Mathematics Courses Online

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{{}} One of the most familiar entities in maths are numbers. From pre-school to advanced levels, numbers are intimately linked with maths. But what are these numbers mathematically speaking? This little book builds numbers the way mathematicians build them up (at least the way most of them do now). We start from the bare axioms of set theory and develop numbers in the mathematical sense. The reader is assumed to be sufficiently mathematically mature, having been exposed at least to high school maths to understand the arguments that follow. Nothing can be built out of nothing. Our arguments can only proceed from some unproven assumptions. We start off with a list of the axioms of set theory, using these as our "unproven truths". We do not bother to focus much on logic here, rather give an intuitive feel of the axioms, so that our end goal of building numbers is not sidetracked. For those interested in a very logically sound discussion of these axioms (specifically in first order logic) the bibliography mentions books which carry further details. 1. Axiom of extensionality

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Mathematical induction is the process of verifying or proving a mathematical statement is true for all values of n {\displaystyle n} within given parameters. For example: We are asked to prove that f ( n ) {\displaystyle f(n)} is divisible by 4. We can test if it's true by giving n {\displaystyle n} values. So, the first 5 values of n are divisible by 4, but what about all cases? That's where mathematical induction comes in. Mathematical induction is a rigorous process, as such all proofs must have the same general format: There will be four types of mathematical induction you will come across in FP1: Proposition: Prove that  4   |   f ( n ) = 5 n + 8 n + 3 ,  for  n ∈ N + {\displaystyle {\text{Prove that }}4~|~f(n)=5^{n}+8n+3,{\text{ for }}n\in \mathbb {N} ^{+}} Note our parameter, for  n ∈ N + {\displaystyle {\text{for }}n\in \mathbb {N} ^{+}} This means it wants us to prove that it's true for all values of n {\displaystyle n} which belong to the set ( ∈ {\displaystyle \in } ) of positive integers ( N + {\displaystyle \mathbb {N} ^{+}} )

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Ratio, as a basic construct in mathematical language, is used to explore the reason why of the sense of beauty, including shapes and music. Thus, ratio is a human notion of geometry and time, as a part of human perception. The reasoning on ratio is proportion, that two ratios are equivalent but different in size, from which a mathematical theory of ratio was born, and variant mathematical subjects were derived. The concept of ratio did become an important tool in the history of mathematics, especially in differential calculus, accompanied with other mathematical concepts such as divisibility. But the modern definition of ratio is mainly arithmetic, that treats a ratio as a value rather than the ratio itself, thus the meaning and possible uses of ratios are lost. This course tries to study ratios from a historical and methodological respect, to recover the possible uses of ratios in the history, thus we may be able to use similar methods and concepts on new things. That is, ratio will be treated as a conceptual tool that can help, rather than merely an arithmetic entity with a value. Before the formal discussion of ratio, we explore the occurrence of the notion itself, thus we can know where it can be used and the reason why of related talk. Let's regard ratio as an innate functionality of human mind. When we say someone is thin, it means the height is relatively larger than the extent of the person, this is where the notion of ratio functions -- the sense of thin is derived from the function of ratio. Here, the word function means an activity of mind that accepts input (height and extent), performs some process (comparison of height and extent and match of the concept of thin) and produces an output (the utterance of thin).

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This course is about the Math Builder (officially called as Equation Editor) tool in Microsoft Word and Outlook 2007 and higher. It also applies to Microsoft PowerPoint and Excel 2010 and higher. Note that this is a different tool than the legacy tool Equation Editor 3.0 (which is still available on 32-bit Office versions until the January 2018 update[1]) and MathType. Typesetting mathematics on a computer has always been a challenge. The mathematical community almost universally accepts a typesetting language called LaTeX. Math Builder is a much easier to use tool that has less functionality than LaTeX but more than typical document processing. Microsoft call this hybrid language the Office Math Markup Language, or OMML for short. It is an appropriate tool for: Note that Math Builder does not perform any mathematics; it is a tool for displaying it. Pros: Cons:

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This course will present images and animations which illustrate points of physics or mathematics. Anyone is invited to contribute. Gravitation: Sound waves : Flow around a cylinder: Flow around a wing: Venturi Effect: Pi and the relationship between a circle's circumference: Heisenberg indeterminacy: Wave-particle duality: Solitary wavelet or particle? Interference of a particle with itself:

