# THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)

Save this PDF as:

Size: px
Start display at page:

Download "THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)"

## Transcription

1 4 Vector fields Last updated: November 26, (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M = M n be a manifold. Consider a vector v T x0 M. Suppose γ : ( ε, ε) M is a curve such that v = dx dt at t = 0. Definition 4.1. For an arbitrary function f : M R the number v f := d dt f(x(t)). (the derivative at t = 0) is called the derivative of f along v. Comparing it with the definition of the differential, we see that v f = df(x)(v). If x 1,..., x n are coordinates near x and v i, i = 1,..., n are the components of v, we have n f v f = x (x) i vi. i=1 Proposition 4.1. The operation v : C (M) R satisfies the following properties: linearity over R and the Leibniz rule Proof. Immediate. v (fg) = v f g(x) + f(x) v g. Note that the map sending a function f C (M) to the number f(x) R is a homomorphism, called the evaluation homomorphism at x. We denote it ev x. There are a fundamental algebraic notions. 1

2 Definition 4.2. For a given homomorphism of algebras α: A 1 linear map D : A 1 A 2 is called an derivation over α if A 2, a D(ab) = D(a) α(b) + α(a) D(b) for all a, b A 1. In the special case of a single algebra A 1 = A 2 = A and α = id (the identity map), a derivation over it is simply called a derivation of the algebra A. Hence for any v T x0 M the operation v : C (M) R is a derivation over the evaluation homomorphism at x 0 M. Remark 4.1. It is possible to consider v on C (V ) for any open set V M s.t. x 0 V, and it is a derivation C (V ) R over ev x0 as well. It turns out that all the derivations of the algebra of functions on a manifold to numbers are of the form v. Let us first explore the case of R n. Theorem 4.1. Let x 0 R n. For an arbitrary derivation D : C (R n ) R over the evaluation homomorphism ev x0 there is a vector v T x0 R n such that D = v. The proof uses the following simple but fundamental statement: Lemma 4.1 (Hadamard s Lemma). For any smooth function f C (R n ) and any x 0 R n there is an expansion f(x) = f(x 0 ) + n (x i x i 0)g i (x) i=1 where g i C (R n ) are some smooth functions. Proof. Consider the segment joining x and x 0 and write f(x) = f(x 0 ) d dt f(x 0 + t(x x 0 )) dt = f(x 0 ) + (x i x i 0) 1 0 f x i (x 0 + t(x x 0 )) dt 2

3 Corollary 4.1. There is an expansion f(x) = f(x 0 ) + n i=1 (x i x i 0) f x i (x 0) + n (x i x i 0)(x j x j 0)g ij (x) (1) i,j=1 where g ij C (R n ). Proof. By iterating the previous expansion we arrive at f(x) = f(x 0 ) + n n (x i x i 0)a i + (x i x i 0)(x j x j 0) g ij (x) i=1 i,j=1 where a i R are numbers and g ij C (R n ), functions. Apply partial derivative at x x i 0 and obtain a i = f (x x i 0 ). Now we can prove the main theorem. Proof of Theorem 4.1. Let D : C (R n ). Consider a point x 0 R n. (We are keeping x as a running point.) Apply D to the expansion (1). First note that derivations kill constants; indeed, D(1) = D(1 1) = D(1) 1+1 D(1) = 2D(1), hence D(1) = 0 and then D(c) = D(c 1) = 0 for any c R. Therefore we obtain D(f) = D(x i ) f x (x 0)+ i i ( ) D(x i )(x j 0 x j 0) g ij (x 0 )+(x i 0 x i 0)D(x j ) g ij (x 0 )+(x i x i 0)(x j x j 0) D(g ij ) = ij for v = (v 1,..., v n ) where v i = D(x i ). i D(x i ) f x i (x 0) = v f Theorem 4.1 can be immediately transferred to open domains of a manifold U M admitting a single coordinate system: any derivation D : C (U) R ev x0, where x 0 U M, is of the form D = v for some v T x0 M if there is a chart φ: V U, where V R n. 3

