This document provides an overview of mathematical formulae across various topics. It includes formulae for series such as arithmetic and geometric progressions. It also covers vector algebra, matrix algebra, vector calculus, complex variables, trigonometric functions, hyperbolic functions, limits, differentiation, integration, differential equations, calculus of variations, functions of several variables, Fourier series and transforms, Laplace transforms, numerical analysis, treatment of random errors, and statistics. Physical constants are also provided as a reference.
The document is a mathematical formula handbook containing formulas and concepts related to series, vector algebra, matrix algebra, vector calculus, complex variables, trigonometric functions, hyperbolic functions, limits, differentiation, integration, differential equations, calculus of variations, functions of several variables, Fourier series and transforms, Laplace transforms, numerical analysis, treatment of random errors, and statistics. It provides the necessary background and formulas for studying physics.
Partial differential equations, graduate level problems and solutions by igor...Julio Banks
The physical world is driving by the laws of mathematics, more specifically PDE (Partial Differential Equations). FEA (Finite Element Analysis) and CFD (Computation Fluid Dynamics) are the numerical methods utilized to model physical events described by PDEs.
This document contains lecture notes on real analysis from Dr. Bernard Mutuku Nzimbi. It covers topics including:
1) The properties of the real number systems and its subsets like natural numbers, integers, rational numbers, irrational numbers. It discusses the field axioms for addition and multiplication.
2) The uncountability of the real number line using concepts like countable and uncountable sets.
3) The structure of the metric space of real numbers including neighborhoods, interior points, open and closed sets.
4) Bounded subsets of real numbers, supremum, infimum and the completeness property.
5) Convergence of sequences, subsequences, Cauchy sequences and
This document provides an introduction to ordinary differential equations. It begins with an example of the "banker's equation" which models the growth of a bank account balance over time. It then introduces the concept of slope fields, which provide a graphical representation of the behavior of solutions to a differential equation. The document emphasizes reading and understanding differential equations through examples rather than providing a comprehensive textbook.
This document provides lecture notes for a course on differential equations. It includes a preface and 13 chapters covering topics such as first-order differential equations, second-order differential equations, systems of equations, nonlinear differential equations, and partial differential equations. The preface describes the purpose of the notes, sources of adapted material, and links for additional resources. An overview of relevant calculus concepts is provided in Chapter 0 as a mathematical review.
This thesis develops mathematical models to analyze the population dynamics of Ateles Hybridus (Brown Spider Monkeys) in fragmented and non-fragmented landscapes. It begins with a single-patch model and analyzes the equilibria and stability. It then integrates this into a multi-patch model accounting for migration between patches. Various parameters are explored, including survival rates, birth gender probability, and reproduction rate. The goal is to provide insights into the endangerment of this species and potential solutions based on modeling their population structure and environment.
This document summarizes Guy Lebanon's PhD thesis on applying concepts from Riemannian geometry to statistical machine learning. It introduces statistical manifolds and examines the geometry of probability model spaces. It explores how this geometric perspective can provide theoretical insights into algorithms like AdaBoost and logistic regression. It also describes developing new algorithms by adapting existing methods like kernels and margin classifiers to non-Euclidean geometries appropriate for data like text documents. The goal is to enhance understanding of machine learning algorithms and develop new approaches by considering the intrinsic geometries of model and data spaces.
This document is a revision of a basic calculus textbook. It covers topics such as exponents, algebraic expressions, solving linear and quadratic equations, inequalities, functions, limits, differentiation, integration, trigonometric functions, exponential and logarithmic functions. The document provides definitions, formulas, examples and explanations of concepts in calculus and precalculus mathematics.
The document is a mathematical formula handbook containing formulas and concepts related to series, vector algebra, matrix algebra, vector calculus, complex variables, trigonometric functions, hyperbolic functions, limits, differentiation, integration, differential equations, calculus of variations, functions of several variables, Fourier series and transforms, Laplace transforms, numerical analysis, treatment of random errors, and statistics. It provides the necessary background and formulas for studying physics.
Partial differential equations, graduate level problems and solutions by igor...Julio Banks
The physical world is driving by the laws of mathematics, more specifically PDE (Partial Differential Equations). FEA (Finite Element Analysis) and CFD (Computation Fluid Dynamics) are the numerical methods utilized to model physical events described by PDEs.
This document contains lecture notes on real analysis from Dr. Bernard Mutuku Nzimbi. It covers topics including:
1) The properties of the real number systems and its subsets like natural numbers, integers, rational numbers, irrational numbers. It discusses the field axioms for addition and multiplication.
2) The uncountability of the real number line using concepts like countable and uncountable sets.
3) The structure of the metric space of real numbers including neighborhoods, interior points, open and closed sets.
4) Bounded subsets of real numbers, supremum, infimum and the completeness property.
5) Convergence of sequences, subsequences, Cauchy sequences and
This document provides an introduction to ordinary differential equations. It begins with an example of the "banker's equation" which models the growth of a bank account balance over time. It then introduces the concept of slope fields, which provide a graphical representation of the behavior of solutions to a differential equation. The document emphasizes reading and understanding differential equations through examples rather than providing a comprehensive textbook.
This document provides lecture notes for a course on differential equations. It includes a preface and 13 chapters covering topics such as first-order differential equations, second-order differential equations, systems of equations, nonlinear differential equations, and partial differential equations. The preface describes the purpose of the notes, sources of adapted material, and links for additional resources. An overview of relevant calculus concepts is provided in Chapter 0 as a mathematical review.
This thesis develops mathematical models to analyze the population dynamics of Ateles Hybridus (Brown Spider Monkeys) in fragmented and non-fragmented landscapes. It begins with a single-patch model and analyzes the equilibria and stability. It then integrates this into a multi-patch model accounting for migration between patches. Various parameters are explored, including survival rates, birth gender probability, and reproduction rate. The goal is to provide insights into the endangerment of this species and potential solutions based on modeling their population structure and environment.
This document summarizes Guy Lebanon's PhD thesis on applying concepts from Riemannian geometry to statistical machine learning. It introduces statistical manifolds and examines the geometry of probability model spaces. It explores how this geometric perspective can provide theoretical insights into algorithms like AdaBoost and logistic regression. It also describes developing new algorithms by adapting existing methods like kernels and margin classifiers to non-Euclidean geometries appropriate for data like text documents. The goal is to enhance understanding of machine learning algorithms and develop new approaches by considering the intrinsic geometries of model and data spaces.
This document is a revision of a basic calculus textbook. It covers topics such as exponents, algebraic expressions, solving linear and quadratic equations, inequalities, functions, limits, differentiation, integration, trigonometric functions, exponential and logarithmic functions. The document provides definitions, formulas, examples and explanations of concepts in calculus and precalculus mathematics.
This document provides an overview and table of contents for a textbook on basic calculus. It discusses the purpose and structure of the book, which aims to explain key concepts in calculus through examples and exercises. The book covers topics like limits, derivatives, integrals, and their applications. It also includes a chapter reviewing prerequisite algebra and geometry topics to refresh students' knowledge before beginning calculus. The overview explains how each chapter builds upon the previous ones to develop an understanding of calculus.
This document contains lecture notes on discrete structures from a course taught at Stanford University in winter 2008. The notes cover topics including sets and notation, induction, proof techniques, divisibility, prime numbers, modular arithmetic, relations and functions, mathematical logic, counting, binomial coefficients, the inclusion-exclusion principle, the pigeonhole principle, asymptotic notation, graphs, and more. The notes begin by defining sets and set notation, discussing formal definitions and the appropriate level of formality for an introductory course.
The document is an introduction to vector spaces, vector algebras, and vector geometries. It aims to promote self-study of some elementary concepts in a manner that emphasizes basic algebraic and geometric structures. The intended primary readers are undergraduate mathematics majors in their junior or senior year. The document contains preface material and a table of contents outlining 8 chapters that cover fundamentals of structure, maps, multilinear transformations, vector algebras, vector affine geometry, basic affine results and methods, projective geometry, and scalar product spaces.
This document provides information about the authors and content of the book "A First Course in Complex Analysis" including:
1) The book was written by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka for an undergraduate complex analysis course. It relies on minimal concepts from real analysis.
2) The goal of the book is to introduce students to the Residue Theorem, including some nontraditional applications from continuous and discrete mathematics.
3) The material should be sufficient for a typical one-semester course, though some sections could be partially omitted or assumed as needed depending on the course.
Principle of Integral Applications - Integral Calculus - by Arun Umraossuserd6b1fd
This book is based on the function analysis of the same writer. Suitable for CBSE board students. This book also helps to understand the physics problems and derivations. Suitable for quick and last minute preparation.
This document is a thesis that examines stochastic differential equations (SDEs). It begins with an introduction that provides background on SDEs and outlines the aims, objectives, and structure of the thesis. The body of the thesis first reviews key concepts in probability, Brownian motion, and stochastic integration. It then defines SDEs and explores numerical methods for solving SDEs such as the Euler-Maruyama and Milstein methods. Applications of SDEs in finance are also discussed, including the Black-Scholes option pricing model. The thesis concludes by summarizing the findings and proposing avenues for further research.
Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zinggRohit Bapat
This document provides an overview of computational fluid dynamics (CFD) and summarizes its key steps and concepts. It discusses the fundamentals of CFD, including conservation laws, governing equations, finite difference approximations, semi-discrete and finite volume methods, and time-marching algorithms. The document is intended to introduce readers to the basic theory and methods in CFD for modeling fluid flow and transport phenomena.
Principle of Angular Motion - Physics - An Introduction by Arun Umraossuserd6b1fd
The document discusses angular motion and rotational dynamics. It defines angular motion as the changing angular position of an object rotating about an axis over time. Key concepts covered include rigid bodies, axis of rotation, moment of force (torque), equilibrium, center of mass, angular velocity, angular momentum, moment of inertia, and angular kinetic energy. Methods for calculating properties like radius of gyration, center of gravity, and moment of inertia are presented for basic shapes.
Notes for GNU Octave - Numerical Programming - for Students 01 of 02 by Arun ...ssuserd6b1fd
The document provides an introduction to GNU Octave, an open-source software program for numerical computations. It covers various topics in mathematics, including expressions, arithmetic operators, comparison operators, evaluation, algebra, complex numbers, geometry, logarithms, and trigonometric functions. For each topic, it lists and briefly describes the relevant Octave commands and functions that can be used for computations and analysis.
