Mathematical Physics, Lecture 9

Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42

Table of contents 1 Differentiable manifolds 2 Differential maps and curve 3 Tangent, cotangent and tensor spaces 4 Tangent map and submanifolds 5 Differential forms 6 Integration of forms Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 2 / 42

Introduction In this lecture we give a short introduction to differentiable manifolds, differential forms and integration on manifolds. This topics are discussed in chapters 15-17 of the main text book: Szekeres, Peter - A Course In Modern Mathematical Physics - Groups, Hilbert Spaces And Differential Geometry (2004) and in Dariusz Chruscinski, Andrzej Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Progress in Mathematical Physics, Birkhäuser, Berlin 2004. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 3 / 42

Differentiable manifolds Differentiable manifolds Differentiable manifolds A topological manifold M of dimension n = dim M is a Hausdorff space in which every point x has a neighboring homeomorphic to an open subset of R n. A coordinate chart at a point of p of M is the pair (U, φ), where domain of the chart U is open subset of M and φ : U φ(u) R n is a homeomorphism between U and its image φ(u) which is also an open subset of R n. Let pr i : R n R be projection maps. Then the map or functions x = pr i φ : U R for i = 1, 2,..., n are called coordinate functions. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 4 / 42

Differentiable manifolds Differentiable manifolds Differentiable manifolds A family of charts A = {(U α, φ α ) : α A} with the coordinate neighborhoods U α that cover M and any pair of charts from A are C -compatible is called an atlas on M. If A and B are two atlas on M, then so is their union A B. Any atlas could be extended to maximal atlas by adding all charts that are C -compatible with charts of A. The maximal atlas is called a differentiable structure on M. A pair (M, A), M is a topological manifold of dimension n and A is a differential structure on M is called differential manifold. Example R n is a trivial manifold, since the charts (U = R n, φ = id) covers it and generates a unique atlas. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 5 / 42

Differentiable manifolds Differentiable manifolds Differentiable manifolds Example Any open subspace of R n is a differentiable manifold formed by giving it the relative topology and the differentiable structure is generated by the chart (U, id U : U R n ). Let (V, ψ) be a charts on R n. Then every charts on U is the restriction of a coordinate neighborhood and coordinate map on R n to the open region U, that is (U V, ψ U V ). This manifold is called open submanifold of R n. Example The circle S 1 R 2 defined by x 2 + y 2 = 1 is a one dimensional manifold (exercise). S 1 is not homeomorphic to R. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 6 / 42

and φ ± i = U ± i R n defined by Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 7 / 42 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example The sphere S 2 R 3 defined by x 2 + y 2 + z 2 = 1 is a two dimensional differentiable manifold (exercise). Example The n-sphere S n R n+1 defined by S n = {x R n+1 : (x 1 ) 2 + (x 2 ) 2 + + (x n+1 ) 2 = 1} is a n dimensional differentiable manifold. Let U + i = {x S n : x i > 0}, U i = {x S n : x i < 0}

which provides φ : M(n, R) with Hausdorff topology inherited from R n2. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 8 / 42 Differentiable manifolds Differentiable manifolds Differentiable manifolds Example φ ± i (x) = (x 1, x 2,..., x i 1, x i+1,..., x n ). Then a set of charts providing an atlas is the set of rectangular on hemispheres (U + i, φ ± i ) and (U i, φ ± i ). Example There is a one-to-one correspondence between the set of n n real matrices M(n, R) and the points of R n2 through the map φ : M(n, R) R n2 defined by φ(a = (a ij )) = (a 11, a 12,..., a 1n, a 21, a 22,..., a nn )

Differentiable manifolds Differentiable manifolds Differentiable manifolds Example Moreover, the differential structure generated by (M(n, R), φ) converts M(n, R) into a differentiable manifolds of dimension n 2. Let M and N be differential manifolds of dimensions m and n respectively. Then their product M N is also differential manifold (exercise). Example The topological 2-torus T 2 = S 1 S 1 has a differential structure as a n times { }} { product manifold. And in general n-torus T n = S 1 S 1 S 1 is a product of n circles. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 9 / 42

