= Differential form
{title2=$\alpha\in\Omega^k(M)$}
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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 $\alpha=x\,dy$ 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 $\int_0^1dy=1$. Taking the left and then the top edge instead gives zero: the vertical leg has $x=0$, and the horizontal leg again has $dy=0$. \b[The two paths have the same endpoints but different integrals.]
Now replace $\alpha$ with the <exact differential> $\beta=d(xy)=y\,dx+x\,dy$. 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>.
\Image[/differential-form-path-integrals.png]
{title=Two routes between the same endpoints distinguish a path-dependent one-form from an exact differential}
The <exterior derivative> records the local obstruction in the first example: $d(x\,dy)=dx\wedge dy$. 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. https://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf[Sjamaar: Manifolds and differential forms]{ref}.
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