This document is Roman Zeyde's 2013 master's thesis from the Technion submitted in partial fulfillment of the requirements for a Master of Science degree in Computer Science. The thesis describes research on computational electrokinetics, which involves developing a numerical scheme to solve the governing equations for electrokinetic phenomena such as electrophoresis and ion exchange. The numerical scheme is based on a finite volume method in spherical coordinates. Results are presented comparing the numerical solutions to asymptotic analytical solutions for steady-state velocity profiles.
This document provides an introduction to differential calculus and its applications using the computer algebra system Sage. It covers topics such as variables, functions, limits, differentiation, rules for differentiating standard functions, and applications of derivatives including geometry, mechanics, and Newton's method for finding roots of equations. The intended audience appears to be students new to calculus.
This thesis investigates the use of meshless methods to solve difficult multibody systems using computational fluid dynamics (CFD). An implicit meshless scheme is developed to solve the Euler, laminar and Reynolds-Averaged Navier-Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The scheme is evaluated for steady and unsteady flows in two and three-dimensions, demonstrating its performance. A meshless preprocessor is developed to handle overlapping point distributions for multibody systems. The preprocessor redefines boundaries, blanks points, and selects stencils for the meshless solver. Results are presented for various 2D and 3D multibody test cases, showing the ability of the
This document is a thesis submitted by Tokelo Khalema to the University of the Free State in partial fulfillment of the requirements for a B.Sc. Honors degree in Mathematical Statistics. The thesis compares the Gaussian linear model, two Bayesian Student-t regression models, and the method of least absolute deviations through a Monte Carlo simulation study. The study aims to evaluate how soon and how severely the least squares regression model starts to lose optimality against these robust alternatives under violations of its assumptions. The document includes sections on robust statistical procedures, literature review of the models considered, research methodology, results and applications of the simulation study, and closing remarks.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
This document contains information about Ravindra Yadav, a first year mechanical engineering student. It discusses traffic engineering, factors affecting traffic such as road users, vehicles, roadways and environment. It covers traffic characteristics, studies, operation, planning, design, management and administration. Specific topics discussed include road user characteristics, vehicular characteristics, density, capacity, time headway and space headway. The document also discusses traffic regulation through devices like licenses, vehicle registration, transport authorities, speed limits and control devices. It describes different types of traffic signs used for regulation, warning and providing information.
How does SnapChat and Social Live Streaming Video Work?Eric T. Tung
The document discusses the rise of social video platforms like Snapchat and how brands can leverage them. It provides statistics on platform usage, such as Snapchat having over 800 million users and 10 billion daily video views. Examples are given of brands using social video successfully, such as Taco Bell's announcements on Snapchat. Advice is given for social video strategies targeting different platforms and live streaming. The potential for future video technologies like VR and 360 video is also mentioned.
5 Easy Ways to Tweak your Pinterest Profile to Increase your PageviewsKathleen Celmins
You've heard that using Pinterest is a good way to bring more views to your blog. But did you know that getting started is simply a matter of editing your profile? Follow these five ways to tweak your Pinterest profile and watch the traffic start to roll in.
This document provides an introduction to computers and their components. It discusses that a computer accepts data from input devices, processes it using its CPU and memory, and outputs processed data. The main components of a computer are the CPU (which contains the ALU and CU), memory, input devices, output devices, and secondary storage. The CPU performs calculations and logical operations. Memory is used to store programs and data. Input devices enter data, output devices display processed data. Secondary storage like hard disks store large amounts of data long-term. The document also covers software, programming languages, limitations of computers, and applications of computers.
Dokumen tersebut membahas tentang persamaan diferensial orde dua homogen dan non homogen. Secara garis besar dibahas tentang bentuk umum persamaan diferensial orde dua, solusi homogen, dan metode penyelesaian persamaan non homogen seperti metode koefisien tak tentu dan metode variasi parameter beserta contoh soalnya.
This document provides an overview and table of contents for a textbook on basic calculus. It discusses the purpose and structure of the book, which aims to explain key concepts in calculus through examples and exercises. The book covers topics like limits, derivatives, integrals, and their applications. It also includes a chapter reviewing prerequisite algebra and geometry topics to refresh students' knowledge before beginning calculus. The overview explains how each chapter builds upon the previous ones to develop an understanding of calculus.
This document contains lecture notes on discrete structures from a course taught at Stanford University in winter 2008. The notes cover topics including sets and notation, induction, proof techniques, divisibility, prime numbers, modular arithmetic, relations and functions, mathematical logic, counting, binomial coefficients, the inclusion-exclusion principle, the pigeonhole principle, asymptotic notation, graphs, and more. The notes begin by defining sets and set notation, discussing formal definitions and the appropriate level of formality for an introductory course.
The document is an introduction to vector spaces, vector algebras, and vector geometries. It aims to promote self-study of some elementary concepts in a manner that emphasizes basic algebraic and geometric structures. The intended primary readers are undergraduate mathematics majors in their junior or senior year. The document contains preface material and a table of contents outlining 8 chapters that cover fundamentals of structure, maps, multilinear transformations, vector algebras, vector affine geometry, basic affine results and methods, projective geometry, and scalar product spaces.
This document provides information about the authors and content of the book "A First Course in Complex Analysis" including:
1) The book was written by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka for an undergraduate complex analysis course. It relies on minimal concepts from real analysis.
2) The goal of the book is to introduce students to the Residue Theorem, including some nontraditional applications from continuous and discrete mathematics.
3) The material should be sufficient for a typical one-semester course, though some sections could be partially omitted or assumed as needed depending on the course.
Principle of Integral Applications - Integral Calculus - by Arun Umraossuserd6b1fd
This book is based on the function analysis of the same writer. Suitable for CBSE board students. This book also helps to understand the physics problems and derivations. Suitable for quick and last minute preparation.
This document is a thesis that examines stochastic differential equations (SDEs). It begins with an introduction that provides background on SDEs and outlines the aims, objectives, and structure of the thesis. The body of the thesis first reviews key concepts in probability, Brownian motion, and stochastic integration. It then defines SDEs and explores numerical methods for solving SDEs such as the Euler-Maruyama and Milstein methods. Applications of SDEs in finance are also discussed, including the Black-Scholes option pricing model. The thesis concludes by summarizing the findings and proposing avenues for further research.
Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zinggRohit Bapat
This document provides an overview of computational fluid dynamics (CFD) and summarizes its key steps and concepts. It discusses the fundamentals of CFD, including conservation laws, governing equations, finite difference approximations, semi-discrete and finite volume methods, and time-marching algorithms. The document is intended to introduce readers to the basic theory and methods in CFD for modeling fluid flow and transport phenomena.
Principle of Angular Motion - Physics - An Introduction by Arun Umraossuserd6b1fd
The document discusses angular motion and rotational dynamics. It defines angular motion as the changing angular position of an object rotating about an axis over time. Key concepts covered include rigid bodies, axis of rotation, moment of force (torque), equilibrium, center of mass, angular velocity, angular momentum, moment of inertia, and angular kinetic energy. Methods for calculating properties like radius of gyration, center of gravity, and moment of inertia are presented for basic shapes.
Notes for GNU Octave - Numerical Programming - for Students 01 of 02 by Arun ...ssuserd6b1fd
The document provides an introduction to GNU Octave, an open-source software program for numerical computations. It covers various topics in mathematics, including expressions, arithmetic operators, comparison operators, evaluation, algebra, complex numbers, geometry, logarithms, and trigonometric functions. For each topic, it lists and briefly describes the relevant Octave commands and functions that can be used for computations and analysis.
This document is Roman Zeyde's 2013 master's thesis from the Technion submitted in partial fulfillment of the requirements for a Master of Science degree in Computer Science. The thesis describes research on computational electrokinetics, which involves developing a numerical scheme to solve the governing equations for electrokinetic phenomena such as electrophoresis and ion exchange. The numerical scheme is based on a finite volume method in spherical coordinates. Results are presented comparing the numerical solutions to asymptotic analytical solutions for steady-state velocity profiles.
This document provides an introduction to differential calculus and its applications using the computer algebra system Sage. It covers topics such as variables, functions, limits, differentiation, rules for differentiating standard functions, and applications of derivatives including geometry, mechanics, and Newton's method for finding roots of equations. The intended audience appears to be students new to calculus.
This thesis investigates the use of meshless methods to solve difficult multibody systems using computational fluid dynamics (CFD). An implicit meshless scheme is developed to solve the Euler, laminar and Reynolds-Averaged Navier-Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The scheme is evaluated for steady and unsteady flows in two and three-dimensions, demonstrating its performance. A meshless preprocessor is developed to handle overlapping point distributions for multibody systems. The preprocessor redefines boundaries, blanks points, and selects stencils for the meshless solver. Results are presented for various 2D and 3D multibody test cases, showing the ability of the
This document is a thesis submitted by Tokelo Khalema to the University of the Free State in partial fulfillment of the requirements for a B.Sc. Honors degree in Mathematical Statistics. The thesis compares the Gaussian linear model, two Bayesian Student-t regression models, and the method of least absolute deviations through a Monte Carlo simulation study. The study aims to evaluate how soon and how severely the least squares regression model starts to lose optimality against these robust alternatives under violations of its assumptions. The document includes sections on robust statistical procedures, literature review of the models considered, research methodology, results and applications of the simulation study, and closing remarks.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
This document contains information about Ravindra Yadav, a first year mechanical engineering student. It discusses traffic engineering, factors affecting traffic such as road users, vehicles, roadways and environment. It covers traffic characteristics, studies, operation, planning, design, management and administration. Specific topics discussed include road user characteristics, vehicular characteristics, density, capacity, time headway and space headway. The document also discusses traffic regulation through devices like licenses, vehicle registration, transport authorities, speed limits and control devices. It describes different types of traffic signs used for regulation, warning and providing information.
How does SnapChat and Social Live Streaming Video Work?Eric T. Tung
The document discusses the rise of social video platforms like Snapchat and how brands can leverage them. It provides statistics on platform usage, such as Snapchat having over 800 million users and 10 billion daily video views. Examples are given of brands using social video successfully, such as Taco Bell's announcements on Snapchat. Advice is given for social video strategies targeting different platforms and live streaming. The potential for future video technologies like VR and 360 video is also mentioned.