4 This can be made slightly more formal by introducing the so-called germs of functions at a given point. Consider functions defined on open neighborhoods U M of x 0. A function f defined on U and a function g defined on U are said to be equivalent if there is an open neighborhood U of x 0 such that f U = g U. The equivalence class of a (local) function f defined near x 0 is called its germ. Germs at x make an algebra, notation: F x ; and there an evaluation homomorphism ev x to R. We get Theorem 4.2. The derivations F R over ε = ev x correspondence with the tangent vectors v T x M. are in one-to-one It is common to identify vectors v with the corresponding derivations v. For example, the coordinate basis vectors e i = x are identified with the x i partial derivatives i =. x i Now what about global functions, i.e., the algebra C (M)? Theorem 4.3. Every derivation C (M) R over ev x0 a tangent vector v T x0 M. has the form v for Proof. Note that if a function vanishes on an open neighborhood U of x 0, it belongs to I 2 x 0. Indeed, consider h such that h = 0 on W U and h = 1 on M U. Then hf = f. Note that both f and h vanish at x 0, and f is the product of such functions. Therefore every derivation C (M) R over ev x0 annihilates h. It follows that all such derivations can be restricted to local functions (defined near x 0 ) or to germs at x 0, because the results of their application to global functions depend only on the behaviour of functions near x 0. Now we apply the previous Theorem. 4.2 Commutator of vector fields Everything what is said below applies to vector fields defined not necessarily on the whole manifold M, but on an open subset U M. We shall speak of vector fields on M for the simplicity of notation. Recall that a tangent vector v at a point x M defines a linear map v : C (M) R satisfying v (fg) = v f g(x) + f(x) v g 4

5 for all f, g C (M), and is frequently identified with this map. (Such linear maps C (M) R are called derivations at x.) Shortly: vectors at a given point are identified with derivations at this point. Now, if we consider a vector field instead of a single vector, that means that we allow x in the formula above to vary. For a vector field v X(M), we arrive at the linear map v : C (M) C (M) of the algebra C (M) to itself satisfying v (fg) = v f g + f v g. Such linear operators on algebras are known as derivations of algebras. Hence every vector field on M defines a derivation of the algebra C (M). Written in local coordinates, v = v i (x) x i if v = v i (x) x x. i Very frequently vector fields and the corresponding derivations are identified: v v, and one writes v = v i (x) x. i There is a convenient abbreviation i =, so one can write v = v i x i i. It is an important fact that not only we can identify vector fields on M with the corresponding derivations of the algebra of functions C (M), but all the derivations of it arise in this way: every derivation D of the algebra C (M) has the form D = v for some vector field v X(M). Recall the following notion: for two linear operators A and B on a vector space V, their commutator is denoted by [A, B] and defined as [A, B] = AB BA. (Here the product of operators is the composition: AB = A B.) Theorem 4.4. The commutator of vector fields considered as linear operators on the algebra of functions is again a vector field. Proof. Calculation in coordinates. Suppose we are given X, Y X(M) and in some local coordinates X = X i i, Y = Y i i. For an arbitrary function f we have X(Y f) = X i i (Y j j f) = X i i Y j j f + X i Y j 2 ijf 5

6 (where we identify vector fields with operators on functions). Here 2 ijf = 2 f x i x j. In the same way we have Y (Xf) = Y i i X j j f + Y i X j 2 ijf. We can rename indices in the second term to get Y (Xf) = Y i i X j j f + Y j X i 2 jif = Y i i X j j f + Y j X i 2 ijf, where we have used the commutativity of the second partial derivatives. Therefore we obtain (XY Y X)f = X(Y f) Y (Xf) = X i i Y j j f Y i i X j j f = (X i i Y j Y i i X j ) j f. We conclude that the commutator of vector fields X and Y is indeed a vector field given by ) [X, Y ] = (X i i Y j Y i i X j j (2) in local coordinates. We have obtained a binary operation on vector fields, called their commutator or (sometimes) the Lie bracket. An explicit formula is coordinates is equation (2). The following form of this formula is convenient for practical calculations: [X, Y ] = X(Y j ) j Y (X j ) j. (3) (It has the appearance of taking the derivative of the vector field Y along the vector field X minus the same with X and Y swapped. Here taking the derivative of one vector field along the other means taking the derivative of the components. Unfortunately, this makes sense only in a fixed coordinate system. However, the difference, which is the commutator, makes good sense independent of coordinates.) Remark 4.2. As one can see from the calculation, the product (composition) XY of two vector fields X, Y as operators on functions is no longer a vector field. (It is a differential operator of the second order.) Hence the significance of the fact that the commutator [X, Y ] is a vector field. Theorem 4.5. The commutator of vector fields has the following properties: 6

