Stephen Wiggins, School of Mathematics, University of Bristol

Link Lecture: Differential Forms

1Tuesday, 16 March 2010

Multivariable calculus: What you have learned?

Domains and Ranges of functions: Rn , where distances between points are measured using the Euclidean distance

scalar valued functions on Rn, e.g. energy

vector valued functions on Rn, e.g. a vector field, such as force

curves in Rn i.e. . vector valued functions of a scalar

2Tuesday, 16 March 2010

What do we want to do with these functions--Calculus

Differentiate them

Integrate them

But the underlying “algebraic structure” is essential, i.e. we (almost) take it for granted that we can “add, subtract, multiply, and divide”

3Tuesday, 16 March 2010

Differentiation of curves• tangent vectors• curvature

Differentiation of scalar valued functions of a vector variable• gradient• Hessian

Differential properties of vector fields• divergence• curl

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Integration of curves• line integrals

Integration of scalar valued functions• multiple integrals

One general theory encompassing all of this: DIFFERENTIAL FORMS

Élie Joseph Cartan (9 April 1869 – 6 May 1951

Jules Henri Poincaré (29 April 1854 – 17 July 1912)

5Tuesday, 16 March 2010

v ! u = "u ! v

u ! u = 0

c(u ! v) = (cu) ! v = u ! (cv)

u ! (v + w) = u ! v + u !w

2 vectors: the wedge product

Some Properties (I left out associativity, but it is)

Hermann Günther Grassmann (April 15, 1809, Stettin (Szczecin) – September 26, 1877, Stettin

Generalize the idea of vectors and vector spaces: “p-vectors”

6Tuesday, 16 March 2010

F (x, y, z)i+G(x, y, z)j+H(x, y, z)k!" F (x, y, z)dx+G(x, y, z)dy+H(x, y, z)dz

Differential 1 forms and vectors

“work form”

7Tuesday, 16 March 2010

F (x, y, z)i + G(x, y, z)j + H(x, y, z)k!"

F (x, y, z)dy # dz + G(x, y, z)dz # dx + H(x, y, z)dx # dy

Differential 2 forms and 2 vectors

“flux form”

8Tuesday, 16 March 2010

F (x, y, z)dx ! dy ! dz

Differential 3 forms and 3 vectors

9Tuesday, 16 March 2010

d(c1! + c2") = c1d! + c2d"

d(! ! ") = d! ! " + ("1)p! ! d", where ! is a p form

d2 = dd = 0

The “d operator”: the exterior derivative

d of a k form is a k+1 form

define recursively: d of a 0 form is a 1 form (the “usual” derivative)

Basic properties

10Tuesday, 16 March 2010

f is a 0 form, ! and " are 1 forms

d(c1! + c2") = c1d! + c2d"

d(f!) = df ! ! + fd!,

d(dx) = d(dy) = d(dz) = 0

For the relevant assigned problems you can assumed(dy ! dz) = d(dz ! dx) = d(dx ! dy) = 0

The “d operator”: Special CaseBasic properties

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d of a 0 form corresponds to the gradient of a vector fieldd of a 1 form corresponds to the curl of a vector fieldd of a 2 form corresponds to the divergence of a vector field

The relationship with classical vector analysis in 3 Dimensions

12Tuesday, 16 March 2010

A di!erential k form, !, is exact if there is a di!erential k-1 form, ", such that ! = d"

A di!erential k form, !, is said to be closed if d! = 0

An exact form is closed, but a closed form is not necessarily exact

A conservative force field on R3 corresponds to an exact di!erential 1 form

F = !"V

F = !Vxi! Vyj! Vzk"# !dV = d(!V ) = !Vxdx! Vydy ! Vzdz

“Closed and Exact” Differential Forms

Example

13Tuesday, 16 March 2010

! b

ag!(x)dx = g(b)! g(a)

!

CPdx + Qdy =

! !

D

"!Q

!x! !P

!y

#dxdy

Integration: Stokes Theorem

1 dimension: Fundamental Theorem of Calculus

2 dimensions: Greenʼs Theorem (on the plane)

George Green (14 July 1793 – 31 May 1841)

14Tuesday, 16 March 2010

!

volume! · F dV =

!

boundary of volume

F · n dS

! : k-1 form, ! : smooth n dimensional surface!

!d! =

!

!!!

3 dimensions: The divergence theorem

n dimensions: Stokes Theorem

n is a unit vector on the surface of the volume pointing “outwards”

Sir George Gabriel Stokes, 1st Baronet FRS (13 August 1819–1 February 1903)

15Tuesday, 16 March 2010

What do I want you to take away from this lecture?

✓A glimpse of how mathematicians synthesize and generalize “simple ideas” into theories and techniques with far reaching implications and applications

✓An idea of how such generalizations can “free your mind” to think more broadly about problems

✓The ability to do some simple calculations with the wedge product and exterior derivative, and relate these to “standard” ideas in multivariable and vector calculus

16Tuesday, 16 March 2010