8 Brief introduction to differential forms “Hamiltonian mechanics cannot be understood without differential forms” V. I. Arnold In this chapter we give a very brief introduction to differential forms fol- lowing the chapter 7 of Ref.[26]. We refer the reader to [26, 9, 8] for more detailed (and more precise) introductions. 8.1 Exterior forms 8.1.1 Definitions Let Rn be an n-dimensional real vector space. Definition An exterior algebraic form of degree k (or k-form) is a function of k vectors which is k-linear and antisymmetric. Namely: ω(λ1ξ1! + λ2ξ1!!,ξ2,...,ξk)=λ1ω(ξ1! ,ξ2,...,ξk)+λ2ω(ξ1!!,ξ2,...,ξk), and ω(ξ ,...,ξ )=( 1)νω(ξ ,...,ξ ) i1 ik − 1 k with ( 1)ν = 1 ( 1) if permutation (i , . , i ) is even (odd). Here ξ Rn − − 1 k i ∈ and λ R. i ∈ The set of all k-forms in Rn form a real vector space with (ω + ω )(ξ)=ω (ξ)+ω (ξ),ξ = ξ ,...,ξ 1 2 1 2 { 1 k} and (λω)(ξ)=λω(ξ). Let us suppose that we have a system of linear coordinates x1, . , xn on n R . We can think of these coordinates as of 1-forms so that xi(ξ)=ξi - the i-th coordinate of vector ξ. These coordinates form a basis of 1-forms in the 65 66 Brief introduction to differential forms n 1-form vector space (which is also called the dual space (R )∗). Any 1-form can be written as a linear combination of basis 1-forms ω1 = a1x1 + ... + anxn. 8.1.2 Exterior products Definition An exterior product of k 1-forms ω1,ω2,...,ωk is a k-form defined by (ω ... ω )(ξ ,...,ξ ) = det ω (ξ ). 1 ∧ ∧ k 1 k i j Exterior products of basis 1-forms x ... x with i < . < i form a i1 ∧ ∧ ik 1 k basis in the space of k-forms. The dimension of the latter space is obviously k Cn. A general k-form can be written as ωk = a x ... x , i1...ik i1 ∧ ∧ ik 1 i1<...<i n ≤ ! k≤ where ai1...ik are real numbers. Definition The exterior product ωk ωl of a k-form with l-form on Rn is ∧ the k + l-form on Rn defined as (ωk ωl)(ξ ,...,ξ )= ( 1)νωk(ξ ,...,ξ )ωl(ξ ,...,ξ ), ∧ 1 k+l − i1 ik j1 jl ! where i1 < . < ik and j1, . , jl and the sum is taken over all permutations (i , . , i ,j , . , j ) with ( 1)ν being +1( 1) for even (odd) permutations. 1 k 1 l − − One can check that this definition is consistent with the exterior product of 1-forms defined above and that the exterior product is distributive, asso- ciative, and skew-commutative. The latter means ωk ωl =( 1)klωl ωk. ∧ − ∧ If ω is a 1-form (or a form of an odd degree) one can easily show that ω ω = 0. ∧ 8.1.3 Behavior under mappings Let f : Rm Rn be a linear map, and ωk an exterior k-form on Rn. Then → k m we can define a k-form (f ∗ω ) on R by k k (f ∗ω )(ξ1,...,ξk)=ω (fξ1,...,fξk). Notice that the obtained mapping of forms f ∗ acts in “opposite” direction to f. Namely, f :Ω (Rn) Ω (Rm), where Ω (Rn) is a vector space of ∗ k → k k k-forms on Rn. 8.2 Differential forms 67 8.2 Differential forms Definition A differential k-form ωk at a point x of a manifold M is an |x exterior k-form on the tangent space TMx to M at x, i.e., a k-linear skew- symmetric function of k vectors ξ1,...,ξk tangent to M at x. If such a form is given at every point x of M and if it is differentiable, we say that we are given a k-form ωk on the manifold M. We introduce the coordinate basis 1-forms dxi with i =1, . , n (n- dimension of M). The notation dxi instead of xi used for exterior forms emphasizes that these basis forms act on TMx at a given point x of M. In the neighborhood of the point x one can always write the general differential k-form as ωk = a (x) dx ... dx , (8.1) i1...ik i1 ∧ ∧ ik i1<...