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This course is about the topic of mathematical analysis, particularly in the field of engineering. It attempts to be a companion piece to high-level engineering texts that will rely on a certain fundamental mathematical background among readers. This will build on topics covered in Probability, Algebra, Linear Algebra, Calculus, and Ordinary Differential Equations. Readers of This course are expected to have background knowledge in all those topics. Topics covered will be inter-disciplinary engineering topics, and will be highly mathematical. However, overlap between This course and other mathematics books, except where necessary, will be minimized. This course is intended to accompany a graduate course of study in engineering analysis.

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Operations research or operational research (OR) is an interdisciplinary branch of mathematics which uses methods like mathematical modeling, statistics, and algorithms to arrive at optimal or good decisions in complex problems which are concerned with optimizing the maxima (profit, faster assembly line, greater crop yield, higher bandwidth, etc) or minima (cost loss, lowering of risk, etc) of some objective function. The eventual intention behind using operations research is to elicit a best possible solution to a problem mathematically, which improves or optimizes the performance of the system. This course is intended for both mathematics students and also for those interested in the subject from a management point of view.

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Mathematical methods of Physics is a course on common techniques of applied mathematics that are often used in theoretical physics. It may be accessible to anyone with beginning undergraduate training in mathematics and physics. It is hoped that the course will be useful for anyone wishing to study advanced Physics. Foreword

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Applied Mathematics is the branch of mathematics which deals with applications of mathematics to the real world problems, often from problems stemming from the fields of engineering or theoretical physics. It is differentiated from Pure Mathematics, which deals with more abstract problems. There is also something called Applicable Mathematics, which deals with real world problems which need the techniques and mindset usually used in Pure Mathematics. These distinctions do not really become apparent during school level mathematics. Examples of topics in Applied Mathematics:

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This course aims to function as a comprehensive reference guide to the measures used in descriptive statistics. For a general introduction to statistics, see Statistics instead. It has a section on all the common measures, and they are categorised according to type. Initially we shall concentrate on univariate statistics. Entries should be brief, information-packed and not overly tutorial in nature. Assume a reader which already has a basic knowledge of statistics, and needs detailed information or a refresher. See Template for how an entry should look.

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This course is a transcribed version of Lectures on the Calculus of Variations (the Weierstrassian theory) by Harris Hancock in 1904. The scanned original is available here from Cornell University. PREFACE CHAPTER I: PRESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS. CHAPTER II: EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLICATIONS TO THE CATENARY. CHAPTER III: PROPERTIES OF THE CATENARY. CHAPTER IV: PROPERTIES OF THE FUNCTION F ( x , y , x ′ , y ′ ) {\displaystyle F(x,y,x',y')} . CHAPTER V: THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION. CHAPTER VI: THE FORM OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATION G = 0 {\displaystyle G=0} . CHAPTER VII: REMOVAL OF CERTAIN LIMITATIONS THAT HAVE BEEN MADE. INTEGRATION OF THE DIFFERENTIAL EQUATION G = 0 {\displaystyle G=0} FOR THE PROBLEMS OF CHAPTER I.

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This course is intended for readers who need to deploy standard statistical techniques for data analysis but do not have statistical training. In particular it might be useful for readers of the wikibook Using SPSS and PASW. It is possible to get by in applied statistics without any real mathematical understanding of what you are doing, but it cannot be recommended. This course starts from the assumption that even flying by the seat of your pants, it is worth knowing how to measure wind-speed. The content was determined by listing what an undergraduate social science student might have to learn on a non-specialist course in statistics or applied statistics and then stripping it to the bare bones and especially avoiding much mathematical detail. There is no coverage of probability, which is one of the foundations of modern statistical thinking, just because it is the author's belief that if really necessary, you can get by without - well almost. Probability rears its head but This course relies only on naive, intuitive ideas of the probable. I want to stress that I do not think anyone should get by without developing a proper understanding of statistical methods but if you find yourself having to analyse data or perform a test and do not know where to start, This course might help.

