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18.156 Differential Analysis (MIT) 18.156 Differential Analysis (MIT)

Description

The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis. The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.

Subjects

Sobolev spaces | Sobolev spaces | Fredholm alternative | Fredholm alternative | Variable coefficient elliptic | parabolic and hyperbolic linear partial differential equations | Variable coefficient elliptic | parabolic and hyperbolic linear partial differential equations | Variational methods | Variational methods | Viscosity solutions of fully nonlinear partial differential equations | Viscosity solutions of fully nonlinear partial differential equations | Schauder theory | Schauder theory | Holder estimates | Holder estimates | linear equations | linear equations | second derivatives | second derivatives | elliptic | elliptic | parabolic | parabolic | nonlinear partial differential equations | nonlinear partial differential equations | linear partial differential equations | linear partial differential equations | harmonic functions | harmonic functions | elliptic equations | elliptic equations | parabolic equations | parabolic equations

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2.29 Numerical Marine Hydrodynamics (13.024) (MIT) 2.29 Numerical Marine Hydrodynamics (13.024) (MIT)

Description

Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects

License

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2.29 Numerical Fluid Mechanics (MIT) 2.29 Numerical Fluid Mechanics (MIT)

Description

This course introduces students to MATLAB®. Numerical methods include number representation and errors, interpolation, differentiation, integration, systems of linear equations, and Fourier interpolation and transforms. Students will study partial and ordinary differential equations as well as elliptic and parabolic differential equations, and solutions by numerical integration, finite difference methods, finite element methods, boundary element methods, and panel methods. This course introduces students to MATLAB®. Numerical methods include number representation and errors, interpolation, differentiation, integration, systems of linear equations, and Fourier interpolation and transforms. Students will study partial and ordinary differential equations as well as elliptic and parabolic differential equations, and solutions by numerical integration, finite difference methods, finite element methods, boundary element methods, and panel methods.

Subjects

numerical methods | numerical methods | interpolation | interpolation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | finite difference methods | finite difference methods | boundary integral equation panel methods | boundary integral equation panel methods | numerical lifting surface computations | numerical lifting surface computations | Fast Fourier Transforms | Fast Fourier Transforms | Numerical representation | Numerical representation | deterministic and random sea waves | deterministic and random sea waves | Integral boundary layer equations | Integral boundary layer equations | numerical solutions | numerical solutions

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10.34 Numerical Methods Applied to Chemical Engineering (MIT) 10.34 Numerical Methods Applied to Chemical Engineering (MIT)

Description

Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®. Acknowledgements The instructor would like to thank Robert Ashcraft, Sandeep Sharma, David Weingeist, and Nikolay Zaborenko for their work in preparing materials for this course site. Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®. Acknowledgements The instructor would like to thank Robert Ashcraft, Sandeep Sharma, David Weingeist, and Nikolay Zaborenko for their work in preparing materials for this course site.

Subjects

Matlab | Matlab | modern computational techniques in chemical engineering | modern computational techniques in chemical engineering | mathematical techniques in chemical engineering | mathematical techniques in chemical engineering | linear systems | linear systems | scientific computing | scientific computing | solving sets of nonlinear algebraic equations | solving sets of nonlinear algebraic equations | solving ordinary differential equations | solving ordinary differential equations | solving differential-algebraic (DAE) systems | solving differential-algebraic (DAE) systems | probability theory | probability theory | use of probability theory in physical modeling | use of probability theory in physical modeling | statistical analysis of data estimation | statistical analysis of data estimation | statistical analysis of parameter estimation | statistical analysis of parameter estimation | finite difference techniques | finite difference techniques | finite element techniques | finite element techniques | converting partial differential equations | converting partial differential equations | Navier-Stokes equations | Navier-Stokes equations

License

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13.002J Introduction to Numeric Analysis for Engineering (MIT) 13.002J Introduction to Numeric Analysis for Engineering (MIT)

Description

An introduction to the formulation, methodology, and techniques for numerical solution of engineering problems. Fundamental principles of digital computing and the implications for algorithm accuracy and stability. Error propagation and stability. The solution of systems of linear equations, including direct and iterative techniques. Roots of equations and systems of equations. Numerical interpolation, differentiation and integration. Fundamentals of finite-difference solutions to ordinary differential equations. Error and convergence analysis. Subject taught first half of term.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc. An introduction to the formulation, methodology, and techniques for numerical solution of engineering problems. Fundamental principles of digital computing and the implications for algorithm accuracy and stability. Error propagation and stability. The solution of systems of linear equations, including direct and iterative techniques. Roots of equations and systems of equations. Numerical interpolation, differentiation and integration. Fundamentals of finite-difference solutions to ordinary differential equations. Error and convergence analysis. Subject taught first half of term.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

digital computing | digital computing | algorithm accuracy | algorithm accuracy | error propagation | error propagation | linear equations | linear equations | iterative techniques | iterative techniques | roots of equations | roots of equations | systems of equations | systems of equations | numerical interpolation | numerical interpolation | differentiation | differentiation | integration | integration | finite-difference solutions | finite-difference solutions | differential equations | differential equations | 10.002J | 10.002J | 13.002 | 13.002 | 10.002 | 10.002

