## $k-forms$

Let ${𝒪}_{{𝔽}^{n}}$ and let ${\Omega }_{{𝔽}^{n}}^{0}={𝒪}_{{𝔽}^{n}}$,

 ${\Omega }_{{𝔽}^{n}}^{1}={𝒪}_{{𝔽}^{n}}\text{-span}\left\{d{x}_{1},\dots ,d{x}_{n}\right\}=\left\{{f}_{1}d{x}_{1}+\cdots +{f}_{n}d{x}_{n}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}{f}_{1},\dots ,{f}_{n}\in {𝒪}_{{𝔽}^{n}}\right\}$
and
 ${\Omega }_{{𝔽}^{n}}^{k}={\Lambda }^{k}{\Omega }_{{𝔽}^{n}}^{1},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}k=1,2,\dots ,n.$
Define
 ${𝒪}_{{𝔽}^{n}}\stackrel{d}{⟶}{\Omega }_{{𝔽}^{n}}^{1}\stackrel{d}{⟶}{\Omega }_{{𝔽}^{n}}^{2}\stackrel{d}{⟶}\cdots \stackrel{d}{⟶}{\Omega }_{{𝔽}^{n}}^{n}\stackrel{d}{⟶}0$
by
 $df=\frac{\partial f}{\partial {x}_{1}}d{x}_{1}+\cdots +\frac{\partial f}{\partial {x}_{n}}d{x}_{n},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}f\in {𝒪}_{{𝔽}^{n}},\phantom{\rule{1em}{0ex}}\text{and}$
by
 $d\left(\omega \wedge \lambda \right)=d\omega \wedge +\left(-1{\right)}^{k}\omega \wedge d\lambda ,\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\omega \in {\Omega }_{{𝔽}^{n}}^{k}.$
HW: Show that ${d}^{2}=0$.

• A closed $k$-form is $\omega \in {\Omega }_{{𝔽}^{n}}^{k}$ such that $d\omega =0$.
• An exact $k$-form is $\lambda \in {\Omega }_{{𝔽}^{n}}^{k}$ such that $\lambda \in \mathrm{im}d$.
• A vector field is an element of ${\Omega }_{{𝔽}^{n}}^{1}$

Formally write, as an operator,

 $\nabla =\frac{\partial \phantom{f}}{\partial {x}_{1}}d{x}_{1}+\frac{\partial \phantom{f}}{\partial {x}_{1}}d{x}_{2}+\frac{\partial \phantom{f}}{\partial {x}_{3}}d{x}_{3}.$
Let $f\in {\Omega }_{{ℝ}^{3}}^{0}={𝒪}_{{ℝ}^{3}}.$ The gradient of $f$ is
 $df=\nabla f=\frac{\partial f}{\partial {x}_{1}}d{x}_{1}+\frac{\partial f}{\partial {x}_{2}}d{x}_{2}+\frac{\partial f}{\partial {x}_{3}}d{x}_{3}$
Let ${\lambda }_{G}={G}_{1}d{x}_{1}+{G}_{2}d{x}_{2}+{G}_{3}d{x}_{3}\in {\Omega }_{{ℝ}^{3}}^{1}.$ The curl of $G$ is
 $\begin{array}{rcl}d{\lambda }_{G}& =& \left(\frac{\partial \phantom{f}}{\partial {x}_{2}}{G}_{3}-\frac{\partial \phantom{f}}{\partial {x}_{3}}{G}_{2}\right)d{x}_{2}\wedge d{x}_{3}+\left(\frac{\partial \phantom{f}}{\partial {x}_{3}}{G}_{1}-\frac{\partial \phantom{f}}{\partial {x}_{1}}{G}_{3}\right)d{x}_{3}\wedge d{x}_{1}+\left(\frac{\partial \phantom{f}}{\partial {x}_{1}}{G}_{2}-\frac{\partial \phantom{f}}{\partial {x}_{2}}{G}_{1}\right)d{x}_{1}\wedge d{x}_{2}\\ & =& \nabla ×G\in {\Omega }_{{ℝ}^{3}}^{2}.\end{array}$
Let ${\omega }_{F}={F}_{1}d{x}_{2}\wedge d{x}_{3}+{F}_{2}d{x}_{3}\wedge d{x}_{1}+{F}_{3}d{x}_{1}\wedge d{x}_{2}\in {\Omega }_{{ℝ}^{3}}^{1}.$ The divergence of $F$ is
 $d{\omega }_{F}=\left(\frac{\partial {F}_{1}}{\partial {x}_{1}}+\frac{\partial {F}_{2}}{\partial {x}_{2}}+\frac{\partial {F}_{3}}{\partial {x}_{3}}\right)d{x}_{1}\wedge d{x}_{2}\wedge d{x}_{3}=\nabla \cdot F\in {\Omega }_{{ℝ}^{3}}^{3}.$
When $D$ is a 3-surface (volume) in ${ℝ}^{3}$ then Stokes theorem is termed the divergence theorem
 ${\iiint }_{D}\mathrm{div}𝐅\phantom{\rule{0.2em}{0ex}}dV={\iint }_{S}𝐅\cdot 𝐧\phantom{\rule{0.2em}{0ex}}d\sigma .$
When $S$ is a 2-surface (surface) in ${ℝ}^{3}$ then Stokes' theorem is termed Stokes' theorem
 ${\iint }_{S}\mathrm{curl}𝐅\cdot 𝐧\phantom{\rule{0.2em}{0ex}}d\sigma ={∳}_{C}𝐅\cdot d𝐑.$

If $\omega =\alpha dx+\beta dy\in {\Omega }_{{ℝ}^{2}}^{1}$ then

 $d\omega =\left(\frac{\partial \beta }{\partial x}-\frac{\partial \alpha }{\partial y}\right)dx\wedge dy\in {\Omega }_{{ℝ}^{2}}^{2}.$
When $\Omega$ is a 2-surface (region) in ${ℝ}^{2}$ Stokes' theorem is termed Green's theorem
 ${\int }_{\partial \Omega }\left(\alpha dx+\beta dy\right)={\int }_{\Omega }\left(\frac{\partial \beta }{\partial x}-\frac{\partial \alpha }{\partial y}\right)dxdy.$

## Notes and References

This is the beginning of the connection between Stokes theorem as it might be covered in a multivariable calculus class and cohomology. This presentation was distilled from [BR, Chapt. 10] and ???? The key computation is in [BR, proof of Theorem 10.43]. Green's theorem is in [TF, §15.5], the divergence theorem is in [TF §15.6] and Stokes' theorem is in [TF §15.7].

## References

[BR] W. Rudin, Principles of Mathematical Analysis, ?????, MR?????.

[TF] Thomas and Finney ??? Calculus and Analytic Geometry, Fifth edition, ?????, MR?????.