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21 Pages

### diff-manifolds2

Course: MATH 2112, Fall 2009
School: UCF
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Word Count: 1444

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very A brief introduction to differentiable manifolds Tom Carter http://cogs.csustan.edu/~ tom/diff-manifolds Santa Fe Institute Complex Systems Summer School June, 2001 1 Our general topics: Why differentiable manifolds 3 Topological spaces 4 Examples of topological spaces 11 Coordinate systems and manifolds 14 Manifolds 19 References 21 2 Why differentiable manifolds Differentiable manifolds...

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very A brief introduction to differentiable manifolds Tom Carter http://cogs.csustan.edu/~ tom/diff-manifolds Santa Fe Institute Complex Systems Summer School June, 2001 1 Our general topics: Why differentiable manifolds 3 Topological spaces 4 Examples of topological spaces 11 Coordinate systems and manifolds 14 Manifolds 19 References 21 2 Why differentiable manifolds Differentiable manifolds can generally be thought of as a generalization of Rn. They are mathematical objects equipped with smooth (local) coordinate systems. Much of physics can be thought of as having a natural home in differentiable manifolds. A particularly valuable aspect of differentiable manifolds is that unlike traditional flat (Euclidean) Rn, they can have (intrinsic) curvature. 3 Topological spaces We need a way to talk about "nearness" of points in a space, and continuity of functions. We can't (yet) talk about the "distance" between pairs of points or limits of sequences we will use a more abstract approach. We start with: Def.: A topological space (X, T) is a set X together with a topology T on X. A topology on a set X is a collection of subsets of X (that is, T P(X)) satisfying: 1. If G1, G2 T, then G1 G2 T. 2. If {G | J} is any collection of sets in T, then G T. J 3. T, and X T. 4 The sets G T are called open sets in X. A subset F X whose complement is open is called a closed set in X. If A is any subset of a topological space X, then the interior of A, denoted by A, is the union of all open sets contained in A. The closure of A, denoted by A, is the intersection of all closed sets containing A. If x X, then a neighborhood of x is any subset A X with x A. If (X, T ) is a topological space, and A is a subset of X, then the induced or subspace topology TA on A is given by TA = {G A | G T }. It is easy to check that TA actually is a topology on A. With this topology, A is called a subspace of X. 5 Suppose X and Y are topological spaces, and f : X Y . Recall that if V Y , we use the notation f -1(V ) = {x X | f (x) V }. We then have the definition: Def.: A function f : X Y is called continuous if f -1(G) is open in X for every open set G in Y . We can also define continuity at a point. Suppose f : X Y , x X, and y = f (x). We say that f is continuous at x if for every neighborhood V of y, there is a neighborhood U of x with f (U ) V . We then say that a function f is continuous if it is continuous at every x X. A homeomorphism from a topological space X to a topological space Y is a 1-1, onto, continuous function f : X Y whose inverse is also continous. 6 A topological space is called separable if there is a countable collection of open sets such that every open set in T can be written as a union of members of the countable collection. A topological space X is called Hausdorff if for every x, y X with x = y, there are neighborhoods U and V of x and y (respectively) with U V = . This is just the barest beginnings of Topology, but it should be enough to get us off the ground . . . 7 Topological spaces - exercises 1. Show that the intersection of a finite number of open sets is open. Give an example to show that the intersection of an infinite number of open sets may not be open. 2. How many distinct topologies are there on a set containing three elements? 3. Show that the interior of a set is open. Show that the closure of a set is closed. Show that A A A. Show that it is possible for A to be empty even when A is not empty. 4. Show that if f : X Y is continuous, and F Y is closed, then f -1(F ) is closed in X. 8 5. Show that a set can be both open and closed. Show that a set can be neither open nor closed. 6. Show that if f : X Y and g : Y Z are both continuous, then g f : X Z is continuous. 7. Show that the two definitions of continuity are equivalent. 8. A subset D X is called dense in X if D = X. Show that it is possible to have a dense subset with D D = . 9. Show that if D is dense in X, then for every open set G X, we have G D = . In particular, every neighborhood of every point in X contains points in D. 9 10. Show that in a Hausdorff space, every set consisting of a single point x (i.e., {x}) is a closed set. 10 Examples of topological spaces For any set X, there are two trivial topologies: Tc = {, X} and Td = P(X). Td is the topology in which each point (considered as a subset) is open (and hence, every subset is open). It is called the discrete topology. Tc is sometimes called the concrete topology. On R, there is the usual topology. We start with open intervals (a, b) = {x | a < x < b}. An open set is then any set which is a union of open intervals. 