Discrete Calculus Brian Hamrick
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Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J
Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J. Keely. All Rights Reserved. Mathematical expressions can be typed online in a number of ways including plain text, ASCII codes, HTML tags, or using an equation editor (see Writing Mathematical Notation Online for overview). If the application in which you are working does not have an equation editor built in, then a common option is to write expressions horizontally in plain text. In doing so you have to format the expressions very carefully using appropriately placed parentheses and accurate notation. This document provides examples and important cautions for writing mathematical expressions in plain text. Section 1. How to Write Exponents Just as on a graphing calculator, when writing in plain text the caret key ^ (above the 6 on a qwerty keyboard) means that an exponent follows. For example x2 would be written as x^2. Example 1a. 4xy23 would be written as 4 x^2 y^3 or with the multiplication mark as 4*x^2*y^3. Example 1b. With more than one item in the exponent you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x2n must be written as x^(2n) and NOT as x^2n. Writing x^2n means xn2 . Example 1c. When using the quotient rule of exponents you often have to perform subtraction within an exponent. In such cases you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x5 The middle step of ==xx52− 3 must be written as x^(5-2) and NOT as x^5-2 which means x5 − 2 . -
Appendix a Short Course in Taylor Series
Appendix A Short Course in Taylor Series The Taylor series is mainly used for approximating functions when one can identify a small parameter. Expansion techniques are useful for many applications in physics, sometimes in unexpected ways. A.1 Taylor Series Expansions and Approximations In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. A.2 Definition A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x ¼ a is given by; f 00ðÞa f 3ðÞa fxðÞ¼faðÞþf 0ðÞa ðÞþx À a ðÞx À a 2 þ ðÞx À a 3 þÁÁÁ 2! 3! f ðÞn ðÞa þ ðÞx À a n þÁÁÁ ðA:1Þ n! © Springer International Publishing Switzerland 2016 415 B. Zohuri, Directed Energy Weapons, DOI 10.1007/978-3-319-31289-7 416 Appendix A: Short Course in Taylor Series If a ¼ 0, the expansion is known as a Maclaurin Series. Equation A.1 can be written in the more compact sigma notation as follows: X1 f ðÞn ðÞa ðÞx À a n ðA:2Þ n! n¼0 where n ! is mathematical notation for factorial n and f(n)(a) denotes the n th derivation of function f evaluated at the point a. -
How to Write Mathematical Papers
HOW TO WRITE MATHEMATICAL PAPERS BRUCE C. BERNDT 1. THE TITLE The title of your paper should be informative. A title such as “On a conjecture of Daisy Dud” conveys no information, unless the reader knows Daisy Dud and she has made only one conjecture in her lifetime. Generally, titles should have no more than ten words, although, admittedly, I have not followed this advice on several occasions. 2. THE INTRODUCTION The Introduction is the most important part of your paper. Although some mathematicians advise that the Introduction be written last, I advocate that the Introduction be written first. I find that writing the Introduction first helps me to organize my thoughts. However, I return to the Introduction many times while writing the paper, and after I finish the paper, I will read and revise the Introduction several times. Get to the purpose of your paper as soon as possible. Don’t begin with a pile of notation. Even at the risk of being less technical, inform readers of the purpose of your paper as soon as you can. Readers want to know as soon as possible if they are interested in reading your paper or not. If you don’t immediately bring readers to the objective of your paper, you will lose readers who might be interested in your work but, being pressed for time, will move on to other papers or matters because they do not want to read further in your paper. To state your main results precisely, considerable notation and terminology may need to be introduced. -
Discrete Calculus with Cubic Cells on Discrete Manifolds 3
