4. Differential forms
A. The Algebra And Integral Calculus Of Forms
4.1 Definition Of Volume – The Geometrical Role Of Differential Forms
4.2 Notation And Definitions For Antisymmetric Tensors4.3 Differential Forms4.4 Manipulating Differential Forms4.5 Restriction Of Forms4.6 Fields Of Forms4.7 Handedness And Orientability 4.8 Volumes And Integration On Oriented Manifolds 4.9 N-vectors, Duals, And The Symbol Ij…k
4.10 Tensor Densities 4.11 Generalized Kronecker Deltas 4.12 Determinants And Ij…k
4.13 Metric Volume Elements.
B. The Differential Calculus Of Forms And Its Applications 4.14 The Exterior Derivative4.15 Notation For Derivatives4-16 Familiar Examples Of Exterior Differentiation4.17 Integrability Conditions For Partial Differential Equations4.18 Exact Forms4.19 Proof Of The Local Exactness Of Closed Forms4.20 Lie Derivatives Of Forms4.21 Lie Derivatives And Exterior Derivatives Commute4.22 Stokes' Theorem4.23 Gauss' Theorem And The Definition Of Divergence 4.24 A Glance At Cohomology Theory4.25 Differential Forms And Differential Equations4.26 Frobenius' Theorem (Differential Forms Version)4.27 Proof Of The Equivalence Of The Two Versions Of Frobenius' Theorem4.28 Conservation Laws4.29 Vector Spherical Harmonics4.30 Bibliography
Concepts that are unified and simplified by forms • Integration on manifolds• Cross-product, divergence & curl of 3-D euclidean geometry• Determinants of matrices• Orientability of manifolds• Integrability conditions for systems of pdes• Stokes' theorem• Gauss' theorem• …
E. Cartan
4.1. Definition Of Volume – The Geometrical Role Of Differential Forms
2 vectors define an area (no metric required).
Different pairs of vectors can have same area.
3. , , ,area a b area a c area a b c
:area V V R , ,a b area a b
area( , ) is a (02) skew-tensor
such that
2. , 0area a a 1. , ,area a b area b a →
Ex. 4.1
,x y
x y
V Varea V W
W WFor vectors in the x-y plane:
4.2. Notation And Definitions For Antisymmetric Tensors
A (0p) tensor is completely antisymmetric if
, , , , , ,U V V U ,U V
Totally antisymmetric part of a (0p) tensor:
1, , ,
2!A U V U V V U
1, , , , , , , ,
3!A U V W U V W V W U W U V
, , , , , ,V U W W V U U W V
Index-notation:
1
2!A i j j ii j
i j
1
3!A i jk j k i k i j j i k k ji j k ii j k i j k
A skew (0p) tensor on an n-D space has at most
!
! !np
nC
p n p
independent components
4.3 Differential Forms
p-form = completely antisymmetric (0p) tensor ( p = degree of form).
0-form = scalar function. 1-form = covariant vector.
Wedge (exterior) product :
p q p q q p
Let , ,p q r be 1-forms. Then
0p p →
be the vector basis & 1-form basis, resp.
ieLet & ie
Then is a basis for 2-forms.
; 1, ,i je e i j n
1
ni j
i ji j
e e
1
2!i j
i j e e
,i j i je e
j i
(antisymmetry)
1,
2!k l l k
k l i je e e e e e 1
2!k l l k
k l i j i je e e e e e e e
1
2!k l l k
k l i j i j 1
2! i j j i i j
2dim nC
= (vector) space of all p-forms at x M px M dim n
pC
2x M
p q r p q r p q r
Grassmann algebra = { all p-forms , +, }
Ex. 4.8:
Show that
(associativity)
Dim = 0
1 1 2n
nn np
p
C
[ ]p q
p i j k li jk lp q C p q
i j k j k i k i ji j kp q p q p q p q [ ]3 i j kp q if
1 2&p q
4.4 Manipulating Differential Forms
Attention: signs
Let ,p q be p- & q-forms, resp. Then pqp q q p
Proof: Let
1 pp
be 1-forms such thatj
1p p qq
Then1 1p p p qp q
1 1 2p p p p p q
2 1 2 1 3p p p p p p q
1 1q p p p q p
pqq p
Proof using basis:
1
1
1
!p
p
iii ip p e e
p 1
1
1
!q
q
iii iq q e e
q
11
1 1
1
! !p p p q
p p p q
i i iii i i ip q p q e e e e
p q
1 21
1 1
1
! !p p p p q
p p p q
p i i i iii i i ip q e e e e e
p q
1 2 31
1 1
2 1
! !p p p p p q
p p p q
p i i i i iii i i ip q e e e e e e
p q
1 1
1 1
1
! !p p q p
p p p q
pq i i iii i i ip q e e e e
p q
pqq p
Contraction:
1
,p empty slots
Let be a vector & a p-form.
i.e., ii j kj k
Define
Example: p q where ,p q are 1-forms
p q p q q p p q q p p q q p
1 1 2 2 1p p pi i ii i i i ie e e e e e e e
1 2][! pii ip e e
1 2 2 1p pi ii i i ie e e e e e
1 2 2 1p pi ii i i ie e e e [ p! terms]
[ ]! i j kp e e
1
1
1
!p
p
iii i e e
p 1 2
1 2
]p
p
ii ii i i e e →
1 1 2][!p pi ii i ie e p e e
1 1p pi i i i
1 2
1 2
p
p
ii i
i i ie e
1 2
1 2
1
1 !p
p
ii ii i i e e
p
pp p p In general
1
1 !i j k
i j k e ep
= (p–1)-form with components 1
1 22
pp
ii i ii i
4.5. Restriction of Forms
A p-form is a (0p) tensor → its domain is
, , , ,W
X Y X Y
Wof
, ,X Y W
dimW p
p
p factors
V V V
The restriction (section) to a subspace W of V is
0W
→
dimW p → W is 1-D
(annulled by W)
4.6. Fields of Forms
A field Ωp(M) of p-forms on a manifold M
= a rule that gives a p-form at each point of M.