11 On Rn, there is the usual topology. One way to get this is to begin with the open balls with center a and radius r, where a Rn can be any point in Rn, and r is any positive real number: Bn(a, r) = {x Rn | |x - a| < r}. An open set is then any set which is a union of open balls. 12 Examples of topological spaces - exercises 1. Check that each of the examples actually is a topological space. 2. For k < n, we can consider Rk to be a subset of Rn. Show that the inherited subspace topology is the same as the usual topology. 3. Show that Rn with the usual topology is separable and Hausdorff. 13 Coordinate systems and manifolds Suppose M is a topological space, U is an open subset of M , and : U Rn. Suppose further that (U ) is an open subset of Rn, and that is a homeomorphism between U and (U ). We call a local coordinate system of dimension n on U . For each point m U , we then have that (m) = (1(m), . . . , n(m)), the coordinates of m with respect to . Now suppose that we have another open subset V of M , and is a local coordinate system on V . We say that and are C compatible if the composite functions -1 and -1 are C functions on (U ) (V ). Remember that a function on Rn is C if it is continuous, and all its partial derivatives are also continuous. 14 A topological manifold of dimension n is a separable Hausdorff space M such that every point in M is in the domain of a local coordinate system of dimension n. These spaces are sometimes called locally Euclidean spaces. A C differentiable structure on a topological manifold M is a collection F of local coordinate systems on M such that: 1. The union of the domains of the local coordinate systems is all of M . 2. If 1 and 2 are in F , then 1 and 2 are C compatible. 3. F is maximal with respect to 2. That is, if is C compatible with all F , then F . 15 A C differentiable manifold of dimension n is a topological manifold M of dimension n, together with a C differentiable structure F on M . Notes: 1. It is possible for a topological manifold to have more than one distinct differentiable structures. 2. In this discussion, we have limited ourselves to C differentiable structures. With somewhat more work, we could define C k structures for k < . 3. We have limited the domains of our local coordinate systems to be open subsets of M. This means that the usual spherical and cylindrical coordinate systems on R3 do not count as local coordinate systems by our definition. 16 4. With somewhat more work, we could define differentiable manifolds with boundaries. 5. We have limited ourselves to manifolds of finite dimension. With somewhat more work, we could define infinite dimensional differentiable manifolds. 17 Coordinate systems and manifolds - exercises 1. 18 Manifolds 19 Manifolds - exercises 1. 20 References [1] Auslander, Louis, and MacKenzie, Robert E., Introduction to Differentiable Manifolds, Dover Publications, New York, 1977. [2] Bishop, Richard L., and Goldberg, Samuel I., Tensor Analysis on Manifolds, Dover Publications, New York, 1980. [3] Warner, Frank W., Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, Illinois, 1971. To top 21
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UCF - MATH - 2112
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UCF - MATH - 2112
Econ 102(Random walksand high nance)Tom Carterhttp:/astarte.csustan.edu/ tom/SFI-CSSSFall, 20081Our general topics: Financial ModelingSome random (variable) backgroundWhat is a random walk?Some Intuitive Derivations2Financial Modeling Lets u
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The Logistic Flow(continuous)Tom CarterComplex Systems Summer SchoolJune, 20091Discrete logistic mapWe all know that the discrete logistic map Pn+1 = rPn (1 - Pn ) exhibits interesting behavior of various sorts for various values of the parameter r
UCF - MATH - 2112
Some Fractals and Fractal Dimensions The Cantor set: we take a line segment,and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely. To calculate the fractal / Hausdorff /capacity / box-counti
UCF - MATH - 2112
Some Fractals and Fractal DimensionsThe Cantor set:we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely.To calculate the fractal / Hausdorff /capacity / box-counting
UCF - MATH - 2112
UCF - MATH - 2112
UCF - MATH - 2112
An introduction to information theory and entropyTom Carterhttp:/astarte.csustan.edu/~ tom/SFI-CSSS Complex Systems Summer School Santa FeJune, 20071ContentsMeasuring complexity Some probability ideas Basics of information theory Some entropy theory
UCF - MATH - 2112
UCF - MATH - 2112
What is Interdisciplinary?Discipline (and punish? :-)Physics ChemistryBiologyMathematicsEconomicsPsychologyEtc.Or . . .Physics Chemistry Biology Social SciencesEtc. qOr . . .MathematicsReal WorldBut is this really . . .MathematicsReal Worl
UCF - MATH - 2112
Some Fractals and Fractal Dimensions The Cantor set:we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely. To calculate the fractal / Hausdorff /capacity / box-counti