7 1. bilinearity over numbers: [cx, Y ] = c[x, Y ], [X + Y, Z] = [X, Z] + [Y, Z], (and the same w.r.t. the second argument); 2. antisymmetry: [X, Y ] = [Y, X] ; 3. Jacobi identity: for all X, Y, Z X(M) and c R. [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0, Proof. Bilinearity and antisymmetry are obvious from the definition. Let us check the Jacobi identity. By expanding the commutators we have [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = as claimed. X(Y Z ZY ) (Y Z ZY )X + Z(XY Y X) (XY Y X)Z + Y (ZX XZ) (ZX XZ)Y = XY Z XZY Y ZX + ZY X + ZXY ZY X XY Z + Y XZ + Y ZX Y XZ ZXY + XZY = 0 Definition 4.3. A vector space endowed with a bilinear antisymmetric operator satisfying the Jacobi identity is called a Lie algebra. Therefore the space X(M) with the operation of commutator is a Lie algebra. Lie algebras play fundamental role in many areas of mathematics. Typical examples of Lie algebras, besides the Lie algebras of vector fields X(M), include various matrix Lie algebras, where the operation is the matrix commutator. See examples below. Remark 4.3. Vector fields can also be multiplied by functions, but the commutator of vector fields is not bilinear w.r.t. this multiplication (one cannot take functions out). Check that [X, fy ] = f[x, Y ] + (Xf) Y, where Xf = X f = X i i f is the extra term. 7

8 Our proof of Theorem 4.4 was based on a direct coordinate calculation. A more elucidating approach is possible. For an arbitrary algebra A consider the space of all derivations of A, i.e., the linear operators D : A A satisfying the Leibniz identity D(ab) = D(a) b + a D(b) for all elements a, b A. It is denoted Der A. One can consider commutators of derivations (as linear operators). Theorem 4.6. The commutator of derivations is a derivation. Therefore the space Der A is a Lie algebra w.r.t. the commutator. Proof. Consider two derivations D 1 and D 2. We have D 1 (D 2 (ab)) = D 1 ( D2 (a) b + a D 2 (b) ) = D 1 (D 2 (a)) b + D 2 (a) D 1 (b)+ and similarly D 1 (a) D 2 (b) + a D 1 (D 2 (b)) D 2 (D 1 (ab)) = D 2 (D 1 (a)) b + D 1 (a) D 2 (b) + D 2 (a) D 1 (b) + a D 2 (D 1 (b)). After subtracting, the cross terms such as D 1 (a) D 2 (b) are cancelled and we arrive at [D 1, D 2 ](ab) = D 1 (D 2 (ab)) D 2 (D 1 (ab)) = D 1 (D 2 (a)) b D 2 (D 1 (a)) b + a D 1 (D 2 (b)) a D 2 (D 1 (b)) = Hence [D 1, D 2 ] is a derivation as claimed. [D 1, D 2 ](a) b + a [D 1, D 2 ](b). Now we see that Theorem 4.4 follows from Theorem 4.6 if we identify the space X(M) with Der A for the algebra A = C (M). Example 4.1. Check that the following subspaces of the space of all n n matrices are closed under the commutator: the space of all antisymmetric matrices; the space of all trace-free matrices; 8

9 (for matrices with complex entries) the space of all anti-hermitian matrices, i.e., satisfying A T = A; the space of all upper-triangular matrices (unlike the previous examples, it is already closed under the matrix product). Therefore all these spaces give examples of Lie algebras. Example 4.2. The ordinary Euclidean three-space becomes a Lie algebra w.r.t. the operation of the cross-product (or vector product) of vectors: (u, v) u v, defined, e.g., via a determinant. It is not easy to establish the Jacobi identity directly. However, one can check that the following linear transformation 0 u 3 u 2 u = (u 1, u 2, u 3 ) u 0 u 1 u 2 u 1 0 maps the vector product on R 3 to the commutator of matrices (it is sufficient to check this for the standard basis vectors i, j, k), and the Jacobi identity follows. The Jacobi identity for the matrix Lie algebras is checked in the same way as we did for vector fields. Since the check uses nothing but the associativity of the composition, it carries through to arbitrary associative algebras: each such an algebra gives rise to a Lie algebra by setting [a, b] = ab ba. 4.3 Flow of a vector field Vector fields on a manifold have the following geometrical interpretation. We have a manifold (or its open subset) and to each point of it a tangent vector is attached. If we imagine our manifold as sitting in some large R N, we have a picture of tangent arrows at each point. Interpreting a tangent vector as the velocity of a curve, we have a flow of some fluid flowing on our manifold so that the velocity of the flow at each point is given by the vector attached to that point. (In particular, this is a stationary flow, in the sense that the velocities of particles traveling through any given point are the same and do not depend on the time, so the velocity is a function of a point only.) Such a hydrodynamic interpretation of vector fields as the velocity fields is very 9