5 Easy Ways to Tweak your Pinterest Profile to Increase your PageviewsKathleen Celmins
You've heard that using Pinterest is a good way to bring more views to your blog. But did you know that getting started is simply a matter of editing your profile? Follow these five ways to tweak your Pinterest profile and watch the traffic start to roll in.
This document provides an introduction to computers and their components. It discusses that a computer accepts data from input devices, processes it using its CPU and memory, and outputs processed data. The main components of a computer are the CPU (which contains the ALU and CU), memory, input devices, output devices, and secondary storage. The CPU performs calculations and logical operations. Memory is used to store programs and data. Input devices enter data, output devices display processed data. Secondary storage like hard disks store large amounts of data long-term. The document also covers software, programming languages, limitations of computers, and applications of computers.
Dokumen tersebut membahas tentang persamaan diferensial orde dua homogen dan non homogen. Secara garis besar dibahas tentang bentuk umum persamaan diferensial orde dua, solusi homogen, dan metode penyelesaian persamaan non homogen seperti metode koefisien tak tentu dan metode variasi parameter beserta contoh soalnya.
Makalah ini membahas tentang rangkuman materi Persamaan Diferensial Linier orde n dengan koefisien konstan dan variable serta sistem Persamaan Diferensial Linier simultan. Terdapat penjelasan mengenai bentuk umum PDL, jenis-jenisnya, dan langkah penyelesaian menggunakan metode invers operator dan variasi parameter.
The document discusses how to make UX (user experience) a strategic priority for a company rather than just a deliverable. It recommends telling a story about the value of UX, embedding UX processes in each project team, and defining the value UX brings. It also suggests growing the UX team and their expertise, bringing the user's voice to guide products, and continuing to educate others about UX.
The spotlight is on pollsters in the UK, following the performance of the polls at the 2015 General Election. Are we alone in facing this challenge, or is it a global issue? Does the experience in other countries point to what we should be doing in the UK?
Ipsos has many of the leading polling experts from around the world, and we brought them together in London to provide unique combined insight. Our panel members from the US, Canada, Italy and Sweden talked us through the role and challenges of polling in their countries and what we need to do to get it right. They also updated us on the political landscape of their countries, with outlines of the major elections they have recently had, and in the case of the US, the on-going race to the White House.
We need a bigger definition of creativity in education. It goes beyond simply creating something new. It also means tweaking things, fixing things, mixing them up, etc. So here are the types of creative teachers.
The document provides information about the Zika virus outbreak, including how it spreads and symptoms. It notes that Zika virus is mainly spread by Aedes mosquitoes and infected mothers can pass it to fetuses. While there is no vaccine currently, people can protect themselves by preventing mosquito bites, using repellent and eliminating standing water where mosquitoes breed. The most important advice is to stay informed but remain calm.
How to Stream to Facebook Live Like a ProLeslie Samuel
In this presentation of Become a Blogger Live, we talk about the professional way to stream to Facebook Live. What extra software and equipment do you need to stream to Facebook Live?
UX Design + UI Design: Injecting a brand persona!Jayan Narayanan
It is my try to shed light on two often heard but little understood or confused acronyms and its impact on overall brand experience. The presentation originally designed to address a group of entrepreneurs who have little knowledge in design and it's technical jargons.
https://www.linkedin.com/in/jayan-narayanan/
Hi! We're the creative team behind Hypothesis's reports, presentations, and infographics, and we're sharing out our best tips. Please share with someone you think would enjoy this slideshow.
www.hypothesisgroup.com
www.linkedin.com/companies/hypothesis-group
www.instagram.com/hypothesisgroup
The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
Despite the myth of "digital natives," most of my students have very little experience using technology as anything more than a consumer device. It doesn't have to be this way. By using the design thinking cycle, teachers can foster creative thinking in every content area.
The document contains 20 quotes from Prince ranging from short phrases to full sentences on a variety of topics including music, life, freedom, spirituality, happiness, and more. Some of the quotes say music should make you feel good, life is a party that doesn't last, too much freedom can lead to soul decay, and a strong spirit transcends rules.
This document provides an abstract and table of contents for a book titled "Essentials of Applied Mathematics for Scientists and Engineers" by Robert G. Watts. The book covers topics in partial differential equations that are common in engineering fields, including the heat equation, vibrating strings/membranes, and elastic bars. It uses techniques like separation of variables and orthogonal functions to solve examples related to heat transfer, fluid flow, and mechanical vibrations. The table of contents provides an overview of the chapters and sections that will be covered in the book.
This document contains lecture notes on quantum mechanics. It introduces key concepts like the Schrodinger equation, ket vectors, operators, and Hamiltonians. The notes are divided into multiple chapters that will cover topics such as the harmonic oscillator, angular momentum, perturbation theory, and other quantum systems. References are provided to textbooks where more of the material in the notes is based on. The notes are intended to review physical and mathematical concepts needed to formulate the theory of quantum mechanics.
This document is a set of lecture notes on the lambda calculus. It covers topics such as the untyped and typed lambda calculus, the Church-Rosser theorem, combinatory algebras, the simply-typed lambda calculus and its connections to propositional logic via the Curry-Howard isomorphism, polymorphism, type inference, denotational semantics using complete partial orders, and the programming language PCF. The notes were developed from courses taught by the author at various universities.
This document provides a preface and table of contents for a book titled "I do like CFD, VOL.1" by Katate Masatsuka. It discusses governing equations and exact solutions for computational fluid dynamics. The preface notes that it is the intellectual property of the author and protected by copyright, with permission required for modification or reproduction. It provides contact information for the author and notes that the PDF version is hyperlinked for ease of navigation. A hard copy version is also available for purchase.
Fundamentals of computational fluid dynamicsAghilesh V
This document provides an introduction to computational fluid dynamics (CFD) and outlines the key steps in the CFD process. It covers topics like conservation laws, finite difference approximations, finite volume methods, semi-discrete and time-marching approaches. It also discusses concepts like stability analysis and choice of numerical methods. The document contains chapters on modeling equations, spatial and temporal discretization techniques, stability analysis of linear systems, and considerations for choosing time-marching methods. It aims to provide fundamentals of CFD modeling and numerical methods.
This document is an introduction to representation theory. It begins with basic notions such as what representation theory is, definitions of algebras, representations, ideals, quotients, and examples of algebras like quivers and Lie algebras. It then covers general results in representation theory, including representations of direct sums, filtrations, characters, and the Jordan-Holder and Krull-Schmidt theorems. Subsequent sections discuss representations of finite groups and quiver representations. The document concludes with an introduction to category theory concepts used in representation theory.
This document contains lecture notes from a 1951 course on relativistic quantum mechanics taught by F.J. Dyson at Cornell University. The notes cover topics including the Dirac theory, scattering problems, quantum field theory, and examples of quantized field theories such as quantum electrodynamics. The notes were originally written by Dyson and edited into a second edition by Michael J. Moravcsik, who is responsible for changes made in the re-editing process.
This document is the thesis of Alessandro Adamo submitted for a PhD in Mathematics and Statistics for Computational Sciences. The thesis proposes a new algorithm called LIMAPS (Lipschitzian Mappings for Sparse recovery) for solving underdetermined linear systems based on nonconvex Lipschitzian mappings. Chapter 1 provides theoretical foundations on sparse recovery and compressive sensing. Chapter 2 introduces LIMAPS and its iterative scheme for sparse representation and sparsity minimization. Chapters 3 and 4 apply LIMAPS to face recognition and ECG signal compression respectively, demonstrating its effectiveness on real-world applications.
Coulomb gas formalism in conformal field theoryMatthew Geleta
This thesis examines the Coulomb gas formalism in conformal field theory. It begins by developing the Coulomb gas formalism starting from a bosonic string theory. Some applications in statistical field theory are demonstrated by constructing minimal model conformal field theories and relating these to critical lattice models like the Ising model. The Coulomb gas formalism is then used to analytically compute primary three-point constants and operator product expansion coefficients for the minimal conformal field theories. Comparisons are made to results from other techniques like bosonization of the free fermion conformal field theory. The main original contribution is conjecturing and verifying a connection between the Coulomb gas formalism and the monodromy theory of certain Fuchsian
Introduction to Abstract Algebra by
D. S. Malik
Creighton University
John N. Mordeson
Creighton University
M.K. Sen
Calcutta University
It includes the most important sections of abstract mathematics like Sets, Relations, Integers, Groups, Permutation Groups, Subgroups and Normal Subgroups, Homomorphisms and Isomorphisms of Groups, Rings etc.
This book is grate not only for those who study in mathematics departments, but for all who want to start with abstract mathematics. Abstract mathematics is very important for computer sciences and engineering.
Lecture notes on planetary sciences and orbit determinationErnst Schrama
This document contains lecture notes on planetary sciences and satellite orbit determination. It covers topics such as the two-body problem, potential theory, Fourier frequency analysis, reference systems, observation techniques like satellite laser ranging and GPS, applications like satellite altimetry and gravimetry, parameter estimation, and modeling two-dimensional functions and data with polynomials. The notes provide mathematical background and explanations of concepts relevant to orbit determination and modeling orbital motion.
Reading Materials for Operational Research Derbew Tesfa
This document provides an introduction to deterministic operations research models. It covers linear programming formulations including the simplex method for solving linear programs. Additional topics include sensitivity analysis, duality, network flows, integer programming, and dynamic programming. Mathematical modeling concepts are illustrated through examples involving product mix optimization, material blending to minimize costs, and network problems. Linear programming and other optimization techniques are presented as tools for decision making under deterministic conditions.
This document provides an overview of mathematical modeling and ordinary differential equations. It covers topics such as first-order single differential equations, population dynamics modeling, techniques for solving single first-order equations, vector fields, existence and uniqueness theorems, numerical methods, second-order linear equations, linear oscillators, 2x2 linear systems, nonlinear systems in two dimensions, linear systems with constant coefficients, Laplace transforms, calculus of variations, Hamiltonian systems, gradient flows, the simple pendulum, and planetary orbits. The document contains examples and applications in multiple areas including physics, engineering, biology, and mechanics.