Differential maps and curve Differential maps and curve Differential maps and curve Let M be a differential manifold of dimension n. Then a map f : M R is called differentiable at a point p M if for some coordinate chart (U, φ; x i ) at p the function f = f φ 1 : φ(u) R is differentiable at φ(p) = (x 1 (p), x 2 (p),..., x n (p)) = x(p). The definition is independent of choice of the chart at p. The set of all real- valued functions on M that are differentiable at p M are denoted by F p (M). Let V be an open subset of M. Then a real-valued function f : M R is called differentiable or smooth if it is differentiable at every point p V and it is denoted by F(V ). One can show that F(V ) is both a ring and a real vector space (exercise). Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 10 / 42

Differential maps and curve Differential maps and curve Differential maps and curve If M and N are differential manifolds, then a map α : M N is differentiable at p M if for any pair of coordinate charts (U, φ; x i ) and (V, ψ; y j ) covering p and α(p) resp., its coordinate representation ψ α φ 1 : φ(u) ψ(v ) is differentiable at φ(p). A diffeomorphism is map α : M N that is injective and α and its inverse are differentiable. Two manifolds M and N are said to bee diffeomorphic, M = N if there exists a diffeomorphism α and m = dim M = dim N = n. A smooth parametrized curve on an manifold M is a differentiable map Λ : (a, b) M, where (a, b) R is open interval. The curve is said to pass through p at t = t 0 if Λ(t 0 ) = p for a < t 0 < b. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 11 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors A directional derivative of a differentiable function f : R n R along the curve at x 0 is defined by Xf = df (x(t)) dt t=t0 = dx i (t) f (x) t=t0 dt x i x=x0, where X is a linear differential operator X = dx i (t) t=t0 dt x i x=x 0 X is a real-valued map on the algebra of differentiable function at x 0. The map X : F x0 (R n ) R has the following properties: Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 12 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors 1 It is linear on the space F x0 (R n ): for any pair of functions f and g we have X (af + bg) = axf + bxg, where a, b R. 2 It satisfies the Leibnitz rule X (fg) = f (x 0 )Xg + g(x 0 )Xf. A tangent vector X p at any point p of a differential manifold M is a linear map X p : F x0 (R n ) R that satisfies 1 Linearity: X p (af + bg) = ax p f + bx p g, where a, b R. 2 Leibnitz rule: X p (fg) = f (p)x p g + g(p)x p f. The set of tangent vector at p form a vector space T p (M) called tangent space at p. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 13 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors If (U, φ) is any chart at p with coordinate functions x i, then the operators defined by ( x i ) p = x i p : F p (M) R ( x i ) p f = x i pf = f (x 1, x 2,..., x n ) x i x=φ(p), where f = f φ 1 : R n R. Thus any linear combination X p = X i x i p = n i=1 X i x i p, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 14 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tangent vectors where X i R is a tangent vector. The coefficient X j is computed by the action of X on coordinate functions x j X p x j = X i x j x i x=φ(p) = X i δ j i = X j. Theorem If (U, φ; x i ) is a chart at p M, then the operators ( x i ) p defined by ( x i ) p f = x i p f = f (x 1,x 2,...,x n ) x i x=φ(p), form a basis of tangent space T p M and its dimension is n = dim M. For the proof see page 431-432. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 15 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Cotangent and tensor spaces The cotangent space at p is the dual space T p (M) associated to the tangent space at p M and it consists of all linear functionals on T p (M), also called covectors or 1-forms at p. The action of covector ω p at p on tangent vector X p is denoted by ω p (X p ), ω p, X p or X p, ω p. Note also that dim T p (M) = dim T p (M) = dim M. If f is differentiable function at p, then its differential at p is defined by (df ) p which acts on tangent vector X p as follows (df ) p, X p = X p f. For a chart (U, φ; x i ) at p, the differential of the coordinate functions have the following property (dx i ) p, X p = X p x i = X i, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 16 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Cotangent and tensor spaces where X i are components of X p = X i ( x i ) p. Applying (dx i ) p to ( x j ) p we get (dx i ) p, ( x j ) p = x j px i = x i x j φ(p) = δj i. Thus the linear functional (dx 1 ) p, (dx 2 ) p,..., (dx n ) p are the dual basis that span cotangent space. Every covector ω p has a unique expansion where w i = ω p, ( x i ) p. ω p = w i (dx i ) p, Vector and tensor field A vector field X is an assignment of tangent vector X p at each point Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 17 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Vector and tensor field p M. Or X is a map from M to the set p M T p(m) with the property that the image of every point, X (p) belong to the tangent space T p (M) at p. The vector field is called differentiable or smooth if for every differentiable function f F(M) the function Xf defined by defined by (Xf )(p) = X p f is differentiable, that is f F(M) = Xf F(M). The set of all differentiable vector fields on M will be denoted by T (M). Every smooth vector field defines a map X : F(M) F(M) which satisfies the following properties Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 18 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Vector and tensor field 1 Linearity: X (af + bg) = axf + bxg, where a, b R and f, g F(M). 2 Leibnitz rule: X (fg) = fxg + gxf. Conversely any map X with above properties defines a smooth vector field. The constructions of tangent and cotangent spaces enable we to introduce an arbitrary tensor field on M. Thus we call a smooth map x T (M) T (k,l) x M = T x M T x M T x M T x M, a tensor field of type (k, l). A vector field is a tensor field of type (1, 0). Any tensor field is uniquely defined by its components T = i 1i 2 i k j 1 j 2 j l x i 1 x i 2 x i k dx j 1 dx j l. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 19 / 42