10 important. Now we shall elaborate it and in particular define the flow of a vector field as a precise mathematical notion. If u = u(x) is a vector field on M, we can associate with it the following ordinary differential equation on M, which will become a system of ODEs when written in coordinates: dx dt = u(x). (4) Here x = x(t) and the parameter t (the time) runs over some interval, e.g., t ( ε, ε). In coordinates, dx i dt = ui (x), (5) where i = 1,..., n. This is a system of non-linear (in general) ordinary differential equations, with the RHS not depending on the time explicitly. (Such systems are called autonomous.) We shall make use of the two main facts concerning such systems. Firstly, if an initial value x(t 0 ) = x 0 is fixed for some moment of time t 0, there is a unique solution x = x(t) with this initial value for t in some interval around t = t 0 ( the existence and uniqueness theorem for solutions of ODEs ). The solutions x = x(t) are called the trajectories or the integral curves of the vector field u. Secondly, for this unique solution of our ODE with a given initial value x(t 0 ) = x 0, it depends smoothly on x 0 M ( the smooth dependence on initial value theorem ). It follows that we have a well-defined smooth map g t : M M, g t : x 0 g t (x 0 ) = x(t), for each t, where x(t) is the solution of (4) with the initial value x(0) = x 0. Here t belongs to some interval around zero, depending, in principle, on x 0. To avoid complicated notation we shall neglect this fact and write all the formulas as if t can take any value. The uniqueness of solution implies that the family of maps g t : M M has the following properties: g 0 = id (6) (indeed, this simply restates that g 0 (x 0 ) = x(0) = x 0 for any x 0 M taken as the initial value for x(t) at t = 0); g t = g 1 t (7) 10

11 (indeed, this simply says that if we travel along a trajectory from any given point backward in time for the time interval t and then take the result as an initial value and travel in the forward direction for the same t, we shall return to the original point; and the same if we do it other way round: first moving forward and then, back); g t+s = g t g s (8) (this means that, starting from any point, if we travel along a trajectory for the time interval s and then, t, it is the same as to travel for the interval t + s; again follows from the uniqueness of solution). Any family of transformations of a manifold M satisfying (6), (7), (8), which are automatically invertible due to (7), is called a one-parameter group of transformations (or diffeomorphisms) of M or, shortly, a flow on M. We see that any vector field on M gives rise to a flow, which is called the flow of a vector field (or: generated by a vector field), with the vector field called as the generator of a flow. Finding the flow for a given vector field u is the same as solving equation (4) for all initial values. Example 4.3. Let M = R n and u(x) = a (a constant vector). Then (4) will be dx dt = a, which has the general solution x = at + x 0, where x 0 = x(0). Hence the flow g t consists of parallel shifts of all points in the direction of the constant vector a: g t : x x + at for any t R. The trajectories are the straight lines parallel to a. Example 4.4. Let M = R 2 and suppose a vector field X is given in Cartesian coordinates as X = ye x + xe y. To find its flow we have to solve the system { x = y y = x (the time derivative denoted by the dot). This can be written in the matrix form as ( ) ( ) ( ) d x 0 1 x =. dt y 1 0 y 11

12 The solution is given by the matrix exponential: where A = x = e At x 0 ( ) Since in our case A 2 = E, A 3 = A, A 4 = E, etc. (E is the identity matrix), we have ( ) cos t sin t e At =. sin t cos t Therefore the flow g t : R 2 R 2 is ( ) ( ) ( ) x cos t sin t x g t :, y sin t cos t y i.e., is the rotations around the origin through angle t. The trajectories are the circles with the center at the origin, and the origin itself (the whole trajectory is one point). 12

### Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.