This document is the contents page for a CK-12 physics workbook. It lists the chapter titles and topics covered in the workbook. CK-12 Foundation is a nonprofit organization that creates open educational resources to reduce textbook costs. It pioneers web-based and customizable textbooks called FlexBooks that can be printed, modified, and shared freely. The workbook covers topics in physics from scientific inquiry to circuits.
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3. Introduction
This Mathematical Formaulae handbook has been prepared in response to a request from the Physics Consultative
Committee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similar
document issued by the Department of Engineering, but obviously reflects the particular interests of physicists.
There was discussion as to whether it should also include physical formulae such as Maxwell’s equations, etc., but
a decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainly
because, in its present form, clean copies can be made available to candidates in exams.
There has been wide consultation among the staff about the contents of this document, but inevitably some users
will seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments to
the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. The
Secretary will also be grateful to be informed of any (equally inevitable) errors which are found.
This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by
Dr Dave Green, using the TEX typesetting package.
Version 1.5 December 2005.
Bibliography
Abramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965.
Gradshteyn, I.S. & Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980.
Jahnke, E. & Emde, F., Tables of Functions, Dover, 1986.
Nordling, C. & ¨Osterman, J., Physics Handbook, Chartwell-Bratt, Bromley, 1980.
Speigel, M.R., Mathematical Handbook of Formulas and Tables.
(Schaum’s Outline Series, McGraw-Hill, 1968).
Physical Constants
Based on the “Review of Particle Properties”, Barnett et al., 1996, Physics Review D, 54, p1, and “The Fundamental
Physical Constants”, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standard-
deviation uncertainties in the last digits.)
speed of light in a vacuum c 2·997 924 58 × 108 m s−1 (by definition)
permeability of a vacuum µ0 4π × 10−7 H m−1 (by definition)
permittivity of a vacuum 0 1/µ0c2 = 8·854 187 817 . . . × 10−12 F m−1
elementary charge e 1·602 177 33(49) × 10−19 C
Planck constant h 6·626 075 5(40) × 10−34 J s
h/2π ¯¯h 1·054 572 66(63) × 10−34 J s
Avogadro constant NA 6·022 136 7(36) × 1023 mol−1
unified atomic mass constant mu 1·660 540 2(10) × 10−27 kg
mass of electron me 9·109 389 7(54) × 10−31 kg
mass of proton mp 1·672 623 1(10) × 10−27 kg
Bohr magneton eh/4πme µB 9·274 015 4(31) × 10−24 J T−1
molar gas constant R 8·314 510(70) J K−1 mol−1
Boltzmann constant kB 1·380 658(12) × 10−23 J K−1
Stefan–Boltzmann constant σ 5·670 51(19) × 10−8 W m−2 K−4
gravitational constant G 6·672 59(85) × 10−11 N m2 kg−2
Other data
acceleration of free fall g 9·806 65 m s−2 (standard value at sea level)
1
4. 1. Series
Arithmetic and Geometric progressions
A.P. Sn = a + (a + d) + (a + 2d) + · · · + [a + (n − 1)d] =
n
2
[2a + (n − 1)d]
G.P. Sn = a + ar + ar2
+ · · · + arn−1
= a
1 − rn
1 − r
, S∞ =
a
1 − r
for |r| < 1
(These results also hold for complex series.)
Convergence of series: the ratio test
Sn = u1 + u2 + u3 + · · · + un converges as n → ∞ if lim
n→∞
un+1
un
< 1
Convergence of series: the comparison test
If each term in a series of positive terms is less than the corresponding term in a series known to be convergent,
then the given series is also convergent.
Binomial expansion
(1 + x)n
= 1 + nx +
n(n − 1)
2!
x2
+
n(n − 1)(n − 2)
3!
x3
+ · · ·
If n is a positive integer the series terminates and is valid for all x: the term in xr
is n
Crxr
or
n
r
where n
Cr ≡
n!
r!(n − r)!
is the number of different ways in which an unordered sample of r objects can be selected from a set of
n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series is
convergent for |x| < 1.
Taylor and Maclaurin Series
If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series,
y(x) = y(a + u) = y(a) + u
dy
dx
+
u2
2!
d2
y
dx2
+
u3
3!
d3
y
dx3
+ · · ·
where u = x − a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with
a = 0,
y(x) = y(0) + x
dy
dx
+
x2
2!
d2
y
dx2
+
x3
3!
d3
y
dx3
+ · · ·
Power series with real variables
ex
= 1 + x +
x2
2!
+ · · · +
xn
n!
+ · · · valid for all x
ln(1 + x) = x −
x2
2
+
x3
3
+ · · · + (−1)n+1 xn
n
+ · · · valid for −1 < x ≤ 1
cos x =
eix
+ e−ix
2
= 1 −
x2
2!
+
x4
4!
−
x6
6!
+ · · · valid for all values of x
sin x =
eix
− e−ix
2i
= x −
x3
3!
+
x5
5!
+ · · · valid for all values of x
tan x = x +
1
3
x3
+
2
15
x5
+ · · · valid for −
π
2
< x <
π
2
tan−1
x = x −
x3
3
+
x5
5
− · · · valid for −1 ≤ x ≤ 1
sin−1
x = x +
1
2
x3
3
+
1.3
2.4
x5
5
+ · · · valid for −1 < x < 1
2
5. Integer series
N
∑
1
n = 1 + 2 + 3 + · · · + N =
N(N + 1)
2
N
∑
1
n2
= 12
+ 22
+ 32
+ · · · + N2
=
N(N + 1)(2N + 1)
6
N
∑
1
n3
= 13
+ 23
+ 33
+ · · · + N3
= [1 + 2 + 3 + · · · N]2
=
N2
(N + 1)2
4
∞
∑
1
(−1)n+1
n
= 1 −
1
2
+
1
3
−
1
4
+ · · · = ln 2 [see expansion of ln(1 + x)]
∞
∑
1
(−1)n+1
2n − 1
= 1 −
1
3
+
1
5
−
1
7
+ · · · =
π
4
[see expansion of tan−1
x]
∞
∑
1
1
n2
= 1 +
1
4
+
1
9
+
1
16
+ · · · =
π2
6
N
∑
1
n(n + 1)(n + 2) = 1.2.3 + 2.3.4 + · · · + N(N + 1)(N + 2) =
N(N + 1)(N + 2)(N + 3)
4
This last result is a special case of the more general formula,
N
∑
1
n(n + 1)(n + 2) . . . (n + r) =
N(N + 1)(N + 2) . . . (N + r)(N + r + 1)
r + 2
.
Plane wave expansion
exp(ikz) = exp(ikr cosθ) =
∞
∑
l=0
(2l + 1)il
jl(kr)Pl(cosθ),
where Pl(cosθ) are Legendre polynomials (see section 11) and jl(kr) are spherical Bessel functions, defined by
jl(ρ) =
π
2ρ
Jl+1/2
(ρ), with Jl(x) the Bessel function of order l (see section 11).
2. Vector Algebra
If i, j, k are orthonormal vectors and A = Axi + Ay j + Azk then |A|2
= A2
x + A2
y + A2
z. [Orthonormal vectors ≡
orthogonal unit vectors.]
Scalar product
A · B = |A| |B| cosθ where θ is the angle between the vectors
= AxBx + AyBy + AzBz = [ Ax Ay Az ]
Bx
By
Bz
Scalar multiplication is commutative: A · B = B · A.
Equation of a line
A point r ≡ (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a + λb
with λ a real number.
3
6. Equation of a plane
A point r ≡ (x, y, z) is on a plane if either
(a) r · d = |d|, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
A×B = n |A| |B| sinθ, whereθ is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A × B in determinant form
i j k
Ax Ay Az
Bx By Bz
A × B in matrix form
0 −Az Ay
Az 0 −Ax
−Ay Ax 0
Bx
By
Bz
Vector multiplication is not commutative: A × B = −B × A.
Scalar triple product
A × B · C = A · B × C =
Ax Ay Az
Bx By Bz
Cx Cy Cz
= −A × C · B, etc.
Vector triple product
A × (B × C) = (A · C)B − (A · B)C, (A × B) × C = (A · C)B − (B · C)A
Non-orthogonal basis
A = A1e1 + A2e2 + A3e3
A1 = · A where =
e2 × e3
e1 · (e2 × e3)
Similarly for A2 and A3.
Summation convention
a = aiei implies summation over i = 1 . . . 3
a · b = aibi
(a × b)i = εijkajbk where ε123 = 1; εijk = −εikj
εijkεklm = δilδjm − δimδjl
4
7. 3. Matrix Algebra
Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., (I)ij = δij. If A is a square matrix of order n, then AI = IA = A. Also I = I−1
.
I is sometimes written as In if the order needs to be stated explicitly.
Products
If A is a (n × l) matrix and B is a (l × m) then the product AB is defined by
(AB)ij =
l
∑
k=1
AikBkj
In general AB = BA.
Transpose matrices
If A is a matrix, then transpose matrix AT
is such that (AT
)ij = (A)ji.
Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A−1
is such that AA−1
= A−1
A = I.
(A−1
)ij =
transpose of cofactor of Aij
|A|
where the cofactor of Aij is (−1)i+j
times the determinant of the matrix A with the j-th row and i-th column deleted.
Determinants
If A is a square matrix then the determinant of A, |A| (≡ det A) is defined by
|A| = ∑
i,j,k,...
ijk...A1i A2j A3k . . .
where the number of the suffixes is equal to the order of the matrix.
2×2 matrices
If A =
a b
c d
then,
|A| = ad − bc AT
=
a c
b d
A−1
=
1
|A|
d −b
−c a
Product rules
(AB . . . N)T
= NT
. . . BT
AT
(AB . . . N)−1
= N−1
. . . B−1
A−1
(if individual inverses exist)
|AB . . . N| = |A| |B| . . . |N| (if individual matrices are square)
Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns qi form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q−1
= QT
, |Q| = ±1, QT
is also orthogonal.
5
8. Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution x = A−1
b if A−1
exists, i.e., if |A| = 0.
If A is square then Ax = 0 has a non-trivial solution if and only if |A| = 0.
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number
of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the
error |Ax − b|) is the solution of the n equations AT
Ax = AT
b. If the columns of A are orthonormal vectors then
x = AT
b.
Hermitian matrices
The Hermitian conjugate of A is A† = (A∗
)T
, where A∗
is a matrix each of whose components is the complex
conjugate of the corresponding components of A. If A = A† then A is called a Hermitian matrix.
