grad, curl and div

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 11 June 2011

$k-forms$

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

${\Omega }_{{𝔽}^n}^1={𝒪}_{{𝔽}^n}\text{-span}\{d{x}_1,\dots ,d{x}_n\}=\{f_{1}d{x}_1+\cdots +f_{n}d{x}_n|f_1,\dots ,f_n\in {𝒪}_{{𝔽}^n}\}$

and

${\Omega }_{{𝔽}^n}^k={\Lambda }^k{\Omega }_{{𝔽}^n}^1,\text{for}k=1,2,\dots ,n.$

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

$df={\displaystyle \frac{\partial f}{\partial x_1}}d{x}_1+\cdots +{\displaystyle \frac{\partial f}{\partial x_n}}d{x}_n,\text{for}f\in {𝒪}_{{𝔽}^n},\text{and}$

by

$d(\omega \wedge \lambda )=d\omega \wedge +(-1{)}^k\omega \wedge d\lambda ,\text{for}\omega \in {\Omega }_{{𝔽}^n}^k.$

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 ${\Omega }_{{𝔽}^n}^1$

grad, curl and div

Formally write, as an operator,

$\nabla ={\displaystyle \frac{\partial \phantom{f}}{\partial x_1}}d{x}_1+{\displaystyle \frac{\partial \phantom{f}}{\partial x_1}}d{x}_2+{\displaystyle \frac{\partial \phantom{f}}{\partial x_3}}d{x}_3.$

Let $f\in {\Omega }_{{ℝ}^3}^0={𝒪}_{{ℝ}^3}.$ The gradient of $f$ is

$df=\nabla f={\displaystyle \frac{\partial f}{\partial x_1}}d{x}_1+{\displaystyle \frac{\partial f}{\partial x_2}}d{x}_2+{\displaystyle \frac{\partial f}{\partial x_3}}d{x}_3$

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

$\begin{array}{rcl}d{\lambda }_G& =& ({\displaystyle \frac{\partial \phantom{f}}{\partial x_2}}{G}_3-{\displaystyle \frac{\partial \phantom{f}}{\partial x_3}}{G}_2)d{x}_2\wedge d{x}_3+({\displaystyle \frac{\partial \phantom{f}}{\partial x_3}}{G}_1-{\displaystyle \frac{\partial \phantom{f}}{\partial x_1}}{G}_3)d{x}_3\wedge d{x}_1+({\displaystyle \frac{\partial \phantom{f}}{\partial x_1}}{G}_2-{\displaystyle \frac{\partial \phantom{f}}{\partial x_2}}{G}_1)d{x}_1\wedge d{x}_2\\ & =& \nabla \times G\in {\Omega }_{{ℝ}^3}^2.\end{array}$

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

$d{\omega }_F=({\displaystyle \frac{\partial F_1}{\partial x_1}}+{\displaystyle \frac{\partial F_2}{\partial x_2}}+{\displaystyle \frac{\partial F_3}{\partial x_3}})d{x}_1\wedge d{x}_2\wedge d{x}_3=\nabla \cdot F\in {\Omega }_{{ℝ}^3}^3.$

When $D$ is a 3-surface (volume) in ${ℝ}^3$ then Stokes theorem is termed the divergence theorem

${\iiint }_D\mathrm{div}𝐅dV={\iint }_S𝐅\cdot 𝐧d\sigma .$

When $S$ is a 2-surface (surface) in ${ℝ}^3$ then Stokes' theorem is termed Stokes' theorem

${\iint }_S\mathrm{curl}𝐅\cdot 𝐧d\sigma ={∳}_C𝐅\cdot d𝐑.$

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

$d\omega =({\displaystyle \frac{\partial \beta }{\partial x}}-{\displaystyle \frac{\partial \alpha }{\partial y}})dx\wedge dy\in {\Omega }_{{ℝ}^2}^2.$

When $\Omega$ is a 2-surface (region) in ${ℝ}^2$ Stokes' theorem is termed Green's theorem

${\int }_{\partial \Omega }(\alpha dx+\beta dy)={\int }_{\Omega }({\displaystyle \frac{\partial \beta }{\partial x}}-{\displaystyle \frac{\partial \alpha }{\partial y}})dxdy.$

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?????.

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