DISCRETE CALCULUS WITH CUBIC CELLS ON DISCRETE MANIFOLDS LEONARDO DE CARLO Abstract. This work is thought as an operative guide to ”discrete exterior calculus” (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on ”cubic” cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered. Keywords: Discrete Calculus, discrete differential geometry AMS 2010 Subject Classification: 97N70 1. Introduction Discrete exterior calculus (DEC) is motivated by potential applications in com- putational methods for field theories (elasticity, fluids, electromagnetism) and in areas of computer vision/graphics, for these applications see for example [DDT15], [DMTS14] or [BSSZ08]. DEC is developing as an alternative approach for com- putational science to the usual discretizing process from the continuous theory. It considers the discrete mesh as the only thing given and develops an entire calcu- lus using only discrete combinatorial and geometric operations. The derivations may require that the objects on the discrete mesh, but not the mesh itself, are interpolated as if they come from a continuous model. Therefore DEC could have interesting applications in fields where there isn’t any continuous underlying struc- ture as Graph theory [GP10] or problems that are inherently discrete since they are defined on a lattice [AO05] and it can stand in its own right as theory that parallels the continuous one. The language of DEC is founded on the concept of discrete differential form, this characteristic allows to preserve in the discrete con- text some of the usual geometric and topological structures of continuous models, arXiv:1906.07054v2 [cs.DM] 8 Jul 2019 in particular the Stokes’theorem dω = ω. -
High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements. -
9.4 Arithmetic Series Notes
9.4 Arithmetic Series Notes PreAP Algebra 2 9.4 Arithmetic Series *Objective: Define arithmetic series and find their sums When you know two terms and the number of terms in a finite arithmetic sequence, you can find the sum of the terms. A series is the indicated sum of the terms of a sequence. A finite series has a first terms and a last term. An infinite series continues without end. Finite Sequence Finite Series 6, 9, 12, 15, 18 6 + 9 + 12 + 15 + 18 = 60 Infinite Sequence Infinite Series 3, 7, 11, 15, ... 3 + 7 + 11 + 15 + ... An arithmetic series is a series whose terms form an arithmetic sequence. When a series has a finite number of terms, you can use a formula involving the first and last term to evaluate the sum. The sum Sn of a finite arithmetic series a1 + a2 + a3 + ... + an is n Sn = /2 (a1 + an) a1 : is the first term an : is the last term (nth term) n : is the number of terms in the series Finding the Sum of a finite arithmetic series Ex1) a. What is the sum of the even integers from 2 to 100 b. what is the sum of the finite arithmetic series: 4 + 9 + 14 + 19 + 24 + ... + 99 c. What is the sum of the finite arithmetic series: 14 + 17 + 20 + 23 + ... + 116? Using the sum of a finite arithmetic series Ex2) A company pays $10,000 bonus to salespeople at the end of their first 50 weeks if they make 10 sales in their first week, and then improve their sales numbers by two each week thereafter. -
Discrete Mechanics and Optimal Control: an Analysis ∗
ESAIM: COCV 17 (2011) 322–352 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2010012 www.esaim-cocv.org DISCRETE MECHANICS AND OPTIMAL CONTROL: AN ANALYSIS ∗ Sina Ober-Blobaum¨ 1, Oliver Junge 2 and Jerrold E. Marsden3 Abstract. The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system’s motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties Rapide Not from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton’s equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated. Mathematics Subject Classification. 49M25, 49N99, 65K10. Received October 8, 2008. Revised September 17, 2009. Highlight Published online March 31, 2010. Introduction In order to solve optimal control problems for mechanical systems, this paper links two important areas of research: optimal control and variational mechanics. The motivation for combining these fields of investigation is twofold. -
Summation and Table of Finite Sums