Ditto vector field.
A submanifold S of M picks a subspace VP of TP PS.
→ Restriction of p-form field to S
= restriction of p-form at P to VP PS.
4.7. Handedness and Orientability
1n nnDim M C Dim M n →
Let nP M
If ie is a basis for TP(M), then 1, , 0ne e iff 0 at P.
1
1
, , 0
, , 0n
in
e eright handede is if
e eleft handed
Relative handedness is independent of choice of
M is orientable if it is possible to define handedness continuously over it,
i.e., a continuous basis with the same handedness everywhere on M.
i.e., 0P P M E.g. En is orientable.
The Mobius band is not.
Absolute handedness is fixed by the choice of the coordinate chart.
4.8. Volumes and Integration on Oriented Manifolds
Integration of a Function
11
, , nn
x xx x
1 1 11
, ,n n nn
dV x x dx dx x xx x
( parallelepiped / cell )
(volume of cell)
:f M R (function)
1 1, , n n n
cell
f x x d x f x x
Integration of f over cell :
1 11
, ,n nn
f dx dx x xx x
11
, , nn
x xx x
cell
1 nf dx dx ( n-form )
Integration of f over U M :
n
cellU
f d x cell U
1 n
U
f dx dx
Change of Variables
is independent of coordinates up to an overall sign.
,f d d ,f d d
d dx dyx y
, ,x y
dx dyx y y x
E.g., M is 2-D :
Changing coordinates
d dx dyx y
d d dx dy dx dyx y x y
→
,
,
x yJ
x y
x y
,
,dx dy
x y
= Jacobian
J dx dy J dy dx
d d J dxdy
Riemannian integration:
11, , n
nx e x e 11
, , nn
cell x xx x
Orientability
1 nf dx dx
1 , , ne e
nf d x cell
cell
Let be another basis which differs from
1 11 , ,n n
ncell f dx dx x e x e
11 , , n
ncell x e x e
1, , ne e only in handedness.
'cell cell
cell cell
cell
Let the entire region of integration be orientable, then
By convention, a right-handed basis is always assumed in
nf d x cell
cell
Integration on Submanifold
1, , n pn n
1, , n pn n
is defined only for n-form on an n-D manifold M,
or p-form over a p-D submanifold S.
Relation between the orientabilities of M and S ?
( Domain must be internally orientable )
Let M be orientable and a right-handed n-form at PS.
the p-form is a right- handed restriction of to S
not tangent to S at P,
Given n–p independent normal vectors
1, , n pn n determines an external orientation for S at P.
S is externally orientable if it is possible to define an external orientation continuously over it.
If U M is orientable, then S U is either both internally and externally orientable, or it is neither.
Otherwise, S may be one but not both.
Mobius strip embedded in R3.
M is not externally orientable in R3.
A curve is always internally orientable
→ it can't be externally orientable inside a nonorientable submanifold
C1 is not orientable in M
But C2 is both internally & externally orientable in M
4.9. N-vectors, Duals, and the Symbol
Dual Maps
n-form
p vector n p forms
g g
p forms n p vectors
* T T T
* T T
1 1
1 1
1
!q n
n q
j j iii i j jT e e e e
n T
g = metric tensor
: dual map
Dual of a q-vector T
*: *q n q n q V V
1
1
q
q
i i
i iT e e T
1
1
1
!n
n
iii i e e
n
1
1
1
!q
q
i i
i iT e eq
1
1
q
q
i i
i iT e e
1 1
* *n q n qi i i i
T T
1 1
1 1
1
!q q n
q q n
j j i inq j j i iC T e e
n
T
1 1
1 1
1
!q n
n q
j j iii i j jT e e e e
n T
1 1 11 1
1 1 1 1
q q q qn n
n q n q
j j j j i ii ii ii i j j i i j jT e e e e T e e
1 1
1 1
q q n
q q n
j j i ij j i i T e e
11
1 1
1
! !q n q
q n q
ij j ij j i i T e e
q n q
*T is an (n-q)-form with components
1 n qi i
T
1
1 1
1
!q
q n q
j jj j i i T
q
,U V
123 1
U V
i j j iU U U U
i ji jU V e e
2 3 32 1dim * 3 dimC C V V
Example: Cross Products in E3
Then
be vectors &Let ,U V the associated 1-forms.
12!
2!i j
i jU V e e
2!i j i jU V
2 3 3 1 1 22 3 3 2 3 1 1 3 1 2 2 1U U U U e e U U U U e e U U U U e e
W U V i j kj k iU V e * i j k
j k iW U V e Let →
Setting gives *W U V * U V
The cross product exists only in E3 , where
cell cell
1 1
1 1
1
! !q n q
q n q
j j i i
j j i iT S e e e eq n q
ST
cell cell cell
1 1
1 1
1* ,
!q n q
q n q
j j i ij j i i T S
q
T S
1 1
1
1
! !q q n
n
j j j jj jT S e e
q n q
1 1
1
1
!q q n
n
j j j jnq j jC T S e e
n
= n-vector with components 1 1 1n q q nj j j j j jnqC T S T S
11! n qqn q T S if
12 1n
cell cell cell cell cell cell
cell cell cell cell cell cell
Top Related