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This course aims to be a high quality calculus course through which users can master the discipline. Standard topics such as limits, differentiation and integration are covered, as well as several others. Please contribute wherever you feel the need. You can simply help by rating individual sections of the course that you feel were inappropriately rated! 1.1 Algebra 1.2 Trigonometric functions 1.3 Functions 1.4 Graphing linear functions 1.5 Exercises 1.6 Hyperbolic logarithm and angles 2.1 An Introduction to Limits 2.2 Finite Limits 2.3 Infinite Limits 2.4 Continuity 2.5 Formal Definition of the Limit 2.6 Proofs of Some Basic Limit Rules 2.7 Exercises 3.1 Differentiation Defined 3.2 Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 Chain Rule 3.5 Higher Order Derivatives: an introduction to second order derivatives 3.6 Implicit Differentiation 3.7 Derivatives of Exponential and Logarithm Functions 3.8 Some Important Theorems 3.9 Exercises 3.10 L'Hôpital's Rule 3.11 Extrema and Points of Inflection 3.12 Newton's Method 3.13 Related Rates 3.14 Optimization 3.15 Euler's Method 3.16 Exercises 4.1 Definite integral 4.2 Fundamental Theorem of Calculus 4.3 Indefinite integral 4.4 Improper Integrals

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This online course is intended for, but not limited to, high school students without a rigorous understanding and knowledge of university-level mathematics. Therefore, the text's language reflects the expected mathematical maturity of the intended audience. This course introduces several interesting topics not covered in the standard high school curriculum of most countries. The materials presented can be challenging, but at the same time, we strive to make This course readable to all who are a few years from applying to higher education. From the authors It is our firm belief that math courses should not just be a collection of mathematical facts carefully laid out for rote memorization and cram sessions. A math course, especially for the youth, should be full of questions, not just exercises. These questions require some thought to answer and spark curiosity. After all, the questions keep the students engaged, not the answers. We sincerely hope to interest, stimulate, and challenge all those who read This course.

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This course introduces three-dimensional vectors as mathematical entities, though their application will be found, very likely, in physical science. For examples, velocity and acceleration of a particle in a reference frame are usually defined as vectors. As this is an elementary mathematical textbook, it is useful to state the prerequisites for readers expecting to benefit from what is written. When faced with listing prerequisites for a similar textbook in 1965, James A. Hummel of University of Maryland gave this list: Basic concepts of trigonometry, of Cartesian coordinates in the plane, and of set theory and notation. For his text he also required knowledge of the definition of a function, of the definition and properties of determinants of orders two and three, the absolute value, the field axioms, and the order axioms for real numbers. Study of the vector algebra in This course is good preparation for Linear Algebra. For students that have studied calculus of one variable, the chapter on vector analysis provides an introduction to the tools physicists use to study vector fields dependent on position in space. Consider the set of directed line segments in the plane. If two such segments are parallel, equal in length, and similarly directed, they are said to be equipollent. Such segments can be considered equivalent, and the collection of equivalence classes of directed segments in the plane provides an illustration of a planar space of vectors.

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Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. You've been asked to buy peanuts for you and your friends at a football game. You've collected $12.50. One bag is $2.75. You want to know how many bags you can buy. This is an algebra problem! Related problems are how much money will be left over, and what should you buy or in what proportions should you return the extra money to your friends. Algebra helps us to predict things that we don't yet know and to determine the relationships between the things we do. Algebra is a powerful and rich branch of mathematics that is useful in everyday life as well as business, engineering, and other technical fields. How can we use numbers and variables to find out unknown information? Math is a method of solving problems. You take information you know, and by manipulating it using mathematical principles, you can find information you don't know. Functions are the mathematical framework for solving problems. They have parameters, rules, and ways of being solved. This section will introduce you to common functions and how to use them.