5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that

### 1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

### CALCULUS ON MANIFOLDS

CALCULUS ON MANIFOLDS 1. Introduction. Crash course on the Multivariate Calculus 1.1. Linear algebra. Field of real numbers R and its completeness. Real line R 1 and real spaces R n. Linear functionals,

### 7. Baker-Campbell-Hausdorff formula

7. Baker-Campbell-Hausdorff formula 7.1. Formulation. Let G GL(n,R) be a matrix Lie group and let g = Lie(G). The exponential map is an analytic diffeomorphim of a neighborhood of 0 in g with a neighborhood

### 5 Constructions of connections

[under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M

### 1 Introduction: connections and fiber bundles

[under construction] 1 Introduction: connections and fiber bundles Two main concepts of differential geometry are those of a covariant derivative and of a fiber bundle (in particular, a vector bundle).

### with a given direct sum decomposition into even and odd pieces, and a map which is bilinear, satisfies the associative law for multiplication, and

Chapter 2 Rules of calculus. 2.1 Superalgebras. A (commutative associative) superalgebra is a vector space A = A even A odd with a given direct sum decomposition into even and odd pieces, and a map A A

### 2 Constructions of manifolds. (Solutions)

2 Constructions of manifolds. (Solutions) Last updated: February 16, 2012. Problem 1. The state of a double pendulum is entirely defined by the positions of the moving ends of the two simple pendula of

### A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström

A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal

### Chap. 1. Some Differential Geometric Tools

Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U

### z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.

Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These

### Differentiation. f(x + h) f(x) Lh = L.

Analysis in R n Math 204, Section 30 Winter Quarter 2008 Paul Sally, e-mail: sally@math.uchicago.edu John Boller, e-mail: boller@math.uchicago.edu website: http://www.math.uchicago.edu/ boller/m203 Differentiation

### Formal Groups. Niki Myrto Mavraki

Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

### CHAPTER 7. Connectedness

CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

### Lecture 8. Connections

Lecture 8. Connections This lecture introduces connections, which are the machinery required to allow differentiation of vector fields. 8.1 Differentiating vector fields. The idea of differentiating vector

### Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds

MA 755 Fall 05. Notes #1. I. Kogan. Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds Definition 1 An n-dimensional C k -differentiable manifold

### The prototypes of smooth manifolds

The prototypes of smooth manifolds The prototype smooth manifolds are the open subsets of R n. If U is an open subset of R n, a smooth map from U to R m is an m-tuple of real valued functions (f 1, f 2,...,

### TOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology

TOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology DONALD M. DAVIS Abstract. We use ku-cohomology to determine lower bounds for the topological complexity of mod-2 e lens spaces. In the

### are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

### Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups

### Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

### Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

### RIEMANN SURFACES. ω = ( f i (γ(t))γ i (t))dt.

RIEMANN SURFACES 6. Week 7: Differential forms. De Rham complex 6.1. Introduction. The notion of differential form is important for us for various reasons. First of all, one can integrate a k-form along

### Notes on the Riemannian Geometry of Lie Groups

Rose- Hulman Undergraduate Mathematics Journal Notes on the Riemannian Geometry of Lie Groups Michael L. Geis a Volume, Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre

### Lie algebra cohomology

Lie algebra cohomology Relation to the de Rham cohomology of Lie groups Presented by: Gazmend Mavraj (Master Mathematics and Diploma Physics) Supervisor: J-Prof. Dr. Christoph Wockel (Section Algebra and

### THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

### Or, more succinctly, lim

Lecture 7. Functions and Stuff PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 7. Functions and differentiable and analytic manifolds. The implicit function theorem, cotangent spaces

### Chern forms and the Fredholm determinant

CHAPTER 10 Chern forms and the Fredholm determinant Lecture 10: 20 October, 2005 I showed in the lecture before last that the topological group G = G (Y ;E) for any compact manifold of positive dimension,

### Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

### Terse Notes on Riemannian Geometry

Terse Notes on Riemannian Geometry Tom Fletcher January 26, 2010 These notes cover the basics of Riemannian geometry, Lie groups, and symmetric spaces. This is just a listing of the basic definitions and