Eigenvalues and eigenvectors
The n eigenvalues λi and eigenvectors ui of an n × n matrix A are the solutions of the equation Au = λu. The
eigenvalues are the zeros of the polynomial of degree n, Pn(λ) = |A − λI|. If A is Hermitian then the eigenvalues
λi are real and the eigenvectors ui are mutually orthogonal. |A − λI| = 0 is called the characteristic equation of the
matrix A.
Tr A = ∑
i
λi, also |A| = ∏
i
λi.
If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the
matrix whose columns are the normalized eigenvectors of A, then
UT
SU = Λ and S = UΛUT
.
If x is an approximation to an eigenvector of A then xT
Ax/(xT
x) (Rayleigh’s quotient) is an approximation to the
corresponding eigenvalue.
Commutators
[A, B] ≡ AB − BA
[A, B] = −[B, A]
[A, B]† = [B†, A†]
[A + B, C] = [A, C] + [B, C]
[AB, C] = A[B, C] + [A, C]B
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
Hermitian algebra
b† = (b∗
1, b∗
2, . . .)
Matrix form Operator form Bra-ket form
Hermiticity b∗
· A · c = (A · b)∗
· c
Z
ψ∗
Oφ =
Z
(Oψ)∗
φ ψ|O|φ
Eigenvalues, λ real Aui = λ(i)ui Oψi = λ(i)ψi O |i = λi |i
Orthogonality ui · uj = 0
Z
ψ∗
i ψj = 0 i|j = 0 (i = j)
Completeness b = ∑
i
ui(ui · b) φ = ∑
i
ψi
Z
ψ∗
i φ φ = ∑
i
|i i|φ
Rayleigh–Ritz
Lowest eigenvalue λ0 ≤
b∗
· A · b
b∗
· b
λ0 ≤
Z
ψ∗
Oψ
Z
ψ∗
ψ
ψ|O|ψ
ψ|ψ
6
9. Pauli spin matrices
σx =
0 1
1 0
, σy =
0 −i
i 0
, σz =
1 0
0 −1
σxσy = iσz, σyσz = iσx, σzσx = iσy, σxσx = σyσy = σzσz = I
4. Vector Calculus
Notation
φ is a scalar function of a set of position coordinates. In Cartesian coordinates
φ = φ(x, y, z); in cylindrical polar coordinates φ = φ(ρ,ϕ, z); in spherical
polar coordinates φ = φ(r,θ,ϕ); in cases with radial symmetry φ = φ(r).
A is a vector function whose components are scalar functions of the position
coordinates: in Cartesian coordinates A = iAx + jAy + kAz, where Ax, Ay, Az
are independent functions of x, y, z.
In Cartesian coordinates (‘del’) ≡ i
∂
∂x
+ j
∂
∂y
+ k
∂
∂z
≡
∂
∂x
∂
∂y
∂
∂z
gradφ = φ, div A = · A, curl A = × A
Identities
grad(φ1 + φ2) ≡ gradφ1 + gradφ2 div(A1 + A2) ≡ div A1 + div A2
grad(φ1φ2) ≡ φ1 gradφ2 + φ2 gradφ1
curl(A + A) ≡ curl A1 + curl A2
div(φA) ≡ φ div A + (gradφ) · A, curl(φA) ≡ φ curl A + (gradφ) × A
div(A1 × A2) ≡ A2 · curl A1 − A1 · curl A2
curl(A1 × A2) ≡ A1 div A2 − A2 div A1 + (A2 · grad)A1 − (A1 · grad)A2
div(curl A) ≡ 0, curl(gradφ) ≡ 0
curl(curl A) ≡ grad(div A) − div(grad A) ≡ grad(div A) − 2
A
grad(A1 · A2) ≡ A1 × (curl A2) + (A1 · grad)A2 + A2 × (curl A1) + (A2 · grad)A1
7
10. Grad, Div, Curl and the Laplacian
Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates
Conversion to
Cartesian
Coordinates
x = ρ cosϕ y = ρ sinϕ z = z
x = r cosϕ sinθ y = r sinϕ sinθ
z = r cosθ
Vector A Axi + Ay j + Azk Aρρ + Aϕϕ + Azz Arr + Aθθ + Aϕϕ
Gradient φ
∂φ
∂x
i +
∂φ
∂y
j +
∂φ
∂z
k
∂φ
∂ρ
ρ +
1
ρ
∂φ
∂ϕ
ϕ +
∂φ
∂z
z
∂φ
∂r
r +
1
r
∂φ
∂θ
θ +
1
r sinθ
∂φ
∂ϕ
ϕ
Divergence
· A
∂Ax
∂x
+
∂Ay
∂y
+
∂Az
∂z
1
ρ
∂(ρAρ)
∂ρ
+
1
ρ
∂Aϕ
∂ϕ
+
∂Az
∂z
1
r2
∂(r2
Ar)
∂r
+
1
r sinθ
∂Aθ sinθ
∂θ
+
1
r sinθ
∂Aϕ
∂ϕ
Curl × A
i j k
∂
∂x
∂
∂y
∂
∂z
Ax Ay Az
1
ρ
ρ ϕ
1
ρ
z
∂
∂ρ
∂
∂ϕ
∂
∂z
Aρ ρAϕ Az
1
r2
sinθ
r
1
r sinθ
θ
1
r
ϕ
∂
∂r
∂
∂θ
∂
∂ϕ
Ar rAθ rAϕ sinθ
Laplacian
2
φ
∂2
φ
∂x2
+
∂2
φ
∂y2
+
∂2
φ
∂z2
1
ρ
∂
∂ρ
ρ
∂φ
∂ρ
+
1
ρ2
∂2
φ
∂ϕ2
+
∂2
φ
∂z2
1
r2
∂
∂r
r2 ∂φ
∂r
+
1
r2
sinθ
∂
∂θ
sinθ
∂φ
∂θ
+
1
r2
sin2
θ
∂2
φ
∂ϕ2
Transformation of integrals
L = the distance along some curve ‘C’ in space and is measured from some fixed point.
S = a surface area
τ = a volume contained by a specified surface
t = the unit tangent to C at the point P
n = the unit outward pointing normal
A = some vector function
dL = the vector element of curve (= t dL)
dS = the vector element of surface (= n dS)
Then
Z
C
A · t dL =
Z
C
A · dL
and when A = φ
Z
C
( φ) · dL =
Z
C
dφ
Gauss’s Theorem (Divergence Theorem)
When S defines a closed region having a volume τ
Z
τ
( · A) dτ =
Z
S
(A · n) dS =
Z
S
A · dS
also
Z
τ
( φ) dτ =
Z
S
φ dS
Z
τ
( × A) dτ =
Z
S
(n × A) dS
8
11. Stokes’s Theorem
When C is closed and bounds the open surface S,
Z
S
( × A) · dS =
Z
C
A · dL
also
Z
S
(n × φ) dS =
Z
C
φ dL
Green’s Theorem
Z
S
ψ φ · dS =
Z
τ
· (ψ φ) dτ
=
Z
τ
ψ 2
φ + ( ψ) · ( φ) dτ
Green’s Second Theorem
Z
τ
(ψ 2
φ − φ 2
ψ) dτ =
Z
S
[ψ( φ) − φ( ψ)] · dS
5. Complex Variables
Complex numbers
The complex number z = x + iy = r(cosθ + i sinθ) = r ei(θ+2nπ)
, where i2
= −1 and n is an arbitrary integer. The
real quantity r is the modulus of z and the angle θ is the argument of z. The complex conjugate of z is z∗
= x − iy =
r(cosθ − i sinθ) = r e−iθ
; zz∗
= |z|2
= x2
+ y2
De Moivre’s theorem
(cosθ + i sinθ)n
= einθ
= cos nθ + i sin nθ
Power series for complex variables.
ez
= 1 + z +
z2
2!
+ · · · +
zn
n!
+ · · · convergent for all finite z
sin z = z −
z3
3!
+
z5
5!
− · · · convergent for all finite z
cos z = 1 −
z2
2!
+
z4
4!
− · · · convergent for all finite z
ln(1 + z) = z −
z2
2
+
z3
3
− · · · principal value of ln(1 + z)
This last series converges both on and within the circle |z| = 1 except at the point z = −1.
tan−1
z = z −
z3
3
+
z5
5
− · · ·
This last series converges both on and within the circle |z| = 1 except at the points z = ±i.
(1 + z)n
= 1 + nz +
n(n − 1)
2!
z2
+
n(n − 1)(n − 2)
3!
z3
+ · · ·
This last series converges both on and within the circle |z| = 1 except at the point z = −1.
9
12. 6. Trigonometric Formulae
cos2
A + sin2
A = 1 sec2
A − tan2
A = 1 cosec2
A − cot2
A = 1
sin 2A = 2 sin A cos A cos 2A = cos2
A − sin2
A tan 2A =
2 tan A
1 − tan2
A
.
sin(A ± B) = sin A cos B ± cos A sin B cos A cos B =
cos(A + B) + cos(A − B)
2
cos(A ± B) = cos A cos B sin A sin B sin A sin B =
cos(A − B) − cos(A + B)
2
tan(A ± B) =
tan A ± tan B
1 tan A tan B
sin A cos B =
sin(A + B) + sin(A − B)
2
sin A + sin B = 2 sin
A + B
2
cos
A − B
2
sin A − sin B = 2 cos
A + B
2
sin
A − B
2
cos A + cos B = 2 cos
A + B
2
cos
A − B
2
cos A − cos B = −2 sin
A + B
2
sin
A − B
2
cos2
A =
1 + cos 2A
2
sin2
A =
1 − cos 2A
2
cos3
A =
3 cos A + cos 3A
4
sin3
A =
3 sin A − sin 3A
4
Relations between sides and angles of any plane triangle
In a plane triangle with angles A, B, and C and sides opposite a, b, and c respectively,
a
sin A
=
b
sin B
=
c
sin C
= diameter of circumscribed circle.
a2
= b2
+ c2
− 2bc cos A
a = b cos C + c cos B
cos A =
b2
+ c2
− a2
2bc
tan
A − B
2
=
a − b
a + b
cot
C
2
area =
1
2
ab sin C =
1
2
bc sin A =
1
2
ca sin B = s(s − a)(s − b)(s − c), where s =
1
2
(a + b + c)
Relations between sides and angles of any spherical triangle
In a spherical triangle with angles A, B, and C and sides opposite a, b, and c respectively,
sin a
sin A
=
sin b
sin B
=
sin c
sin C
cos a = cos b cos c + sin b sin c cos A
cos A = − cos B cos C + sin B sin C cos a
10
13. 7. Hyperbolic Functions
cosh x =
1
2
( ex
+ e−x
) = 1 +
x2
2!