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13.024 Numerical Marine Hydrodynamics (MIT) 13.024 Numerical Marine Hydrodynamics (MIT)

Description

This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™ This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects | 2.29 | 2.29

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RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler (MIT) RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler (MIT)

Description

Includes audio/video content: AV lectures. Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler is an in-depth series of videos about differential equations and the MATLAB® ODE suite. These videos are suitable for students and life-long learners to enjoy.About the Instructors Gilbert Strang is the MathWorks Professor of Mathematics at MIT. His research focuses on mathematical analysis, linear algebra and PDEs. He has written textbooks on linear algebra, computational science, finite elements, wavelets, GPS, and calculus.Cleve Moler is chief mathematician, chairman, and cofounder of MathWorks. He was a professor of math and computer science for almost 20 years at the University of Michigan, Stanford University, and the University of New Mexico. These videos w Includes audio/video content: AV lectures. Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler is an in-depth series of videos about differential equations and the MATLAB® ODE suite. These videos are suitable for students and life-long learners to enjoy.About the Instructors Gilbert Strang is the MathWorks Professor of Mathematics at MIT. His research focuses on mathematical analysis, linear algebra and PDEs. He has written textbooks on linear algebra, computational science, finite elements, wavelets, GPS, and calculus.Cleve Moler is chief mathematician, chairman, and cofounder of MathWorks. He was a professor of math and computer science for almost 20 years at the University of Michigan, Stanford University, and the University of New Mexico. These videos w

Subjects

differential equations | differential equations | ODE | MATLAB | ODE | MATLAB | first order equations | first order equations | second order equations | second order equations | matrices | matrices | Laplace transform | Laplace transform | linear algebra | linear algebra | eigenvalues | eigenvalues | eigenvectors | eigenvectors | Fourier series | Fourier series | Runge-Kutta | Runge-Kutta | Tumbling box | Tumbling box | predator-prey equations | predator-prey equations | Lorenz Attractor | Lorenz Attractor

License

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2.993J Introduction to Numerical Analysis for Engineering (13.002J) (MIT) 2.993J Introduction to Numerical Analysis for Engineering (13.002J) (MIT)

Description

This course is offered to undergraduates and introduces students to the formulation, methodology, and techniques for numerical solution of engineering problems. Topics covered include: fundamental principles of digital computing and the implications for algorithm accuracy and stability, error propagation and stability, the solution of systems of linear equations, including direct and iterative techniques, roots of equations and systems of equations, numerical interpolation, differentiation and integration, fundamentals of finite-difference solutions to ordinary differential equations, and error and convergence analysis. The subject is taught the first half of the term. This subject was originally offered in Course 13 (Department of Ocean Engineering) as 13.002J. In 2005, ocean engineering This course is offered to undergraduates and introduces students to the formulation, methodology, and techniques for numerical solution of engineering problems. Topics covered include: fundamental principles of digital computing and the implications for algorithm accuracy and stability, error propagation and stability, the solution of systems of linear equations, including direct and iterative techniques, roots of equations and systems of equations, numerical interpolation, differentiation and integration, fundamentals of finite-difference solutions to ordinary differential equations, and error and convergence analysis. The subject is taught the first half of the term. This subject was originally offered in Course 13 (Department of Ocean Engineering) as 13.002J. In 2005, ocean engineering

Subjects

digital computing | digital computing | algorithm accuracy | algorithm accuracy | error propagation | error propagation | linear equations | linear equations | iterative techniques | iterative techniques | roots of equations | roots of equations | systems of equations | systems of equations | numerical interpolation | numerical interpolation | differentiation | differentiation | integration | integration | finite-difference solutions | finite-difference solutions | differential equations | differential equations | convergence analysis | convergence analysis

License

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18.156 Differential Analysis (MIT)

Description

The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.

