\mathrm{tr}(Y\mapsto {\nabla }_{Y}X)$. In coordinates it equals $(\det g{)}^{-1/2}{\partial }_i((\det g{)}^{1/2}{X}^i)$. On an oriented manifold it satisfies ${\mathcal{L}}_X{\omega }_g=({\mathrm{div}}_{g}X){\omega }_g$. The codifferential on a one-form satisfies $\delta \theta =-{\mathrm{div}}_g({\theta }^{♯})$, where $♯$ is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

The exterior derivative is the unique graded derivation $d:{\Omega }^p(M)\to {\Omega }^{p+1}(M)$ that agrees with the differential on functions and satisfies $d^2=0$.

On a smooth planar one-form $\alpha =Pdx+Qdy$, the product rule and antisymmetry of the wedge product of differential forms giveFor $\alpha =xdy$, the coefficient is $1$: there is a nonzero integral around the boundary of every positively oriented rectangle. For $\beta =d(xy)$, the coefficient is ${\partial }_{x}x-{\partial }_{y}y=0$. More generally, the identity $d^2=0$ makes every exact differential form a closed differential form. It expresses cancellation between mixed partial derivatives locally; it does not imply that every closed differential form has a global potential.

A differential 1-form is a smooth section of the cotangent bundle; it assigns a covector to every point of a smooth manifold. In planar coordinates, $\alpha =P(x,y)dx+Q(x,y)dy$ evaluates a tangent vector $v$ as $P(x,y)v_x+Q(x,y)v_y$.

Along a parametric curve $\gamma (t)=(x(t),y(t))$, its line integral isReversing the direction reverses the integral. A force expressed as a one-form therefore accumulates work along a directed path. In Euclidean coordinates its components can be displayed as arrows, but identifying a one-form with a vector field uses an inner product; the covector field itself does not require a Riemannian metric.