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The word geometry originates from the Greek words (geo meaning world, metri meaning measure) and means, literally, to measure the earth. It is an ancient branch of mathematics, but its modern meaning depends largely on context. Geometry largely encompasses forms of non-numeric mathematics, such as those involving measurement, area and perimeter calculation, and work involving angles and position. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. This course is dedicated to high school geometry and geometry in general. The outline of topics reflects the California curriculum content standards. Template:Download version [1]

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This course aims to provide many solutions and explanations to questions and assignments posed in mathematics textbooks. Few mathematics textbooks (especially towards university / college level mathematics) have the complete set of solutions available, often favouring answers to selected problems, or perhaps odd or even numbered questions exclusively. Even where answers are provided, they may be succinct in order to save printed space. New pages devoted to an entire book should take the following naming convention (note the leading slash, which makes each page of This course a subpage of this one): For instance: If there is not an edition number, omit it. The ISBN and name must appear.

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This course is on associative composition algebras, structures that have been used in kinematics and mathematical physics. 1. Introduction 2. Transcendental paradigm 3. Binarions 4. Quaternions 5. Homographies This text expands the repertoire of algebra beyond real numbers R and complex numbers C to just five more algebras. The prospective reader will be well-acquainted with the utility of R and C in science, and might like to know (more) about quaternions H and related algebras, and what have been the historical invocations of these algebras. Some group theory and matrix multiplication are prerequisites from linear and abstract algebra. Attention to this text will show some concrete instances of mathematical objects, thus nailing down the abstruse nature of abstract algebra. Whereas linear algebra characteristically is concerned with n-dimensional space and n × n matrices, for this text n = 2 is the limit. Some of the content of this text was summarized in 1914 by Leonard Dickson when he noted that the complex quaternion and complex matrix algebras are equivalent, but their real subalgebras are not ! For more history of these algebras see Abstract Algebra/Hypercomplex numbers,  w:Composition algebra#History and w:History of quaternions.

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The purpose of This course is to show how the computer can draw technically perfect pictures of Julia and Mandelbrot sets. All the definitions, results and formulas required will be stated. If you know what a complex rational function is, then you certainly have enough knowledge of mathematics to follow this account.

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This course is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more. Readers of This course are expected to have read and understood the information presented in the Linear Algebra book, or an equivalent alternative. This course shall give an introduction to the fundamental concepts of abstract algebra, such as groups, rings and ideals, and fields and Galois theory. Sources

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This Intermediate Algebra course is designed as a chronological course to guide you through High School Algebra (sometimes called Algebra II in some locations). This course assumes you have completed Arithmetic and Algebra. Although not required, Intermediate Algebra is normally taken the year after Geometry. Polynomials Complex Numbers Conic Sections y = 1 x {\displaystyle y={\frac {1}{x}}} This is a hyperbola with center at (0,0) and asymptotes are y = 0 {\displaystyle y=0} and x = 0 {\displaystyle x=0} . The transverse axis is y = x {\displaystyle y=x} . y = 2 x {\displaystyle y=2^{x}}

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As the title suggests, we assume you have prior knowledge of differential equations and linear algebra separately. the course is structured into three main chapters, each with an important introduction that itself introduces material (So don't just skim over it thinking it's part of an outline) and then leads into method-heavy subsections. Other than that, enjoy!

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The subject of real analysis is concerned with studying the behavior and properties of functions, sequences, and sets on the real number line, which we denote as the mathematically familiar R. Concepts that we wish to examine through real analysis include properties like Limits, Continuity, Derivatives (rates of change), and Integration (amount of change over time). Many of these ideas are, on a conceptual or practical level, dealt with at lower levels of mathematics, including a regular First-Year Calculus course, and so, to the uninitiated reader, the subject of Real Analysis may seem rather senseless and trivial. However, Real Analysis is at a depth, complexity, and arguably beauty, that it is because under the surface of everyday mathematics, there is an assurance of correctness, that we call rigor, that permeates the whole of mathematics. Thus, Real Analysis can, to some degree, be viewed as a development of a rigorous, well-proven framework to support the intuitive ideas that we frequently take for granted. Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of mathematical ideas you have come across throughout your story of mathematics. However, instead of relying on sometimes uncertain intuition (which we have all felt when we were solving a problem we did not understand), we will anchor it to a rigorous set of mathematical theorems. Throughout this course, we will begin to see that we do not need intuition to understand mathematics - we need a manual. The overarching thesis of This course is how to define the real numbers axiomatically. How would that work? This course will read in this manner: we set down the properties which we think define the real numbers. We then prove from these properties - and these properties only - that the real numbers behave in the way which we have always imagined them to behave. We will then rework all our elementary theorems and facts we collected over our mathematical lives so that it all comes together, almost as if it always has been true before we analyzed it; that it was in fact rigorous all along - except that now we will know how it came to be.