UCF - MATH - 2112
A brief survey of linear algebraTom Carterhttp:/astarte.csustan.edu/~ tom/linear-algebraSanta Fe Institute Complex Systems Summer SchoolJune, 20011Our general topics: Why linear algebra Vector spaces (ex) Examples of vector spaces (ex) Subspaces (e
UCF - MATH - 2112
The Logistic Flow(Continuous)Tom Carterhttp:/astarte.csustan.edu/ tom/SFI-CSSSComplex Systems Summer SchoolJune, 20081Logistic ow . . .We all know that the discrete logistic mapPn+1 = rPn(1 Pn)exhibits interesting behavior of various sortsfor v
UCF - MATH - 2112
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UCF - MATH - 2112
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AIMS Exercise Set # 1Peter J. Olver1. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest positive number n1 ? The second
UCF - MATH - 5485
AIMS Exercise Set # 2Peter J. Olver1. Explain why the equation e- x = x has a solution on the interval [ 0, 1 ]. Use bisection to find the root to 4 decimal places. Can you prove that there are no other roots? 2. Find 6 3 to 5 decimal places by setting
UCF - MATH - 5485
AIMS Exercise Set # 3Peter J. Olver1. Which of the following matrices are regular? If reguolar, write down its L U 1 -2 3 2 1 0 -1 factorization. (a) , (b) , (c) -2 4 -1 . 1 4 3 -2 3 -1 2 2. In each of the following problems, find the A = L U factorizat
UCF - MATH - 5485
AIMS Exercise Set # 4Peter J. Olver1. Find the explicit formula for the solution to the following linear iterative system: u(k+1) = u(k) - 2 v (k) , v (k+1) = - 2 u(k) + v (k) , u(0) = 1, v (0) = 0.2. Determine whether or not the following matrices are
UCF - MATH - 5485
AIMS Exercise Set # 5Peter J. Olver1. Use the power method to find the dominant eigenvalue and associated 4 1 0 1 -2 0 1 1 4 1 0 eigenvector of the following matrices: (a) -3 -2 0 , (b) . 0 1 4 1 -2 5 4 1 0 1 4 2. Use Newton's Method to find all points
UCF - MATH - 5485
AIMS Exercise Set # 6Peter J. Olver1. Prove that the Midpoint Method (10.58) is a second order method. 2. Consider the initial value problem du = u(1 - u), dt for the logistic differential equation. solution for t &gt; 0. (b) Use the Euler Method with step
UCF - MATH - 5485
AIMS Exercise Set # 7Peter J. Olver1. In this exercise, you are asked to find &quot;one-sided&quot; finite difference formulas for derivatives. These are useful for approximating derivatives of functions at or near the boundary of their domain. (a) Construct a se
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver6. Eigenvalues and Singular ValuesIn this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint, the eigenvectors indicate the directions of pure stretch and t
UCF - MATH - 5485
Chapter 16 Complex AnalysisThe term &quot;complex analysis&quot; refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. On the surface, it may seem that this subject should merely be a simple reworking of standard real v
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver1. Computer ArithmeticThe purpose of computing is insight, not numbers. - R.W. Hamming, [23]The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathemati
UCF - MATH - 5485
Chapter 21 The Calculus of VariationsWe have already had ample opportunity to exploit Nature's propensity to minimize. Minimization principles form one of the most wide-ranging means of formulating mathematical models governing the equilibrium configurat
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver3. Review of Matrix AlgebraVectors and matrices are essential for modern analysis of systems of equations - algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithm
UCF - MATH - 5485
Chapter 13 Fourier AnalysisIn addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An
UCF - MATH - 5485
Chapter 12 Fourier SeriesJust before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver4. Gaussian EliminationIn this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor of one of the all-time mathematical greats - the early nin
UCF - MATH - 5485
Chapter 14 Vibration and Diffusion in OneDimensional MediaIn this chapter, we study the solutions, both analytical and numerical, to the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffusion of thermal
UCF - MATH - 5485
Math 5485 September 11, 2006Homework #1Problems: 1.3 1.4 2.1 1b, 3, 4a,b(single precision only), 9. 1a, 2, 7, 13. 1d, 3, 8, 11, 16a.Due: Wednesday, September 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 20, 2006Homework #2Problems: 2.2 2.3 2.4 1d, 5 (only do parts 1 &amp; 3), 13. 1, 5, 7, 11. 1d, 4, 9, 14a.Due: Wednesday, September 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 27, 2006Homework #3Problems: 2.5 2.6 1d, 6, 11a. 1, 5, 8.Due: Wednesday, October 4 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 October 4, 2006Homework #4Problems: 3.1 3.2 3.5 1, 8, 10, 12b. 7a, 14, 18b. 10a, 11.Due: Friday, October 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover chapters 1, 2, 3. You
UCF - MATH - 5485
Math 5485 October 13, 2006Homework #5Problems: 3.3 3.6 3.7 2a, 3a, b(a), c, 5b, d (also, what is the spectral radius?), 6b, c, 7a, 10. 2,10. 14b, 19.Due: Friday, October 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm:
