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V × . . . × V (k times), multilinear and skew-symmetric: for any choice of k vectors (v 1 , . . . , v k ) ∈ V k , v
1 ∈ V and two scalars λ 1 , λ
2 ∈ R we have ω(λ 1
1 + λ
2 v 1 , v 2 , . . . , v k ) = λ
1 ω(v
1 , . . . , v k ) + λ
2 ω(v
1 , . . . , v k )
and ω(v
i 1 , . . . , v i k ) = ( −1) ν ω(v 1 , . . . , v k ),
where ν = 0 if the permutation (i 1 , . . . , i k ) of (1, . . . , k) is even, and ν = 1 if it is odd. Recall that a permutation is even if it is obtained by an even number of exchanges of pairs of indices. Example A4.1 The oriented area of the parallelogram in R 2 with sides v 1 , v
2 is given by ω(v 1
2 ) = det
v 1 1 v 2 1 v 1 2 v 2 2 . This is clearly an algebraic 2-form. Similarly the oriented volume of the solid with parallel sides v 1 , . . . , v l in R
l is an algebraic l-form, while the oriented volume of the projection of such a solid onto x 1 , . . . , x k is a k-form. Example A4.2 A symplectic vector space V is endowed with a skew-symmetric linear form ω which is clearly an example of a 2-form.
A4.1 Algebraic forms, differential forms, tensors 717 The set of all the k-forms is a vector space, if we introduce the operations of sum and product with a scalar λ ∈ R:
(ω 1 + ω 2 )(v
1 , . . . , v k ) = ω
1 (v 1 , . . . , v k ) + ω 2 (v 1 , . . . , v k ), (λω)(v 1 , . . . , v k ) = λω(v
1 , . . . , v k ).
We denote this space by Λ k (V ). D efinition A4.3 Let α ∈ Λ r , β ∈ Λ s . The exterior product of α and β, denoted by α
∧ β, is the (r + s)-form given by (α ∧ β)(v 1 , . . . , v r +s
σ ∈P ν(σ)α(v σ 1 , . . . , v σ r )β(v σ r+1
, . . . , v σ r+s ), (A4.10)
where σ = (σ 1 , . . . , σ r +s ), P denotes the set of all possible permutations of (1, . . . , r + s) and ν(σ) = ±1 according to whether σ is even or odd. It is not difficult to check that the exterior product satisfies the following properties: if α ∈ Λ
, β ∈ Λ s and γ
∈ Λ t , we have α ∧ (β ∧ γ) = (α ∧ β) ∧ γ, α ∧ (β + γ) = α ∧ β + α ∧ γ (t = s), α ∧ β = (−1) rs β ∧ α. (A4.11) Hence it is associative, distributive and anticommutative. Example A4.3 Let V = R 2l , ω =
l i =1 e i ∗ ∧ e (i+l)∗
, where (e 1 , . . . , e 2l ) denotes the canonical basis of R 2l . It is immediate to check that for every k = 1, . . . , l, setting Ω k = ω ∧ . . . ∧ ω (k times), we have Ω k
−1) k − 1 k! 1≤i 1 < ··· k ≤l e i 1 ∗ ∧ . . . ∧ e i k ∗ ∧ e
(i 1 +l)∗ ∧ . . . ∧ e (i k +l)∗ . (A4.12) Example A4.4 Let ω be a 2-form on R 3 . If (e
1 , e
2 , e
3 ) is a basis of R 3 it can be checked that for every v, w ∈ R
3 we have
ω(v, w) = ω(v i e i , w
j e j ) = (v 1 w 2 − v
2 w 1 )ω(e 1 , e 2 ) + (v 2 w 3 − v 3 w 2 )ω(e
2 , e
3 ) + (v
3 w 1 − v 1 w 3 )ω(e
3 , e
1 ) = (ω 12 e 1∗ ∧ e 2∗ + ω 23 e 2∗ ∧ e 3∗ + ω 31 e 3∗ ∧ e 1∗ )(v, w), (A4.13) where clearly ω 12 = ω(e
1 , e
2 ), ω
23 = ω(e
2 , e
3 ) and ω
31 = ω(e
3 , e
1 ). Therefore dim Λ
(R 3 ) = 3. 718 Algebraic forms, differential forms, tensors A4.1 Example A4.5 Let V = R 3 . Because of the Euclidean space structure of R 3 , we can associate with each vector in R 3 , a 1-form ϑ v and a 2-form ω v by setting ϑ v (w) = v · w, ω v (w 1 , w
2 ) = v
· w 1 × w 2 , where, as usual, w 1 × w
2 denotes the vector product of w 1 and w
2 . We can