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This course aims to cover algebraic structures and methods that play basic roles in other fields of mathematics such as algebraic geometry and representation theory. More precisely, the first chapter covers the rudiments of non-commutative rings and homological language that provide foundations for subsequent chapters. The second chapter covers commutative algebra, which we view as the local theory of algebraic geometry; the emphasis will be on (historical) connections to several complex variables. The third chapter is devoted to field theory, and the fourth to Linear algebra. The fifth chapter studies Lie algebra with emphasis on applications to arithmetic problems. Part I. Foundations Part II. Applications Part III. Appendix

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General Topology is based solely on set theory and concerns itself with structures of sets. It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalizes many distance-related concepts, such as continuity, compactness, and convergence. In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning. Have a question? Why not ask the very course that you are learning from? 1. What is the difference between topology, algebra and analysis? 2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed? The reason we have both definitions is because these two things have different properties. The most useful fact about a base is that it determines the topology. A basis must have "arbitrarily small" sets, that is, any open set contains a basis element. On the other hand, an open cover does not determine the topology at all. It can be used to build things such as partitions of unity, and often draws on the compactness property. Topology Expert (talk) 04:17, 8 June 2008 (UTC)

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Surreal numbers are a fascinating mathematical structure, built from a few simple rules but giving rise to marvellous complexity. The surreal numbers contain all the real numbers with which we are familiar, as well as an infinitude of new quantities. We will discover surreal numbers that are greater than any positive integer, and ones that are infinitesimally small. Concepts like the square root and the reciprocal of infinite quantities will not only be defined, but we will find that they show logical and beautiful behaviour. The Surreal numbers were invented by mathematician John H. Conway as part of an investigation into endgames in the game of Go, an oriental board game that also produces complex behaviour from a small set of simple rules. They were presented to the world in the form of a small novelette by Donald E. Knuth, in which a young couple on holiday discover a rock inscribed with Conway's rules and proceed to derive the entire theory. We will begin the same way, beginning with the initial axioms and working our way up to the entire vast structure. Along the way we will prove that all the familiar properties of real numbers (such as the transitive law of inequality, and the commutative law of addition) all hold. A basic familiarity with set theory is assumed; for a refresher, see Set Theory. The Beginning

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A key application of calculus is in optimization: finding maximum and minimum values of a function, and which points realize these extrema. Formally, the field of mathematical optimization is called mathematical programming, and calculus methods of optimization are basic forms of nonlinear programming. We will primarily discuss finite-dimensional optimization, illustrating with functions in 1 or 2 variables, and algebraically discussing n variables. We will also indicate some extensions to infinite-dimensional optimization, such as calculus of variations, which is a primary application of these methods in physics. Basic techniques include the first and second derivative test, and their higher-dimensional generalizations. A more advanced technique is Lagrange multipliers, and generalizations as Karush–Kuhn–Tucker conditions and Lagrange multipliers on Banach spaces. Optimization, particularly via Lagrange multipliers, is particularly used in the following fields: Further, several areas of mathematics can be understood as generalizations of these methods, notably Morse theory and calculus of variations.

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This course is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. The course contains many examples drawn from various branches of math. If you are not familiar with some of the kinds of math mentioned, don’t worry. If all the examples are unfamiliar, it may be wise to research a few before continuing. A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics. (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that unifies many concepts in different parts of math. In more detail, a category has objects and morphisms or arrows. (It is best to think of the morphisms as arrows: the word “morphism” makes you think they are set maps, and they are not always set maps. The formal definition of category is given in the chapter on categories.)