+
x4
4!
+ · · · valid for all x
sinh x =
1
2
( ex
− e−x
) = x +
x3
3!
+
x5
5!
+ · · · valid for all x
cosh ix = cos x cos ix = cosh x
sinh ix = i sin x sin ix = i sinh x
tanh x =
sinh x
cosh x
sech x =
1
cosh x
coth x =
cosh x
sinh x
cosech x =
1
sinh x
cosh2
x − sinh2
x = 1
For large positive x:
cosh x ≈ sinh x →
ex
2
tanh x → 1
For large negative x:
cosh x ≈ − sinh x →
e−x
2
tanh x → −1
Relations of the functions
sinh x = − sinh(−x) sech x = sech(−x)
cosh x = cosh(−x) cosech x = − cosech(−x)
tanh x = − tanh(−x) coth x = − coth(−x)
sinh x =
2 tanh (x/2)
1 − tanh2
(x/2)
=
tanh x
1 − tanh2
x
cosh x =
1 + tanh2
(x/2)
1 − tanh2
(x/2)
=
1
1 − tanh2
x
tanh x = 1 − sech2
x sech x = 1 − tanh2
x
coth x = cosech2
x + 1 cosech x = coth2
x − 1
sinh(x/2) =
cosh x − 1
2
cosh(x/2) =
cosh x + 1
2
tanh(x/2) =
cosh x − 1
sinh x
=
sinh x
cosh x + 1
sinh(2x) = 2 sinh x cosh x tanh(2x) =
2 tanh x
1 + tanh2
x
cosh(2x) = cosh2
x + sinh2
x = 2 cosh2
x − 1 = 1 + 2 sinh2
x
sinh(3x) = 3 sinh x + 4 sinh3
x cosh 3x = 4 cosh3
x − 3 cosh x
tanh(3x) =
3 tanh x + tanh3
x
1 + 3 tanh2
x
11
14. sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
tanh(x ± y) =
tanh x ± tanh y
1 ± tanh x tanh y
sinh x + sinh y = 2 sinh
1
2
(x + y) cosh
1
2
(x − y) cosh x + cosh y = 2 cosh
1
2
(x + y) cosh
1
2
(x − y)
sinh x − sinh y = 2 cosh
1
2
(x + y) sinh
1
2
(x − y) cosh x − cosh y = 2 sinh
1
2
(x + y) sinh
1
2
(x − y)
sinh x ± cosh x =
1 ± tanh (x/2)
1 tanh(x/2)
= e±x
tanh x ± tanh y =
sinh(x ± y)
cosh x cosh y
coth x ± coth y = ±
sinh(x ± y)
sinh x sinh y
Inverse functions
sinh−1 x
a
= ln
x + x2 + a2
a
for −∞ < x < ∞
cosh−1 x
a
= ln
x + x2 − a2
a
for x ≥ a
tanh−1 x
a
=
1
2
ln
a + x
a − x
for x2
< a2
coth−1 x
a
=
1
2
ln
x + a
x − a
for x2
> a2
sech−1 x
a
= ln
a
x
+
a2
x2
− 1
for 0 < x ≤ a
cosech−1 x
a
= ln
a
x
+
a2
x2
+ 1
for x = 0
8. Limits
nc
xn
→ 0 as n → ∞ if |x| < 1 (any fixed c)
xn
/n! → 0 as n → ∞ (any fixed x)
(1 + x/n)n
→ ex
as n → ∞, x ln x → 0 as x → 0
If f (a) = g(a) = 0 then lim
x→a
f (x)
g(x)
=
f (a)
g (a)
(l’Hˆopital’s rule)
12
15. 9. Differentiation
(uv) = u v + uv ,
u
v
=
u v − uv
v2
(uv)(n)
= u(n)
v + nu(n−1)
v(1)
+ · · · + n
Cru(n−r)
v(r)
+ · · · + uv(n)
Leibniz Theorem
where n
Cr ≡
n
r
=
n!
r!(n − r)!
d
dx
(sin x) = cos x
d
dx
(sinh x) = cosh x
d
dx
(cos x) = − sin x
d
dx
(cosh x) = sinh x
d
dx
(tan x) = sec2
x
d
dx
(tanh x) = sech2
x
d
dx
(sec x) = sec x tan x
d
dx
(sech x) = − sech x tanh x
d
dx
(cot x) = − cosec2
x
d
dx
(coth x) = − cosech2
x
d
dx
(cosec x) = − cosec x cot x
d
dx
(cosech x) = − cosech x coth x
10. Integration
Standard forms
Z
xn
dx =
xn+1
n + 1
+ c for n = −1
Z
1
x
dx = ln x + c
Z
ln x dx = x(ln x − 1) + c
Z
eax
dx =
1
a
eax
+ c
Z
x eax
dx = eax x
a
−
1
a2
+ c
Z
x ln x dx =
x2
2
ln x −
1
2
+ c
Z
1
a2
+ x2
dx =
1
a
tan−1 x
a
+ c
Z
1
a2
− x2
dx =
1
a
tanh−1 x
a
+ c =
1
2a
ln
a + x
a − x
+ c for x2
< a2
Z
1
x2
− a2
dx = −
1
a
coth−1 x
a
+ c =
1
2a
ln
x − a
x + a
+ c for x2
> a2
Z
x
(x2
± a2
)n
dx =
−1
2(n − 1)
1
(x2
± a2
)n−1
+ c for n = 1
Z
x
x2
± a2
dx =
1
2
ln(x2
± a2
) + c
Z
1
a2 − x2
dx = sin−1 x
a
+ c
Z
1
x2 ± a2
dx = ln x + x2 ± a2 + c
Z
x
x2 ± a2
dx = x2 ± a2 + c
Z
a2 − x2 dx =
1
2
x a2 − x2 + a2
sin−1 x
a
+ c
13
16. Z ∞
0
1
(1 + x)xp dx = π cosec pπ for p < 1
Z ∞
0
cos(x2
) dx =
Z ∞
0
sin(x2
) dx =
1
2
π
2
Z ∞
−∞
exp(−x2
/2σ2
) dx = σ
√
2π
Z ∞
−∞
xn
exp(−x2
/2σ2
) dx =
1 × 3 × 5 × · · · (n − 1)σn+1
√
2π
0
for n ≥ 2 and even
for n ≥ 1 and odd
Z
sin x dx = − cos x + c
Z
sinh x dx = cosh x + c
Z
cos x dx = sin x + c
Z
cosh x dx = sinh x + c
Z
tan x dx = − ln(cos x) + c
Z
tanh x dx = ln(cosh x) + c
Z
cosec x dx = ln(cosec x − cot x) + c
Z
cosech x dx = ln [tanh(x/2)] + c
Z
sec x dx = ln(sec x + tan x) + c
Z
sech x dx = 2 tan−1
( ex
) + c
Z
cot x dx = ln(sin x) + c
Z
coth x dx = ln(sinh x) + c
Z
sin mx sin nx dx =
sin(m − n)x
2(m − n)
−
sin(m + n)x
2(m + n)
+ c if m2
= n2
Z
cos mx cos nx dx =
sin(m − n)x
2(m − n)
+
sin(m + n)x
2(m + n)
+ c if m2
= n2
Standard substitutions
If the integrand is a function of: substitute:
(a2
− x2
) or a2 − x2 x = a sinθ or x = a cosθ
(x2
+ a2
) or x2 + a2 x = a tanθ or x = a sinhθ
(x2
− a2
) or x2 − a2 x = a secθ or x = a coshθ
If the integrand is a rational function of sin x or cos x or both, substitute t = tan(x/2) and use the results:
sin x =
2t
1 + t2
cos x =
1 − t2
1 + t2
dx =
2 dt
1 + t2
.
If the integrand is of the form: substitute:
Z
dx
(ax + b) px + q
px + q = u2
Z
dx
(ax + b) px2 + qx + r
ax + b =
1
u
.
14
17. Integration by parts
Z b
a
u dv = uv
b
a
−
Z b
a
v du
Differentiation of an integral
If f (x,α) is a function of x containing a parameter α and the limits of integration a and b are functions of α then
d
dα
Z b(α)
a(α)
f (x,α) dx = f (b,α)
db
dα
− f (a,α)
da
dα
+
Z b(α)
a(α)
∂
∂α
f (x,α) dx.
Special case,
d
dx
Z x
a
f (y) dy = f (x).
Dirac δ-‘function’
δ(t − τ) =
1
2π
Z ∞
−∞
exp[iω(t − τ)] dω.
If f (t) is an arbitrary function of t then
Z ∞
−∞
δ(t − τ) f (t) dt = f (τ).
δ(t) = 0 if t = 0, also
Z ∞
−∞
δ(t) dt = 1
Reduction formulae
Factorials
n! = n(n − 1)(n − 2) . . . 1, 0! = 1.
Stirling’s formula for large n: ln(n!) ≈ n ln n − n.
For any p > −1,
Z ∞
0
xp
e−x
dx = p
Z ∞
0
xp−1
e−x
dx = p!. (−1/2)! =
√
π, (1/2)! =
√
π/2, etc.
For any p, q > −1,
Z 1
0
xp
(1 − x)q
dx =
p!q!
(p + q + 1)!
.
Trigonometrical
If m, n are integers,
Z π/2
0
sinm
θ cosn
θ dθ =
m − 1
m + n
Z π/2
0
sinm−2
θ cosn
θ dθ =
n − 1
m + n
Z π/2
0
sinm
θ cosn−2
θ dθ
and can therefore be reduced eventually to one of the following integrals
Z π/2
0
sinθ cosθ dθ =
1
2
,
Z π/2
0
sinθ dθ = 1,
Z π/2
0
cosθ dθ = 1,
Z π/2
0
dθ =
π
2
.