check then that, for a fixed orthonormal basis (e 1 , e 2 , e
3 ) of R
3 , we have ϑ v
1 e 1∗ + v 2 e 2∗ + v
3 e 3∗ , ω v = v 1 e 2∗ ∧ e
3∗ + v
2 e 3∗ ∧ e 1∗ + v 3 e 1∗ ∧ e 2∗ . T heorem A4.1 Let (e 1 , . . . , e l ) be a basis of V . A basis of Λ k
⎧ ⎨ ⎩ 1≤i 1
2
··· k ≤l e i 1 ∗ ∧ . . . ∧ e i k ∗ ⎫ ⎬ ⎭ . Therefore dim Λ k (V ) = l k and every k-form α can be uniquely expressed as follows: α =
1≤i 1
2
··· k ≤l α i 1 ...i k e i 1 ∗ ∧ . . . ∧ e i k ∗ , (A4.14)
where α i 1 ...i
k = α(e
i 1 , . . . , e i k ). (A4.15) The proof of this theorem is a good exercise, that we leave to the reader. Such k-forms have additional transformation properties under changes of basis, or under the action of a linear map. These properties generalise the properties of covectors. If (e 1 , . . . , e l ) and (e
1 , . . . , e l ) are two bases of V , M is the matrix of the change of basis, and A is given by (A4.5), for every k-form we have the representations ω = ω i
...i k e i 1 ∗ ∧ e i k ∗ = ω i 1 ...i k e i 1 ∗ ∧ e i k ∗ , where ω i 1 ...i
k = M
j 1 i 1 . . . M
j k i k ω j 1 ...j
k . (A4.16) Every linear map f : V → V induces a linear map f ∗ on
k (V ):
(f ∗ (α))(v 1 , . . . , v k ) = α(f (v 1 ), . . . , f (v k )).
(A4.17) If (f
j i ) is the matrix representing f , f (v) = f v i e i = v i f j i e j , if α = α i 1 ,...i
k e i 1 ∗ ∧ e i k ∗ , setting (f ∗ α) = 1≤i
1 <...k ≤l (f ∗ α) i 1 ...i k e i 1 ∗ ∧ . . . ∧ e i k ∗ , (A4.18)
A4.2 Algebraic forms, differential forms, tensors 719 we find
(f ∗ α) i 1 ...i k = f
j 1 i 1 . . . f
j k i k α j 1 ...j
k . (A4.19) Equation (A4.19) is immediately verified, once one shows that f ∗ preserves the exterior product: f ∗ (α ∧ β) = (f ∗ α)
∗ β).
(A4.20) A4.2
Differential forms Let M be a connected differentiable manifold of dimension l. D efinition A4.4 The dual space T ∗ P M of the tangent space T P M to M in P is called the cotangent space to M in P . The elements ϑ ∈ T
∗ P M are called cotangent vectors to M in P . It is possible to identify the tangent vectors with differentiations (along a curve), so that if (x 1 , . . . , x l ) is a local parametrisation of M a basis of T P M is given by ∂/∂x 1 , . . . , ∂/∂x l . In the same way, every cotangent vector is identified with the differential of a function. Therefore a basis of T ∗ P M is given by (dx 1 , . . . , dx l ) and every cotangent vector ϑ ∈ T ∗ P M can be written as ϑ = ϑ
i dx i . (A4.21)
It is immediate to check that, if (x 1 , . . . , x l ) is a different local parametrisation of M in P , setting ϑ = ϑ
i dx i , we have
ϑ i = ϑ j ∂x j ∂x i . Hence the components of a cotangent vector are covariant. D efinition A4.5 We call the cotangent bundle T ∗ M of the manifold M the union of the cotangent spaces to M at all of its points: T ∗ M = P ∈M {P } × T ∗ P M. (A4.22)
Remark A4.1 The cotangent bundle T ∗ M is naturally endowed with the structure of a dif- ferentiable manifold of dimension 2l. If (x 1 , . . . , x l ) is a local parametrisation of M , and (ϑ 1 , . . . , ϑ l ) are the components of a covector with respect to the basis (dx 1 , . . . , dx l ) of T
∗ P M , a local parametrisation of T ∗ M can be obtained by considering (x 1 , . . . , x l , ϑ
1 , . . . , ϑ l ).
If M = R l , T ∗ M R 2l ; if M = T l , T
∗ M T l × R
l .