Tangent, cotangent and tensor spaces Tangent, cotangent and tensor spaces Tensor bundles The tangent bundle on a manifold is defined by TM = T p (M). p M And there is a natural projection map π : TM M defined by π(u) = p for u T p (M). Moreover, for chart (U, φ; x i ) on M we define a chart (π 1 (U), φ) on TM where the coordinate map φ : π 1 (U) R 2n is defined by φ(v) = (x 1 (p),..., x n (p), v 1,..., v n ), where p = π(v) and v = n i=1 v i x i p. The cotangent bundle Tp (M) is defined in similar manner. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 20 / 42

Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Tangent map Let α : M N be a differentiable map. Then α induces a map α : T p (M) T α(p) (N) called the tangent map of α, where the tangent vector Y α(p) = α X p is defined by Y α(p) f = (α X p )f = X p (f α) for any function f F α(p) (N). Pullback of a map The map α : M N also induces a map Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 21 / 42

Submanifolds Let α : M N be a differentiable map such that m = dim M n = dim N. Then the map α is called immersion if the tangent map Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 22 / 42 Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Pullback of a map α : T α(p) (N) T p (M) between cotangent spaces which is called the pullback induced by α. The pullback of a 1-form ω α(p) is defined by for arbitrary tangent vectors X p. α ω α(p), X p = ω α(p), α X p

Tangent map and submanifolds Tangent map and submanifolds The tangent map and pullback of a map Submanifolds α : T p (M) T α(p) (N) is injective at every point p M, e.g., α is non-degenerate linear map everywhere. If the map α and the tangent map α are injective, then the map is called an embedding and the pair (M, α) are called an embedded submanifold of N. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 23 / 42

Differential forms Differential forms Differential forms Next we will review a class of tensor fields on differential manifold that play an important role in physical applications. A skew-symmetric tensor of type (0, k) is called a differential form of order k or a k-form. Let Λ k (M) be the space of k-form on M with Λ k (M) = { } for k > n. Then, the space of differential forms on M is defined by Λ k (M) = n Λ k (M) with Λ 0 (M) = C (M). The space Λ(M) equipped with the following operations: k=0 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 24 / 42

Differential forms Differential forms Differential forms 1 A wedge product: : Λ k (M) Λ l (M) Λ k+l (M). 2 The exterior derivative: d : Λ k (M) Λ k+1 (M) defined by dα = 1 k! α i1 i k x j dx j dx i 1 dx i k. Example A function on a differential manifold M is a zero-form which has the following one form as its exterior derivative df = f dx i. note also that x i d(df ) = 0. If M = R n and (x 1, x 2,..., x n ) are cartesian coordinates, then df gives the components of gradf. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 25 / 42