<i! k where ai1...ik (x) are smooth functions of x. A simple example of a differential 1-form is the differential of some scalar function f(x) defined on the manifold M. We introduce n ∂f df = dxk, ∂xk x k=1 " ! " " where dxk are basis differential 1-forms. The value of this 1-form on a vector ξ TM is given by ∈ x n ∂f df (ξ)= ξk. ∂xk x k=1 " ! " " 8.2.1 Behavior under mappings Let f : M N be a differentiable map of a smooth manifold M to a → smooth manifold N, and let ω be a differential k-form on N. The mapping f induces the mapping f : TMx TMf(x) of tangent spaces. The latter ∗ → mapping f is called the differential of the map f. The mapping f is a ∗ ∗ mapping of linear spaces and gives rise to the mapping of forms defined on corresponding tangent spaces (see Sec. 8.1.3). As a result a well-defined differential k-form (f ∗ω) exists on M: (f ∗ω)(ξ1,...,ξk)=ω(f ξ1, . , f ξk). (8.2) ∗ ∗ 68 Brief introduction to differential forms 8.3 Integration of differential forms over chains 8.3.1 Integration of k-form over k-dimensional cell k Let D be a bounded convex polyhedron in R and x1, . , xk an oriented coordinate system on Rk. Any differential k-form on Rk can be written as ωk = φ(x) dx ... dx , where φ(x) is a differentiable function on Rk. We 1 ∧ ∧ k define the integral of the form ωk over D as the integral of the function φ: k ω = φ(x) dx1dx2 . dxk. (8.3) #D #D Let us define a k-dimensional cell σ of n-dimensional manifold M as a polyhedron D in Rk defined above with a differentiable map f : D M. → One can think about σ as of “curvilinear polyhedron” - the image of D on M. If ω is a differentiable k-form on M we define the integral of a form ω over the cell σ as ω = f ∗ω, (8.4) #σ #D where f ∗ is a mapping of k-forms induced by f (see Sec. 8.2.1). The cell σ inherits an orientation from the orientation of Rk. The k- dimensional cell which differs from σ only by the choice of orientation is called the negative of σ and is denoted by σ or by ( 1)σ. One can show − − that under a change of orientation the integral changes sign: ω = ω. (8.5) σ − σ #− # 8.3.2 Chains and boundary operator It is convenient to generalize our definition of the integral of a form over cell to the integral over chain. Definition A chain of dimension k on a manifold M consists of a fi- nite collection of k-dimensional oriented cells σ1,...,σr in M and integers m1, . , mr, called multiplicities. A chain is denoted ck = m1σ1 + ... + mrσr. (8.6) One can introduce the structure of a commutative group on a set of k-chains on M with natural definitions of addition of chains ck + bk. The boundary of a convex oriented k-polyhedron D on Rk is the (k 1)- − chain ∂D on Rk defined as ∂D = σi, !i 8.4 Exterior differentiation and Stokes formula 69 where the cells σ the (k 1)-dimensional faces of D with orientations in- i − herited from the orientation of Rk (see [26]). One can easily extend this definition to the definition of the boundary of a cell ∂σ on M and then to the boundary of a chain (8.6) as ∂ck = m1∂σ1 + ... + mr∂σr. (8.7) ∂c is a (k 1)-chain on M. Additionally, we define a 0-chain is a collection k − of points with multiplicities and the boundary of an oriented interval AB' is B A. The boundary of a point is empty. − One can show that the boundary of the boundary of the cell is zero ∂(∂σ) = 0 and therefore ∂(∂ck)=0 for any k-chain ck. We write this property as ∂∂ =0. (8.8) 8.3.3 Integration of k-form over k-chains An integral of a k-form over k-chain (8.6) is defined as k k ω = mi ω . (8.9) ck σi # ! # 8.4 Exterior differentiation and Stokes formula An exterior derivative of the differential k-form ωk is a (k+1)-form Ω= dωk. One can define it using a choice of coordinates on M as Ω= dωk = da dx ... dx (8.10) i1...ik ∧ i1 ∧ ∧ ik if ωk is given by (8.1). Here!da is a 1-form of a differential of a function a(x). One can show that the definition (8.10) does not actually depend on the choice of coordinates (see [26] for more rigorous discussion). We think of the 1-form given by the differential of a scalar function (see Sec. 8.2) as of an external derivative of 0-form (scalar function). It is easy to see from the definition (8.10) that d(dxi)=0. Using this property one can show that generally d(dω)=0, 70 Brief introduction to differential forms where ω is a general differential k-form.