UCF - MATH - 5485
Math 5485 October 23, 2006Homework #6Problems: 3.7 3.8 5b, 6. 3a, 9, 11 (for 9), 12 (for 9), 13.Due: Monday, November 6 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover sections 1.24, 2.16,
UCF - MATH - 5485
Math 5485 November 15, 2006Homework #7Problems: 3.10 4.1 4.2 4.3 7 (just do Newton's Method), 11b. 2, 11, 14a, 15a. 2, 8, 10. 1b, 5.Due: Monday, November 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 November 27, 2006Homework #8Problems: 4.4 4.5 1ac, 4a, 5b, 8. 6 (ignore the Wilkinson shift), 12 (compare the convergence rate of the direct QR algorithm with that based on tridiagonalization).Due: Monday, December 4 Text: B. Bradie, A Friend
UCF - MATH - 5485
Math 5485 December 4, 2006Homework #9Problems: 5.1 5.2 5.3 5.4 2, 5, 8. 1b, 4, 9, 12. 4, 8, 11. 2, 3, 10 (only uniform and Chebyshev).Due: Wednesday, December 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis. Second Midterm: Friday, De
UCF - MATH - 5485
Chapter 17 Dynamics of Planar MediaIn this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems - massspring chains, circuits, and struc
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver7. Iterative Methods for Linear SystemsLinear iteration coincides with multiplication by successive powers of a matrix; convergence of the iterates depends on the magnitude of its eigenvalues. We discuss in some det
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver5. Inner Products and NormsThe norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number of other important norms that are used in numerical analysis
UCF - MATH - 5485
Chapter 15 The Planar Laplace EquationThe fundamental partial differential equations that govern the equilibrium mechanics of multi-dimensional media are the Laplace equation and its inhomogeneous counterpart, the Poisson equation. The Laplace equation i
UCF - MATH - 5485
Very Basic MATLABPeter J. Olver January, 2009 Matrices: Type your matrix as follows: Use space or , to separate entries, and ; or return after each row. &gt; A = [4 5 6 -9;5 0 -3 6;7 8 5 0; -1 4 5 1] or &gt; A = [4,5,6,-9;5,0,-3,6;7,8,5,0;-1,4,5,1] or &gt; A = [
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver9. Numerical Solution of Algebraic SystemsIn this part, we discuss basic iterative methods for solving systems of algebraic equations. By far the most common is a vector-valued version of Newton's Method, which will
UCF - MATH - 5485
Chapter 19 Nonlinear SystemsNonlinearity is ubiquitous in physical phenomena. Fluid and plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inhe
UCF - MATH - 5485
Chapter 22 Nonlinear Partial Differential EquationsThe ultimate topic to be touched on in this book is the vast and active field of nonlinear partial differential equations. Leaving aside quantum mechanics, which remains to date an inherently linear theo
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver11. Numerical Solution of the Heat and Wave EquationsIn this part, we study numerical solution methodss for the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffus
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver10. Numerical Solution of Ordinary Differential EquationsThis part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver8. Numerical Computation of EigenvaluesIn this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices. Needless to say, we completely avoid trying to solve (or even write down
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver13. Approximation and InterpolationWe will now apply our minimization results to the interpolation and least squares fitting of data and functions.13.1. Least Squares.Linear systems with more equations than unknow
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver2. Numerical Solution of Scalar EquationsMost numerical solution methods are based on some form of iteration. The basic idea is that repeated application of the algorithm will produce closer and closer approximation
UCF - MATH - 5485
Chapter 20 Nonlinear Ordinary Differential EquationsThis chapter is concerned with initial value problems for systems of ordinary differential equations. We have already dealt with the linear case in Chapter 9, and so here our emphasis will be on nonline
UCF - MATH - 5485
Chapter 18 Partial Differential Equations in ThreeDimensional SpaceAt last we have ascended the dimensional ladder to its ultimate rung (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. A
UCF - MATH - 5485
Orthogonal Bases and the QR Algorithmby Peter J. Olver University of Minnesota1. Orthogonal Bases.Throughout, we work in the Euclidean vector space V = R n , the space of column vectors with n real entries. As inner product, we will only use the dot pr
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver14. Finite ElementsIn this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver12. MinimizationIn this part, we will introduce and solve the most basic mathematical optimization problem: minimize a quadratic function depending on several variables. This will require a short introduction to pos