Differential forms Differential forms Differential forms Proposition The exterior derivative satisfies the following properties d 2 α = d(dα) = 0 for any α Λ(M). d(α β) = dα β + ( 1) k α dβ, for α, β Λ(M). Proposition The pullback operation commutes with exterior derivative and wedge product, that is φ (dα) = d(φ α), φ (α β) = φ α φ β for any differential form α and β on N. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 26 / 42

Integration of forms Integration of forms Integration of forms Let f : R n R be a function and (x 1,..., x n ) be cartesian coordinates of R n. Then, an n-dimensional integral id defined by f fdv = f (x 1,..., x n )dx 1 dx n. After a coordinate transformation we have =J { ( }} ) { x d x 1 d x n i = det x j dx 1 dx n, where J is the Jacobian of transformation. An n-dimensional manifold M is orientable if and only if there exists a nowhere vanishing n-form on it. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 27 / 42

Integration of forms Integration of forms Example A Riemannian manifold (M, g) is differentiable manifold M equipped with a smooth metric tensor g of type (0, 2) such that g is symmetric and for each x M, the bilinear form g x : T x M T x M R is nondegenerate. We call a Riemannian manifold proper if g x (v, v) > 0 v T x M, v 0. Otherwise a manifold is called pseodu-riemannian. The n dimensional Euclidean space R n is a proper Riemannian manifold and the Minkowski space R 1,3 is pseudo-riemannian. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 28 / 42

Integration of forms Integration of forms Hodge star Consider the space of differential forms on M with n = dim M. The so called Hodge operation of Hodge star : Λ k (M) Λ n k (M) defined by ( α) i1 i n k = 1 k! g ɛi1 i n k j 1 j k α j 1 j k, where g = det(g ij ) and α j 1 j k = g j 1m1 g j km k α m1 m k. The form α is called the Hodge dual of α and g il g lj = δj i. Let τ = g dx 1 dx n. Then if M is compact we define volume of M by Vol(M) = τ. M Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 29 / 42

Integration of forms Integration of forms Hodge star Let K be a k dimensional orientable submanifold of M and j : K M be a canonical embedding. If (y 1,..., y k ) are local coordinates on K then j is defined by x 1 = x 1 (y 1,..., y k ) x 2 = x 2 (y 1,..., y k )... x n = x n (y 1,..., y k ) If α is a k-form on M, then j α is a k-form on K. Thus we can define an integral of K over j α (K, α) = j α K = 1 k! x i 1 ik x α i1 i k y 1 y k dy 1 dy k, Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 30 / 42

Integration of forms Integration of forms Hodge star where α = 1 k! α i 1 i k dx i 1 dx i k. Let R n + = {(x 1,..., x n ) R n : x 1 0}. The we define the boundary of R n + by R n + = {(x 1,..., x n ) R n : x 1 = 0}. We call M a manifold with a boundary when there exists an open covering (U i, φ i ) such that φ i (U i ) defines an open subset of R n + and the boundary M is defined by M = i φ 1 i ( R n +). Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 31 / 42

Integration of forms Integration of forms Hodge star Example As an example we consider the unit ball B n in R n B n = {(x 1,..., x n ) R n : n (x i ) 2 1}. i=1 The boundary of B n is a (n 1)-dimensional sphere B n = S n 1. Note also that for any manifold M = ( M) = { } and in particular S n = ( B n+1 ) = {}. A manifold without boundary is called a closed manifold. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 32 / 42

Integration of forms Integration of forms Hodge star Stokes theorem Let M be a n dimensional manifold with boundary M and let ω Λ n 1 (M). The we have dω = ω. M M Example Let Σ be two-dimensional surface in R 3 and A be a vector field. Then we have curla ds = A dl, where ds denotes a surface elements on σ. Σ C= Σ Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 33 / 42

Integration of forms Integration of forms De Rham cohomology We call a k-form α a closed form if dα = 0, and an exact form if α = dβ for any (k 1)-form β. For a manifold M, the set of closed form is defined by Z k = {α Λ k (M) : dα = 0} and the set of exact form is defined by B k = {α Λ k (M) : β Λ k 1 (M), dβ = α}. Next we define the following equivalence relation in Λ k (M) α 1 α 2 β Λ k 1 (M), dβ = α 1 α 2 } Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 34 / 42