Subjects

Sobolev spaces | Fredholm alternative | Variable coefficient elliptic | parabolic and hyperbolic linear partial differential equations | Variational methods | Viscosity solutions of fully nonlinear partial differential equations | Schauder theory | Holder estimates | linear equations | second derivatives | elliptic | parabolic | nonlinear partial differential equations | linear partial differential equations | harmonic functions | elliptic equations | parabolic equations

License

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail. The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

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16.920J Numerical Methods for Partial Differential Equations (SMA 5212) (MIT) 16.920J Numerical Methods for Partial Differential Equations (SMA 5212) (MIT)

Description

A presentation of the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: Mathematical Formulations; Finite Difference and Finite Volume Discretizations; Finite Element Discretizations; Boundary Element Discretizations; Direct and Iterative Solution Methods.This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5212 (Numerical Methods for Partial Differential Equations). A presentation of the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: Mathematical Formulations; Finite Difference and Finite Volume Discretizations; Finite Element Discretizations; Boundary Element Discretizations; Direct and Iterative Solution Methods.This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5212 (Numerical Methods for Partial Differential Equations).

Subjects

numerical methods | numerical methods | differential equations | differential equations | linear | linear | nonlinear | nonlinear | elliptic | elliptic | parabolic | parabolic | hyperbolic | hyperbolic | partial differential equations | partial differential equations | integral equations | integral equations | mathematical formulations | mathematical formulations | mathematics | mathematics | finite difference | finite difference | finite volume | finite volume | discretisation | discretisation | finite element | finite element | boundary element | boundary element | iteration | iteration | 16.920 | 16.920 | 2.097 | 2.097 | 6.339 | 6.339

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

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22.105 Electromagnetic Interactions (MIT) 22.105 Electromagnetic Interactions (MIT)

Description

This course is a graduate level subject on electromagnetic theory with particular emphasis on basics and applications to Nuclear Science and Engineering. The basic topics covered include electrostatics, magnetostatics, and electromagnetic radiation. The applications include transmission lines, waveguides, antennas, scattering, shielding, charged particle collisions, Bremsstrahlung radiation, and Cerenkov radiation. Acknowledgments Professor Freidberg would like to acknowledge the immense contributions made to this course by its previous instructors, Ian Hutchinson and Ron Parker. This course is a graduate level subject on electromagnetic theory with particular emphasis on basics and applications to Nuclear Science and Engineering. The basic topics covered include electrostatics, magnetostatics, and electromagnetic radiation. The applications include transmission lines, waveguides, antennas, scattering, shielding, charged particle collisions, Bremsstrahlung radiation, and Cerenkov radiation. Acknowledgments Professor Freidberg would like to acknowledge the immense contributions made to this course by its previous instructors, Ian Hutchinson and Ron Parker.

Subjects

electrostatics | electrostatics | coulomb's law | coulomb's law | gauss's law | gauss's law | potentials | potentials | laplace equations | laplace equations | poisson equations | poisson equations | capacitors | capacitors | resistors | resistors | child-langmuir law | child-langmuir law | magnetostatics | magnetostatics | ampere's law | ampere's law | biot-savart law | biot-savart law | magnets | magnets | inductors | inductors | superconducting magnets | superconducting magnets | single particle motion | single particle motion | lorentz force | lorentz force | quasi-statics | quasi-statics | faraday's law | faraday's law | maxwell equations | maxwell equations | plane waves | plane waves | reflection | reflection | refraction | refraction | klystrons | klystrons | gyrotrons | gyrotrons | lienard-wiechert potentials | lienard-wiechert potentials | thomson scattering | thomson scattering | compton scattering | compton scattering | synchrotron radiation | synchrotron radiation | bremsstrahlung radiation | bremsstrahlung radiation | cerenkov radiation | cerenkov radiation

License

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

License

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22.105 Electromagnetic Interactions (MIT) 22.105 Electromagnetic Interactions (MIT)

Description

This course is a graduate level subject on electromagnetic theory with particular emphasis on basics and applications to Nuclear Science and Engineering. The basic topics covered include electrostatics, magnetostatics, and electromagnetic radiation. The applications include transmission lines, waveguides, antennas, scattering, shielding, charged particle collisions, Bremsstrahlung radiation, and Cerenkov radiation. Acknowledgments Professor Freidberg would like to acknowledge the immense contributions made to this course by its previous instructors, Ian Hutchinson and Ron Parker. This course is a graduate level subject on electromagnetic theory with particular emphasis on basics and applications to Nuclear Science and Engineering. The basic topics covered include electrostatics, magnetostatics, and electromagnetic radiation. The applications include transmission lines, waveguides, antennas, scattering, shielding, charged particle collisions, Bremsstrahlung radiation, and Cerenkov radiation. Acknowledgments Professor Freidberg would like to acknowledge the immense contributions made to this course by its previous instructors, Ian Hutchinson and Ron Parker.