Other
If In =
Z ∞
0
xn
exp(−αx2
) dx then In =
(n − 1)
2α
In−2, I0 =
1
2
π
α
, I1 =
1
2α
.
15
18. 11. Differential Equations
Diffusion (conduction) equation
∂ψ
∂t
= κ 2
ψ
Wave equation
2
ψ =
1
c2
∂2
ψ
∂t2
Legendre’s equation
(1 − x2
)
d2
y
dx2
− 2x
dy
dx
+ l(l + 1)y = 0,
solutions of which are Legendre polynomials Pl(x), where Pl(x) =
1
2l
l!
d
dx
l
x2
− 1
l
, Rodrigues’ formula so
P0(x) = 1, P1(x) = x, P2(x) =
1
2
(3x2
− 1) etc.
Recursion relation
Pl(x) =
1
l
[(2l − 1)xPl−1(x) − (l − 1)Pl−2(x)]
Orthogonality
Z 1
−1
Pl(x)Pl (x) dx =
2
2l + 1
δll
Bessel’s equation
x2 d2
y
dx2
+ x
dy
dx
+ (x2
− m2
)y = 0,
solutions of which are Bessel functions Jm(x) of order m.
Series form of Bessel functions of the first kind
Jm(x) =
∞
∑
k=0
(−1)k
(x/2)m+2k
k!(m + k)!
(integer m).
The same general form holds for non-integer m > 0.
16
19. Laplace’s equation
2
u = 0
If expressed in two-dimensional polar coordinates (see section 4), a solution is
u(ρ,ϕ) = Aρn
+ Bρ−n
C exp(inϕ) + D exp(−inϕ)
where A, B, C, D are constants and n is a real integer.
If expressed in three-dimensional polar coordinates (see section 4) a solution is
u(r,θ,ϕ) = Arl
+ Br−(l+1)
Pm
l C sin mϕ + D cos mϕ
where l and m are integers with l ≥ |m| ≥ 0; A, B, C, D are constants;
Pm
l (cosθ) = sin|m|
θ
d
d(cosθ)
|m|
Pl(cosθ)
is the associated Legendre polynomial.
P0
l (1) = 1.
If expressed in cylindrical polar coordinates (see section 4), a solution is
u(ρ,ϕ, z) = Jm(nρ) A cos mϕ + B sin mϕ C exp(nz) + D exp(−nz)
where m and n are integers; A, B, C, D are constants.
Spherical harmonics
The normalized solutions Ym
l (θ,ϕ) of the equation
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂ϕ2
Ym
l + l(l + 1)Ym
l = 0
are called spherical harmonics, and have values given by
Ym
l (θ,ϕ) =
2l + 1
4π
(l − |m|)!
(l + |m|)!
Pm
l (cosθ) eimϕ
× (−1)m
for m ≥ 0
1 for m < 0
i.e., Y0
0 =
1
4π
, Y0
1 =
3
4π
cosθ, Y±1
1 =
3
8π
sinθ e±iϕ
, etc.
Orthogonality
Z
4π
Y∗m
l Ym
l dΩ = δll δmm
12. Calculus of Variations
The condition for I =
Z b
a
F(y, y , x) dx to have a stationary value is
∂F
∂y
=
d
dx
∂F
∂y
, where y =
dy
dx
. This is the
Euler–Lagrange equation.
17
20. 13. Functions of Several Variables
If φ = f (x, y, z, . . .) then
∂φ
∂x
implies differentiation with respect to x keeping y, z, . . . constant.
dφ =
∂φ
∂x
dx +
∂φ
∂y
dy +
∂φ
∂z
dz + · · · and δφ ≈
∂φ
∂x
δx +
∂φ
∂y
δy +
∂φ
∂z
δz + · · ·
where x, y, z, . . . are independent variables.
∂φ
∂x
is also written as
∂φ
∂x y,...
or
∂φ
∂x y,...
when the variables kept
constant need to be stated explicitly.
If φ is a well-behaved function then
∂2
φ
∂x ∂y
=
∂2
φ
∂y ∂x
etc.
If φ = f (x, y),
∂φ
∂x y
=
1
∂x
∂φ y
,
∂φ
∂x y
∂x
∂y φ
∂y
∂φ x
= −1.
Taylor series for two variables
If φ(x, y) is well-behaved in the vicinity of x = a, y = b then it has a Taylor series
φ(x, y) = φ(a + u, b + v) = φ(a, b) + u
∂φ
∂x
+ v
∂φ
∂y
+
1
2!
u2 ∂2
φ
∂x2
+ 2uv
∂2
φ
∂x ∂y
+ v2 ∂2
φ
∂y2
+ · · ·
where x = a + u, y = b + v and the differential coefficients are evaluated at x = a, y = b
Stationary points
A function φ = f (x, y) has a stationary point when
∂φ
∂x
=
∂φ
∂y
= 0. Unless
∂2
φ
∂x2
=
∂2
φ
∂y2
=
∂2
φ
∂x ∂y
= 0, the following
conditions determine whether it is a minimum, a maximum or a saddle point.
Minimum:
∂2
φ
∂x2
> 0, or
∂2
φ
∂y2
> 0,
Maximum:
∂2
φ
∂x2
< 0, or
∂2
φ
∂y2
< 0,
and
∂2
φ
∂x2
∂2
φ
∂y2
>
∂2
φ
∂x ∂y
2
Saddle point:
∂2
φ
∂x2
∂2
φ
∂y2
<
∂2
φ
∂x ∂y
2
If
∂2
φ
∂x2
=
∂2
φ
∂y2
=
∂2
φ
∂x ∂y
= 0 the character of the turning point is determined by the next higher derivative.
Changing variables: the chain rule
If φ = f (x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then
∂φ
∂u
=
∂φ
∂x
∂x
∂u
+
∂φ
∂y
∂y
∂u
+ · · ·
∂φ
∂v
=
∂φ
∂x
∂x
∂v
+
∂φ
∂y
∂y
∂v
+ · · ·
etc.
18
21. Changing variables in surface and volume integrals – Jacobians
If an area A in the x, y plane maps into an area A in the u, v plane then
Z
A
f (x, y) dx dy =
Z
A
f (u, v)J du dv where J =
∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
The Jacobian J is also written as
∂(x, y)
∂(u, v)
. The corresponding formula for volume integrals is
Z
V
f (x, y, z) dx dy dz =
Z
V
f (u, v, w)J du dv dw where now J =
∂x
∂u
∂x
∂v
∂x
∂w
∂y
∂u
∂y
∂v
∂y
∂w
∂z
∂u
∂z
∂v
∂z
∂w
14. Fourier Series and Transforms
Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π then
y(x) ≈ c0 +
M
∑
m=1
cm cos mx +
M
∑
m=1
sm sin mx
where the coefficients are
c0 =
1
2π
Z π
−π
y(x) dx
cm =
1
π
Z π
−π
y(x) cos mx dx (m = 1, . . . , M)
sm =
1
π
Z π
−π
y(x) sin mx dx (m = 1, . . . , M )
with convergence to y(x) as M, M → ∞ for all points where y(x) is continuous.
Fourier series for other ranges
Variable t, range 0 ≤ t ≤ T, (i.e., a periodic function of time with period T, frequency ω = 2π/T).
y(t) ≈ c0 + ∑cm cos mωt + ∑sm sin mωt
where
c0 =
ω
2π
Z T
0
y(t) dt, cm =
ω
π
Z T
0
y(t) cos mωt dt, sm =
ω
π
Z T
0
y(t) sin mωt dt.
Variable x, range 0 ≤ x ≤ L,
y(x) ≈ c0 + ∑cm cos
2mπx
L
+ ∑sm sin
2mπx
L
where
c0 =
1
L
Z L
0
y(x) dx, cm =
2
L
Z L
0
y(x) cos
2mπx
L
dx, sm =
2
L
Z L
0
y(x) sin
2mπx
L
dx.
19
22. Fourier series for odd and even functions
If y(x) is an odd (anti-symmetric) function [i.e., y(−x) = −y(x)] defined in the range −π ≤ x ≤ π, then only
sines are required in the Fourier series and sm =
2
π
Z π
0
y(x) sin mx dx. If, in addition, y(x) is symmetric about
x = π/2, then the coefficients sm are given by sm = 0 (for m even), sm =
4
π
Z π/2
0
y(x) sin mx dx (for m odd). If
y(x) is an even (symmetric) function [i.e., y(−x) = y(x)] defined in the range −π ≤ x ≤ π, then only constant
and cosine terms are required in the Fourier series and c0 =
1
π
Z π
0
y(x) dx, cm =
2
π
Z π
0
y(x) cos mx dx. If, in
addition, y(x) is anti-symmetric about x =
π
2
, then c0 = 0 and the coefficients cm are given by cm = 0 (for m even),
cm =
4
π
Z π/2
0
y(x) cos mx dx (for m odd).
[These results also apply to Fourier series with more general ranges provided appropriate changes are made to the
limits of integration.]
Complex form of Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π then
y(x) ≈
M
∑
−M
Cm eimx
, Cm =
1
2π
Z π
−π
y(x) e−imx
dx
with m taking all integer values in the range ±M. This approximation converges to y(x) as M → ∞ under the same
conditions as the real form.
For other ranges the formulae are:
Variable t, range 0 ≤ t ≤ T, frequency ω = 2π/T,
y(t) =
∞
∑
−∞
Cm eimωt
, Cm =
ω
2π
Z T
0
y(t) e−imωt
dt.
Variable x , range 0 ≤ x ≤ L,
y(x ) =
∞
∑
−∞
Cm ei2mπx /L
, Cm =
1
L
Z L
0
y(x ) e−i2mπx /L
dx .
Discrete Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π which is sampled in the 2N equally spaced points xn =
nx/N [n = −(N − 1) . . . N], then
y(xn) = c0 + c1 cos xn + c2 cos 2xn + · · · + cN−1 cos(N − 1)xn + cN cos Nxn
+ s1 sin xn + s2 sin 2xn + · · · + sN−1 sin(N − 1)xn + sN sin Nxn
where the coefficients are
c0 =
1
2N ∑ y(xn)
cm =
1
N ∑ y(xn) cos mxn (m = 1, . . . , N − 1)
cN =
1
2N ∑ y(xn) cos Nxn
sm =
1
N ∑ y(xn) sin mxn (m = 1, . . . , N − 1)
sN =
1
2N ∑ y(xn) sin Nxn
each summation being over the 2N sampling points xn.