### Gravitation: Tensor Calculus

An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### LECTURE 8: THE MOMENT MAP

LECTURE 8: THE MOMENT MAP Contents 1. Properties of the moment map 1 2. Existence and Uniqueness of the moment map 4 3. Examples/Exercises of moment maps 7 4. Moment map in gauge theory 9 1. Properties

### DIFFERENTIAL GEOMETRY. LECTURE 12-13,

DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

### Review of Linear Algebra

Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

### Vector Calculus. Lecture Notes

Vector Calculus Lecture Notes Adolfo J. Rumbos c Draft date November 23, 211 2 Contents 1 Motivation for the course 5 2 Euclidean Space 7 2.1 Definition of n Dimensional Euclidean Space........... 7 2.2

### 7 Planar systems of linear ODE

7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

### be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore

### Elementary realization of BRST symmetry and gauge fixing

Elementary realization of BRST symmetry and gauge fixing Martin Rocek, notes by Marcelo Disconzi Abstract This are notes from a talk given at Stony Brook University by Professor PhD Martin Rocek. I tried

### Implicit Functions, Curves and Surfaces

Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### M.6. Rational canonical form

book 2005/3/26 16:06 page 383 #397 M.6. RATIONAL CANONICAL FORM 383 M.6. Rational canonical form In this section we apply the theory of finitely generated modules of a principal ideal domain to study the

### Let V, W be two finite-dimensional vector spaces over R. We are going to define a new vector space V W with two properties:

5 Tensor products We have so far encountered vector fields and the derivatives of smooth functions as analytical objects on manifolds. These are examples of a general class of objects called tensors which

### Introduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim

Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its

### LIE ALGEBRAS AND LIE BRACKETS OF LIE GROUPS MATRIX GROUPS QIZHEN HE

LIE ALGEBRAS AND LIE BRACKETS OF LIE GROUPS MATRIX GROUPS QIZHEN HE Abstract. The goal of this paper is to study Lie groups, specifically matrix groups. We will begin by introducing two examples: GL n

### DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such

### STOKES THEOREM ON MANIFOLDS

STOKES THEOREM ON MANIFOLDS GIDEON DRESDNER Abstract. The generalization of the Fundamental Theorem of Calculus to higher dimensions requires fairly sophisticated geometric and algebraic machinery. In

### Axioms of Kleene Algebra

Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

### Invariant Manifolds of Dynamical Systems and an application to Space Exploration

Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated

### Math 361: Homework 1 Solutions

January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

### After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

### FIXED POINT ITERATIONS

FIXED POINT ITERATIONS MARKUS GRASMAIR 1. Fixed Point Iteration for Non-linear Equations Our goal is the solution of an equation (1) F (x) = 0, where F : R n R n is a continuous vector valued mapping in

### 1. Classifying Spaces. Classifying Spaces

Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.

### Lecture 4 Super Lie groups

Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

### HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS

MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016 2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical

### Extra Problems for Math 2050 Linear Algebra I

Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as

### THE POINCARE-HOPF THEOREM

THE POINCARE-HOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under

### Lagrange Multipliers

Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

### Math 225B: Differential Geometry, Final

Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of

### CHAPTER 7 DIV, GRAD, AND CURL

CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n

### Fuchsian groups. 2.1 Definitions and discreteness

2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

### Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

### LECTURE 8: THE SECTIONAL AND RICCI CURVATURES

LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to

### Lecture 2 Some Sources of Lie Algebras

18.745 Introduction to Lie Algebras September 14, 2010 Lecture 2 Some Sources of Lie Algebras Prof. Victor Kac Scribe: Michael Donovan From Associative Algebras We saw in the previous lecture that we can

### Module 2: First-Order Partial Differential Equations

Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Topological dynamics: basic notions and examples

CHAPTER 9 Topological dynamics: basic notions and examples We introduce the notion of a dynamical system, over a given semigroup S. This is a (compact Hausdorff) topological space on which the semigroup

### 2.3. VECTOR SPACES 25

2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous

### A Non-Topological View of Dcpos as Convergence Spaces

A Non-Topological View of Dcpos as Convergence Spaces Reinhold Heckmann AbsInt Angewandte Informatik GmbH, Stuhlsatzenhausweg 69, D-66123 Saarbrücken, Germany e-mail: heckmann@absint.com Abstract The category