Integration of forms Integration of forms De Rham cohomology Thus α 1 α 2 = dα 1 = dα 2 and we can define the space of equivalence classes by H k (M) = Z k (M) B k (M), is called the de Rham cohomology group of M and means that H k (M) is the set of closed forms that differ only by an exact k-form. de Rham cohomology group is an abelian group where the operation is the addition of k-forms. [ω] is the equivalence class containing ω and is called a cohomology class of ω. If [ω 1 ] H k (M) and [ω 2 ] H l (M), then we have Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 35 / 42

defined by φ # ([ω]))[φ ω]. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 36 / 42 Integration of forms Integration of forms De Rham cohomology In R n all closed forms are exact. [ω 1 ω 2 ] = [ω 1 ] [ω 2 ] H k+l (M) Poincare lemma Let M be a differentiable manifold. Then any closed form on M is locally exact, that is if dα = 0, then for any x M, there is a neighborhood U containing x such that α = dβ on U. Thus only global properties ofm decide if de Rham cohomology grops are trivial or not. The smooth map φ : M N induces a linear transformation φ # : H K (M) H k (N)

Integration of forms Integration of forms De Rham cohomology If φ is a homeomorphism, then the induced map φ # is an isomorphism. Thus topologically equivalent manifolds have isomorphic cohomology groups and in particular we have b k (M) = dim H k (M) = dim H k (N) = b k (N), k = 0, 1, 2,..., n are called Betti numbers. From Betti numbers we can construct Euler characteristic n χ(m) = ( 1) k b k (M). k=0 The topological equivalent manifolds have same Euler characteristic. Example Let M = R 2. Then H 1 (R 2 ) = H 2 (R 2 ) = 0, H 0 (R 2 ) = R, and χ(r 2 ) = 1. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 37 / 42

Integration of forms Integration of forms De Rham cohomology Example Let M = S 2. Then H 0 (S 2 ) = H 2 (S 2 ) = R and all other cohomology groups are trivial. We have also χ(s 2 ) = 2. Example Let M = T 2 = S 1 S 1. Then H 0 (T 2 ) = H 2 (T 2 ) = R and all other cohomology groups are trivial (exercise). We have also χ(t 2 ) = 0. Proposition A contractible manifold M by which we mean a manifold that may be continuously contracted to single point has trivial de Rham cohomology groups H k (M) for all k 1. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 38 / 42

Integration of forms Integration of forms De Rham cohomology Proposition Let M be a manifold which is compact, connected, and orientable with n = dim M. Then we have H k (M) = H n k (M), k = 0, 1, 2,..., n. This important result is called the Poincare duality. Lie derivative Let X be a vector field on a manifold M. Then the flow of the X is the collection of maps F t : M M which satisfy d dt F t(x) = X (F t (x)), x X, t R. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 39 / 42

Integration of forms Integration of forms Lie derivative By fixing a point x X we obtain a map t R F t (x) M which defines a curve in M called the integral curve of X passing a point x. The flow F t also satisfies the following property F t F s = F t+s. Example Let A : R n R n be a vector field on R n defined by x Ax T x R n = R n, where x R n and the flow satisfies the following equation d dt F t(x) = AF t (x) = F t (x) = e At x. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 40 / 42

where the Lie bracket is defined by [X, Y ] i = X k k Y i Y k k X i. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 41 / 42 Integration of forms Integration of forms Lie derivative Let T be a tensor field on a manifold M and X be a vector field on M. Then the Lie derivative of T with respect to X is defined by (L X T )(x) = d dt (F t T )(x) t=0, where F t is the flow of vector field X. If f is a function on M then L X f is given by L X f = d dt (F t f )(x) t=0 = d dt (f F tt )(x) t=0 = X (f ), is the directional derivative of f along X. If X and Y are vector fields on M, then L X Y = [X, Y ],

Problems Integration of forms Problem 15.4 This is the first problem of the home assignment three. Problem 16.4 This is the second problem of the home assignment three. Problem 17.7 This is the third problem of the home assignment three. Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 42 / 42