20
23. Fourier transforms
If y(x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform y(ω) is defined by the equations
y(t) =
1
2π
Z ∞
−∞
y(ω) eiωt
dω, y(ω) =
Z ∞
−∞
y(t) e−iωt
dt.
If ω is replaced by 2πf, where f is the frequency, this relationship becomes
y(t) =
Z ∞
−∞
y( f ) ei2π f t
d f , y( f ) =
Z ∞
−∞
y(t) e−i2π f t
dt.
If y(t) is symmetric about t = 0 then
y(t) =
1
π
Z ∞
0
y(ω) cosωt dω, y(ω) = 2
Z ∞
0
y(t) cosωt dt.
If y(t) is anti-symmetric about t = 0 then
y(t) =
1
π
Z ∞
0
y(ω) sinωt dω, y(ω) = 2
Z ∞
0
y(t) sinωt dt.
Specific cases
y(t) = a, |t| ≤ τ
= 0, |t| > τ
(‘Top Hat’), y(ω) = 2a
sinωτ
ω
≡ 2aτ sinc(ωτ)
where sinc(x) =
sin(x)
x
y(t) = a(1 − |t|/τ), |t| ≤ τ
= 0, |t| > τ
(‘Saw-tooth’), y(ω) =
2a
ω2
τ
(1 − cosωτ) = aτ sinc2 ωτ
2
y(t) = exp(−t2
/t2
0) (Gaussian), y(ω) = t0
√
π exp −ω2
t2
0/4
y(t) = f (t) eiω0t
(modulated function), y(ω) = f (ω − ω0)
y(t) =
∞
∑
m=−∞
δ(t − mτ) (sampling function) y(ω) =
∞
∑
n=−∞
δ(ω − 2πn/τ)
21
24. Convolution theorem
If z(t) =
Z ∞
−∞
x(τ)y(t − τ) dτ =
Z ∞
−∞
x(t − τ)y(τ) dτ ≡ x(t) ∗ y(t) then z(ω) = x(ω) y(ω).
Conversely, xy = x ∗ y.
Parseval’s theorem
Z ∞
−∞
y∗
(t) y(t) dt =
1
2π
Z ∞
−∞
y∗
(ω) y(ω) dω (if y is normalised as on page 21)
Fourier transforms in two dimensions
V(k) =
Z
V(r) e−ik·r
d2
r
=
Z ∞
0
2πrV(r)J0(kr) dr if azimuthally symmetric
Fourier transforms in three dimensions
Examples
V(r) V(k)
1
4πr
1
k2
e−λr
4πr
1
k2
+ λ2
V(r) ikV(k)
2
V(r) −k2
V(k)
V(k) =
Z
V(r) e−ik·r
d3
r
=
4π
k
Z ∞
0
V(r) r sin kr dr if spherically symmetric
V(r) =
1
(2π)3
Z
V(k) eik·r
d3
k
22
25. 15. Laplace Transforms
If y(t) is a function defined for t ≥ 0, the Laplace transform y(s) is defined by the equation
y(s) = L{y(t)} =
Z ∞
0
e−st
y(t) dt
Function y(t) (t > 0) Transform y(s)
δ(t) 1 Delta function
θ(t)
1
s
Unit step function
tn n!
sn+1
t
1/2
1
2
π
s3
t−1/2
π
s
e−at 1
(s + a)
sinωt
ω
(s2 + ω2
cosωt
s
(s2 + ω2)
sinh ωt
ω
(s2 − ω2)
coshωt
s
(s2 − ω2)
e−at
y(t) y(s + a)
y(t − τ) θ(t − τ) e−sτ
y(s)
ty(t) −
dy
ds
dy
dt
sy(s) − y(0)
dn
y
dtn
sn
y(s) − sn−1
y(0) − sn−2 dy
dt 0
· · · −
dn−1
y
dtn−1
0
Z t
0
y(τ) dτ
y(s)
s
Z t
0
x(τ) y(t − τ) dτ
Z t
0
x(t − τ) y(τ) dτ
x(s) y(s) Convolution theorem
[Note that if y(t) = 0 for t < 0 then the Fourier transform of y(t) is y(ω) = y(iω).]
23
26. 16. Numerical Analysis
Finding the zeros of equations
If the equation is y = f (x) and xn is an approximation to the root then either
xn+1 = xn −
f (xn)
f (xn)
. (Newton)
or, xn+1 = xn −
xn − xn−1
f (xn) − f (xn−1)
f (xn) (Linear interpolation)
are, in general, better approximations.
Numerical integration of differential equations
If
dy
dx
= f (x, y) then
yn+1 = yn + h f (xn, yn) where h = xn+1 − xn (Euler method)
Putting y∗
n+1 = yn + h f (xn, yn) (improved Euler method)
then yn+1 = yn +
h[ f (xn, yn) + f (xn+1, y∗
n+1)]
2
Central difference notation
If y(x) is tabulated at equal intervals of x, where h is the interval, then δyn+1/2 = yn+1 − yn and
δ2
yn = δyn+1/2 − δyn−1/2
Approximating to derivatives
dy
dx n
≈
yn+1 − yn
h
≈
yn − yn−1
h
≈
δyn+1/2
+ δyn−1/2
2h
where h = xn+1 − xn
d2
y
dx2
n
≈
yn+1 − 2yn + yn−1
h2
=
δ2
yn
h2
Interpolation: Everett’s formula
y(x) = y(x0 + θh) ≈ θy0 + θy1 +
1
3!
θ(θ
2
− 1)δ2
y0 +
1
3!
θ(θ2
− 1)δ2
y1 + · · ·
where θ is the fraction of the interval h (= xn+1 − xn) between the sampling points and θ = 1 − θ. The first two
terms represent linear interpolation.
Numerical evaluation of definite integrals
Trapezoidal rule
The interval of integration is divided into n equal sub-intervals, each of width h; then
Z b
a
f (x) dx ≈ h c
1
2
f (a) + f (x1) + · · · + f (xj) + · · · +
1
2
f (b)
where h = (b − a)/n and xj = a + jh.
Simpson’s rule
The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h =
(b − a)/2n; then
Z b
a
f (x) dx ≈
h
3
f (a) + 4 f (x1) + 2 f (x2) + 4 f (x3) + · · · + 2 f (x2n−2) + 4 f (x2n−1) + f (b)
24
27. Gauss’s integration formulae
These have the general form
Z 1
−1
y(x) dx ≈
n
∑
1
ci y(xi)
For n = 2 : xi = ±0·5773; ci = 1, 1 (exact for any cubic).
For n = 3 : xi = −0·7746, 0·0, 0·7746; ci = 0·555, 0·888, 0·555 (exact for any quintic).
17. Treatment of Random Errors
Sample mean x =
1
n
(x1 + x2 + · · · xn)
Residual: d = x − x
Standard deviation of sample: s =
1
√
n
(d2
1 + d2
2 + · · · d2
n)1/2
Standard deviation of distribution: σ ≈
1
√
n − 1
(d2
1 + d2
2 + · · · d2
n)1/2
Standard deviation of mean: σm =
σ
√
n
=
1
n(n − 1)
(d2
1 + d2
2 + · · · d2
n)1/2
=
1
n(n − 1)
∑x2
i −
1
n ∑xi
2
1/2
Result of n measurements is quoted as x ± σm.
Range method
A quick but crude method of estimating σ is to find the range r of a set of n readings, i.e., the difference between
the largest and smallest values, then
σ ≈
r
√
n
.
This is usually adequate for n less than about 12.
Combination of errors
If Z = Z(A, B, . . .) (with A, B, etc. independent) then
(σZ)2
=
∂Z
∂A
σA
2
+
∂Z
∂B
σB
2
+ · · ·
So if
(i) Z = A ± B ± C, (σZ)2
= (σA)2
+ (σB)2
+ (σC)2
(ii) Z = AB or A/B,
σZ
Z
2
=
σA
A
2
+
σB
B
2
(iii) Z = Am
,
σZ
Z
= m
σA
A
(iv) Z = ln A, σZ =
σA
A
(v) Z = exp A,
σZ
Z
= σA
25
28. 18. Statistics
Mean and Variance
A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is
discrete, the probability that X = xi is Pi. When the distribution is continuous, the probability that X lies in an
interval δx is f (x)δx, where f (x) is the probability density function.
Mean µ = E(X) = ∑Pixi or
Z
x f (x) dx.
Variance σ2
= V(X) = E[(X − µ)2
] = ∑Pi(xi − µ)2
or
Z
(x − µ)2
f (x) dx.
Probability distributions
Error function: erf(x) =
2
√
π
Z x
0
e−y2
dy
Binomial: f (x) =
n
x
px
qn−x
where q = (1 − p), µ = np, σ2
= npq, p < 1.
Poisson: f (x) =
µx
x!
e−µ
, and σ2
= µ
Normal: f (x) =
1
σ
√
2π
exp −
(x − µ)2
2σ2
Weighted sums of random variables
If W = aX + bY then E(W) = aE(X) + bE(Y). If X and Y are independent then V(W) = a2
V(X) + b2
V(Y).
Statistics of a data sample x1, . . . , xn
Sample mean x =
1
n ∑xi
Sample variance s2
=
1
n ∑(xi − x)2
=
1
n ∑x2
i − x2
= E(x2
) − [E(x)]2
Regression (least squares fitting)
To fit a straight line by least squares to n pairs of points (xi, yi), model the observations by yi = α + β(xi − x) + i,
where the i are independent samples of a random variable with zero mean and variance σ2
.
Sample statistics: s2
x =
1
n ∑(xi − x)2
, s2
y =
1
n ∑(yi − y)2
, s2
xy =
1
n ∑(xi − x)(yi − y).
Estimators: α = y, β =
s2
xy
s2
x
; E(Y at x) = α + β(x − x); σ2
=
n
n − 2
(residual variance),
where residual variance =
1
n ∑{yi − α − β(xi − x)}2
= s2
y −
s4
xy
s2
x
.
Estimates for the variances of α and β are
σ2
n
and
σ2
ns2
x
.
Correlation coefficient: ρ = r =
s2
xy
sxsy
.
26