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 𝒪𝔽n and let Ω 𝔽n 0 = 𝒪𝔽n ,

Ω 𝔽n 1 = 𝒪𝔽n-span {dx1,, dxn} = {f1 dx1++ fn dxn | f1,, fn 𝒪𝔽n }
and
Ω 𝔽n k = Λk Ω 𝔽n 1 ,for k=1,2,,n.
Define
𝒪𝔽n d Ω 𝔽n 1 d Ω 𝔽n 2 d d Ω 𝔽n n d 0
by
df = f x1 dx1 ++ f xn dxn , for f 𝒪𝔽n ,and
by
d(ωλ) = dω+ (-1)k ωdλ, for ω Ω 𝔽n k .
HW: Show that d2=0.

grad, curl and div

Formally write, as an operator,

= f x1 dx1 + f x1 dx2 + f x3 dx3 .
Let f Ω 3 0 = 𝒪3 . The gradient of f is
df=f= f x1 dx1 + f x2 dx2 + f x3 dx3
Let λG = G1dx1 + G2dx2 + G3dx3 Ω 3 1 . The curl of G is
dλG = ( f x2 G3 - f x3 G2 ) dx2 dx3 + ( f x3 G1 - f x1 G3 ) dx3 dx1 + ( f x1 G2 - f x2 G1 ) dx1 dx2 = ×G Ω 3 2 .
Let ωF = F1 dx2 dx3 + F2 dx3 dx1 + F3 dx1 dx2 Ω 3 1 . The divergence of F is
dωF = ( F1 x1 + F2 x2 + F3 x3 ) dx1 dx2 dx3 = F Ω 3 3 .
When D is a 3-surface (volume) in 3 then Stokes theorem is termed the divergence theorem
D div𝐅 dV = S 𝐅𝐧 dσ .
When S is a 2-surface (surface) in 3 then Stokes' theorem is termed Stokes' theorem
S curl𝐅𝐧 dσ = C 𝐅d𝐑 .

If ω=αdx+ βdy Ω 2 1 then

dω= ( β x - α y ) dxdy Ω 2 2 .
When Ω is a 2-surface (region) in 2 Stokes' theorem is termed Green's theorem
Ω ( αdx+ βdy ) = Ω ( β x - α 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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