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Functional Analysis can mean different things, depending on who you ask. The core of the subject, however, is to study linear spaces with some topology which allows us to do analysis; ones like spaces of functions, spaces of operators acting on the space of functions, etc. Our interest in those spaces is twofold: those linear spaces with topology (i) often exhibit interesting properties that are worth investigating for their own sake, and (ii) have important application in other areas of mathematics (e.g., partial differential equations) as well as theoretical physics; in particular, quantum mechanics. (i) arises because linear vectors spaces that are of interest to analysts are infinite-dimensional in nature, and this requires careful investigation of geometry. (More on this in Chapter 2 and 4.) (ii) was what initially motivated the development of the field; Functional Analysis has its historical roots in linear algebra and the mathematical formulation of quantum mechanics in the early 20 century. (See w:Mathematical formulation of quantum mechanics) the course aims to cover these two interests simultaneously. the course consists of two parts. The first part covers the basics of Banach spaces theory with the emphasis on its applications. The second part covers topological vector spaces, especially locally convex ones, generalization of Banach spaces. In both parts, we give principal results e.g., the closed graph theorem, resulting in some repetition. One reason for doing this organization is that one often only needs a Banach-version of such results. Another reason is that this approach seems more pedagogically sound; the statement of the results in their full generality may obscure its simplicity. Exercises are meant to be an unintegrated part of the course. They can be skipped altogether, and the course should be fully read and understood. Some alternative proofs and additional results are relegated as exercises when their inclusion may disrupt the flow of the exposition. Knowledge of measure theory will not be needed except for Chapter 6, where we formulate the spectrum theorem in the language of measure theory. As for topology, knowledge of metric spaces suffices for Chapter 1 and Chapter 2. The solid background in general topology is required for the ensuing chapters.

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This course covers an elementary introduction to Number Theory, with an emphasis on presenting and proving a large number of theorems. No attempts will be made to derive number theory from set theory and no knowledge of Calculus will be assumed. It is important to convince yourself of the truth of each proof as you work through the course, and make sure you have a complete understanding. For those who wish to use this as a reference book, an index of theorems will be given.

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This course is intended as a general overview of undergraduate mathematics. In any one field, it may not have the widest coverage on this wiki but the idea is to present the most useful results with many exercises that are tied in carefully into the rest of the course. It can be used by readers as a hub to connect their current knowledge to what they want to know, laid out in a traditional course style, and for editors as a source to expand out from and create more specific titles. The project was inspired by the Feynmann Lectures in Physics which feature as recommended reading below, for mathematical physicists. It also owes a debt to the early success of Linear Algebra. We expect that the reader have the level usually required of a student starting a university level course that heavily involves mathematics. For example in the UK an A level equivalent is required. Specifically it would be useful to know skills like this:

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Crystallography is a branch of geometry that deals with indefinitely repeating patterns. Two-dimensional crystallography can be used, for example, to describe the way tiles cover a floor. Extending the field into three dimensions allows a general description of the way atoms or molecules arrange themselves into crystals. The three-dimensional crystallography was proven to be complete over a century ago. The fact that the mathematics itself cannot be advanced without some change of its axioms has meant that it is studied less often as pure mathematics, than as a means of understanding the details of complex structures in matter. Two fields are particularly reliant on it: materials scientist use it to describe the structure of engineering materials, often with particular attention to crystallographic defects; biochemists use it to describe the structure of biopolymers (see proteomics for an example), which usually must be processed laboriously before they form crystals. In mathematical terms, a crystal is an object with translational symmetry, i.e. it can be moved some distance and remain the same. This type of symmetry is fundamentally different from the more familiar mirror symmetry (the human face) or rotational symmetry (an airplane propeller), in that objects we imagine to represent translational symmetry must be larger than ourselves. We can imagine passing Alice through the looking glass, or spinning a propeller by one blade's fraction of a rotation, while we stand still. For an experience of translational symmetry, however, we must move ourselves, and not notice the difference. This can happen in an ocean, a desert, or a large suburb, if every wave, or dune, or tract home looks exactly like the next. Just as no eye is the exact mirror of its opposite, and no propeller is perfectly balanced, no physical crystal is perfect. There will always be a boundary that gets nearer or farther after a unit of translation. Strictly speaking, any true crystal must fill the entire universe. If we imagine a "perfect" housing development (dystopian though it may be) which covers a two-dimensional plane with an infinitely-repeating pattern of homes, and want to apply crystallography to it, we can save a lot of work by eliminating all the geometric complexity of garages and sprinkler heads and such. To make things as simple as possible, we could abstract every house down to a single point, although we need to keep track of each house's orientation.