SUMMATION A!D TABLE OF FI1ITE SUMS by ROBERT DELMER STALLE! A THESIS subnitted to OREGON STATE COLlEGE in partial fulfillment of the requirementh for the degree of MASTER OF ARTS June l94 APPROVED: Professor of Mathematics In Charge of Major Head of Deparent of Mathematics Chairman of School Graduate Committee Dean of the Graduate School ACKOEDGE!'T The writer dshes to eicpreßs his thanks to Dr. W. E. Mime, Head of the Department of Mathenatics, who has been a constant guide and inspiration in the writing of this thesis. TABLE OF CONTENTS I. i Finite calculus analogous to infinitesimal calculus. .. .. a .. .. e s 2 Suniming as the inverse of perfornungA............ 2 Theconstantofsuirrnation......................... 3 31nite calculus as a brancn of niathematics........ 4 Application of finite 5lflITh1tiOfl................... 5 II. LVELOPMENT OF SULTION FORiRLAS.................... 6 ttethods...........................a..........,.... 6 Three genera]. sum formulas........................ 6 III S1ThATION FORMULAS DERIVED B TIlE INVERSION OF A Z FELkTION....,..................,........... 7 s urnmation by parts..................15...... 7 Ratlona]. functions................................ Gamma and related functions........,........... 9 Ecponential and logarithrnic functions...... ... Thigonoretric arÎ hyperbolic functons..........,. J-3 Combinations of elementary functions......,..... 14 IV. SUMUATION BY IfTHODS OF APPDXIMATION..............,. 15 . a a Tewton s formula a a a S a C . a e a a s e a a a a . a a 15 Extensionofpartialsunmation................a... 15 Formulas relating a sum to an ifltegral..a.aaaaaaa. 16 Sumfromeverym'thterm........aa..a..aaa........ 17 V. TABLE OFST.Thß,..,,..,,...,.,,.....,....,,,........... 18 VI. SLThMTION OF A SPECIAL TYPE OF POER SERIES.......... 26 VI BIBLIOGRAPHY. a a a a a a a a a a . a . a a a I a s . -
Discrete Differential Geometry: an Applied Introduction
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane Last updated: April 13, 2020 Contents Chapter 1. Introduction 3 1.1. Disclaimer 6 1.2. Copyright 6 1.3. Acknowledgements 6 Chapter 2. Combinatorial Surfaces 7 2.1. Abstract Simplicial Complex 8 2.2. Anatomy of a Simplicial Complex: Star, Closure, and Link 10 2.3. Simplicial Surfaces 14 2.4. Adjacency Matrices 16 2.5. Halfedge Mesh 17 2.6. Written Exercises 21 2.7. Coding Exercises 26 Chapter 3. A Quick and Dirty Introduction to Differential Geometry 28 3.1. The Geometry of Surfaces 28 3.2. Derivatives and Tangent Vectors 31 3.3. The Geometry of Curves 34 3.4. Curvature of Surfaces 37 3.5. Geometry in Coordinates 41 Chapter 4. A Quick and Dirty Introduction to Exterior Calculus 45 4.1. Exterior Algebra 46 4.2. Examples of Wedge and Star in Rn 52 4.3. Vectors and 1-Forms 54 4.4. Differential Forms and the Wedge Product 58 4.5. Hodge Duality 62 4.6. Differential Operators 67 4.7. Integration and Stokes’ Theorem 73 4.8. Discrete Exterior Calculus 77 Chapter 5. Curvature of Discrete Surfaces 84 5.1. Vector Area 84 5.2. Area Gradient 87 5.3. Volume Gradient 89 5.4. Other Definitions 91 5.5. Gauss-Bonnet 94 5.6. Numerical Tests and Convergence 95 Chapter 6. The Laplacian 100 6.1. Basic Properties 100 6.2. Discretization via FEM 103 6.3. Discretization via DEC 107 1 CONTENTS 2 6.4. Meshes and Matrices 110 6.5. The Poisson Equation 112 6.6. -
List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX. -
MATHEMATICAL INDUCTION SEQUENCES and SERIES
MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES John J O'Connor 2009/10 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 13 Power Series 22 Taylor Series 24 Summary 29 Mathematician's pictures 30 Exercises on these topics are on the following pages: Mathematical Induction 8 Sequences 13 Series 21 Power Series 24 Taylor Series 28 Solutions to the exercises in this booklet are available at the Web-site: www-history.mcs.st-andrews.ac.uk/~john/MISS_solns/ 1 Mathematical induction This is a method of "pulling oneself up by one's bootstraps" and is regarded with suspicion by non-mathematicians. Example Suppose we want to sum an Arithmetic Progression: 1+ 2 + 3 +...+ n = 1 n(n +1). 2 Engineers' induction € Check it for (say) the first few values and then for one larger value — if it works for those it's bound to be OK. Mathematicians are scornful of an argument like this — though notice that if it fails for some value there is no point in going any further. Doing it more carefully: We define a sequence of "propositions" P(1), P(2), ... where P(n) is "1+ 2 + 3 +...+ n = 1 n(n +1)" 2 First we'll prove P(1); this is called "anchoring the induction". Then we will prove that if P(k) is true for some value of k, then so is P(k + 1) ; this is€ c alled "the inductive step". Proof of the method If P(1) is OK, then we can use this to deduce that P(2) is true and then use this to show that P(3) is true and so on.