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdor t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the (algebraic)

### Chap. 3. Controlled Systems, Controllability

Chap. 3. Controlled Systems, Controllability 1. Controllability of Linear Systems 1.1. Kalman s Criterion Consider the linear system ẋ = Ax + Bu where x R n : state vector and u R m : input vector. A :

### u n 2 4 u n 36 u n 1, n 1.

Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then

### Vectors, metric and the connection

Vectors, metric and the connection 1 Contravariant and covariant vectors 1.1 Contravariant vectors Imagine a particle moving along some path in the 2-dimensional flat x y plane. Let its trajectory be given

### M4P52 Manifolds, 2016 Problem Sheet 1

Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

### Key words. n-d systems, free directions, restriction to 1-D subspace, intersection ideal.

ALGEBRAIC CHARACTERIZATION OF FREE DIRECTIONS OF SCALAR n-d AUTONOMOUS SYSTEMS DEBASATTAM PAL AND HARISH K PILLAI Abstract In this paper, restriction of scalar n-d systems to 1-D subspaces has been considered

### Binary Operations. Chapter Groupoids, Semigroups, Monoids

36 Chapter 5 Binary Operations In the last lecture, we introduced the residue classes Z n together with their addition and multiplication. We have also shown some properties that these two operations have.

### The first order quasi-linear PDEs

Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.

### Lie Matrix Groups: The Flip Transpose Group

Rose-Hulman Undergraduate Mathematics Journal Volume 16 Issue 1 Article 6 Lie Matrix Groups: The Flip Transpose Group Madeline Christman California Lutheran University Follow this and additional works

### Diagonalisierung. Eigenwerte, Eigenvektoren, Mathematische Methoden der Physik I. Vorlesungsnotizen zu

Eigenwerte, Eigenvektoren, Diagonalisierung Vorlesungsnotizen zu Mathematische Methoden der Physik I J. Mark Heinzle Gravitational Physics, Faculty of Physics University of Vienna Version 5/5/2 2 version

### Chapter 1. Smooth Manifolds

Chapter 1. Smooth Manifolds Theorem 1. [Exercise 1.18] Let M be a topological manifold. Then any two smooth atlases for M determine the same smooth structure if and only if their union is a smooth atlas.

### Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016

Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on

### DIFFERENTIAL GEOMETRY HW 7

DIFFERENTIAL GEOMETRY HW 7 CLAY SHONKWILER 1 Show that within a local coordinate system x 1,..., x n ) on M with coordinate vector fields X 1 / x 1,..., X n / x n, if we pick n 3 smooth real-valued functions

### Complex Differentials and the Stokes, Goursat and Cauchy Theorems

Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the

### Proposition 5. Group composition in G 1 (N) induces the structure of an abelian group on K 1 (X):

2 RICHARD MELROSE 3. Lecture 3: K-groups and loop groups Wednesday, 3 September, 2008 Reconstructed, since I did not really have notes { because I was concentrating too hard on the 3 lectures on blow-up

### Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.

Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over

### BACKGROUND IN SYMPLECTIC GEOMETRY

BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations

### Abstract Vector Spaces and Concrete Examples

LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.

### Math 396. Linear algebra operations on vector bundles

Math 396. Linear algebra operations on vector bundles 1. Motivation Let (X, O) be a C p premanifold with corners, 0 p. We have developed the notion of a C p vector bundle over X as a certain kind of C

### Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### The Integers. Math 3040: Spring Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers Multiplying integers 12

Math 3040: Spring 2011 The Integers Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers 11 4. Multiplying integers 12 Before we begin the mathematics of this section, it is worth

### THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. This is an edited version of a

### Kinematics of fluid motion

Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx

### Lecture 7. Lie brackets and integrability

Lecture 7. Lie brackets and integrability In this lecture we will introduce the Lie bracket of two vector fields, and interpret it in several ways. 7.1 The Lie bracket. Definition 7.1.1 Let X and Y by

### arxiv: v1 [math.sg] 6 Nov 2015

A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one

### Math 396. Quotient spaces

Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition

### Representations of Matrix Lie Algebras

Representations of Matrix Lie Algebras Alex Turzillo REU Apprentice Program, University of Chicago aturzillo@uchicago.edu August 00 Abstract Building upon the concepts of the matrix Lie group and the matrix