720 Algebraic forms, differential forms, tensors A4.2 Example A4.7 Let (M, (ds) 2 ) be a Riemannian manifold and V : M → R be a regular function. Consider the Lagrangian L : T M → R, L =
1 2 ds dt 2 − V. (A4.23) If (q
1 , . . . , q l , ˙
q 1 , . . . , ˙ q l ) is a local parametrisation of T M and (ds) 2 = g ij (q) dq
i dq j , (A4.24)
we have L(q, ˙q) = 1 2
ij (q) ˙
q i ˙ q j − V (q). (A4.25) The kinetic moments p 1 , . . . , p l conjugate to (q 1 , . . . , q l ): p i = ∂L ∂ ˙ q i = g ij (q) ˙ q j (A4.26) are covariant and can therefore be considered as the components of a cotangent vector to M at the point with coordinates (q 1 , . . . , q l ). The Hamiltonian of the system H(p, q) = 1 2 g ij (q)p
i p j + V (q), (A4.27)
where g ij (q)g jk (q) = δ
i k , is a regular function defined on the cotangent bundle of M : H : T
∗ M → R. (A4.28) It follows that the Hamiltonian phase space of the system coincides with the cotangent bundle of M . The cotangent bundle T ∗ M is endowed with a natural projection: π : T ∗ M → M, (P, ϑ)
→ P. (A4.29)
Note that π − 1 (P ) = T ∗ P M . D efinition A4.6 The field of cotangent vectors (or differential 1-forms on M) of a manifold M is a section of T ∗ M , i.e. a regular map Θ : M
→ T ∗ M (A4.30) such that π ◦
= id M . (A4.31) A4.2 Algebraic forms, differential forms, tensors 721 If (x
1 , . . . , x l , ϑ
1 , . . . , ϑ l ) is a local parametrisation of T ∗ M ,
Θ can be written in the form Θ (x) = ϑ i (x) dx
i , (A4.32) where the functions ϑ i are regular. Remark A4.2 A field of cotangent vectors Θ can be identified with a regular map Θ on the tangent bundle T M with values in R, linear on each tangent space T P M :
Θ : T M
→ R, (P, v)
→ Θ (P, v) = Θ (P )(v).
(A4.33) If (x
1 , . . . , x l ) is a local parametrisation of M , we have Θ (x, v) = ϑ i (x)v
i . (A4.34) Example A4.8 Let f : N → R be a regular function. The differential of f: df (x) =
∂f ∂x i (x) dx i (A4.35) defines a field of cotangent vectors. It is indeed immediate to verify the covariance of its components: ϑ i
∂f ∂x i = ∂f ∂x j ∂x j ∂x i = ∂x j ∂x i ϑ j . D efinition A4.7 Let P ∈ M and denote by Λ k P (M ) the vector space of algebraic k-forms on T P M . We call a differential k-form on M a regular map Ω : M
→ P ∈M {P } × Λ k P (M ),
(A4.36) that associates with every point P ∈ M a k-form on T P M with a reg- ular dependence on P . The space of differential k-forms M is denoted by Λ
(M ). Remark A4.3 Note that P ∈M {P } × Λ 1 P (M ) = T
∗ M .
In local coordinates (x 1 , . . . , x l ) we have Ω (x
, . . . , x l ) = 1≤i 1
···k
Ω i 1 ...i k (x) dx i 1 ∧ . . . ∧ dx i k , (A4.37) and the
l k functions Ω i 1 ...i k : M → R are regular. 722 Algebraic forms, differential forms, tensors A4.2 Given a function f : M → R, which can be considered as a ‘differential 0-form’, its differential df is a 1-form. This procedure can be generalised to an operation called ‘exterior derivation’ that transforms k-forms into (k + 1)-forms. D efinition A4.8 Let Ω ∈ Λ k (M ). The exterior derivative d Ω ∈
k +1 (M ) is the (k + 1)-form d Ω = l j =1 1≤i 1
k ≤l
Ω i 1 ...i k ∂x j dx j ∧ dx i 1 ∧ . . . ∧ dx i k . (A4.38)
Remark A4.4 It is immediate to check that if Ω ∈
0 (M ), i.e. a function, d Ω is its
differential. T heorem A4.2 (properties of exterior differentiation) (1) If Ω , Ω ∈ Λ k (M ), then d( Ω +
) = d Ω + d Ω . (2) If Ω ∈ Λ k (M ),
Ω ∈ Λ j (M ), then d( Ω ∧
) = d Ω ∧ Ω + (
−1) k Ω ∧ d Ω . (3) For any Ω ∈ Λ k (M ), we have d(d Ω ) = 0.
Proof We leave (1) and (2) as an exercise, and we prove (3). Setting i = (i 1 , . . . , i k ), where 1 ≤ i 1
· · · < i k ≤ l, and dx i = dx
i 1 ∧ . . . ∧ dx i k , by (A4.38) we have d(d Ω ) = d l j =1 i ∂ Ω i ∂x j dx j ∧ dx i = l j,k =1 i ∂ 2 Ω i ∂x j ∂x k dx k ∧ dx
j ∧ dx
i = i j ∂ 2 Ω i ∂x j ∂x k − ∂ 2 Ω i ∂x k ∂x j dx k ∧ dx j ∧ dx
i = 0.
Example A4.9: vector calculus in R 3 If f : R
3 → R is a regular function, its differential df = ∂f
1 dx 1 + ∂f ∂x 2 dx 2 + ∂f ∂x 3 dx 3 is identified with the gradient vector field of f : ∇f =
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