Subjects

electrostatics | electrostatics | coulomb's law | coulomb's law | gauss's law | gauss's law | potentials | potentials | laplace equations | laplace equations | poisson equations | poisson equations | capacitors | capacitors | resistors | resistors | child-langmuir law | child-langmuir law | magnetostatics | magnetostatics | ampere's law | ampere's law | biot-savart law | biot-savart law | magnets | magnets | inductors | inductors | superconducting magnets | superconducting magnets | single particle motion | single particle motion | lorentz force | lorentz force | quasi-statics | quasi-statics | faraday's law | faraday's law | maxwell equations | maxwell equations | plane waves | plane waves | reflection | reflection | refraction | refraction | klystrons | klystrons | gyrotrons | gyrotrons | lienard-wiechert potentials | lienard-wiechert potentials | thomson scattering | thomson scattering | compton scattering | compton scattering | synchrotron radiation | synchrotron radiation | bremsstrahlung radiation | bremsstrahlung radiation | cerenkov radiation | cerenkov radiation

License

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations | linear algebra | linear algebra | determinants | determinants | eigenvalues | eigenvalues | similarity | similarity | least-squares approximations | least-squares approximations | stability of differential equations | stability of differential equations | networks | networks

License

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18.307 Integral Equations (MIT) 18.307 Integral Equations (MIT)

Description

This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications. This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.

Subjects

integral equations | integral equations | applied mathematics | applied mathematics | Volterra equation | Volterra equation | Fredholm equation | Fredholm equation | Fredholm theory | Fredholm theory | Hilbert-Schmidt theorem | Hilbert-Schmidt theorem | Wiener-Hopf Method | Wiener-Hopf Method | partial differential equations | partial differential equations | Hilbert Problem | Hilbert Problem | ingular integral equations | ingular integral equations | Cauchy type | Cauchy type | inverse scattering transform | inverse scattering transform | group theory | group theory | fluid mechanics | fluid mechanics | solid mechanics | solid mechanics | acoustics | acoustics | quantum mechanics | quantum mechanics

License

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3.016 Mathematics for Materials Scientists and Engineers (MIT) 3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site. The class will cover mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from 3.012 to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site.

Subjects

energetics | energetics | materials structure and symmetry: applied fields | materials structure and symmetry: applied fields | mechanics and physics of solids and soft materials | mechanics and physics of solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalues | eigenvalues | eigenvectors | eigenvectors | quadratic forms | quadratic forms | tensor operations | tensor operations | symmetry operations | symmetry operations | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walks | random walks | mathematical technicques | mathematical technicques | materials science | materials science | materials engineering | materials engineering | materials structure | materials structure | symmetry | symmetry | applied fields | applied fields | materials response | materials response | solids mechanics | solids mechanics | solids physics | solids physics | soft materials | soft materials | multi-variable calculus | multi-variable calculus | ordinary differential equations | ordinary differential equations | partial differential equations | partial differential equations | applied mathematics | applied mathematics | mathematical techniques | mathematical techniques

License

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10.34 Numerical Methods Applied to Chemical Engineering (MIT) 10.34 Numerical Methods Applied to Chemical Engineering (MIT)

Description

This course focuses on the use of modern computational and mathematical techniques in chemical engineering. Starting from a discussion of linear systems as the basic computational unit in scientific computing, methods for solving sets of nonlinear algebraic equations, ordinary differential equations, and differential-algebraic (DAE) systems are presented. Probability theory and its use in physical modeling is covered, as is the statistical analysis of data and parameter estimation. The finite difference and finite element techniques are presented for converting the partial differential equations obtained from transport phenomena to DAE systems. The use of these techniques will be demonstrated throughout the course in the MATLAB® computing environment. This course focuses on the use of modern computational and mathematical techniques in chemical engineering. Starting from a discussion of linear systems as the basic computational unit in scientific computing, methods for solving sets of nonlinear algebraic equations, ordinary differential equations, and differential-algebraic (DAE) systems are presented. Probability theory and its use in physical modeling is covered, as is the statistical analysis of data and parameter estimation. The finite difference and finite element techniques are presented for converting the partial differential equations obtained from transport phenomena to DAE systems. The use of these techniques will be demonstrated throughout the course in the MATLAB® computing environment.

Subjects

Matlab | Matlab | modern computational techniques in chemical engineering | modern computational techniques in chemical engineering | mathematical techniques in chemical engineering | mathematical techniques in chemical engineering | linear systems | linear systems | scientific computing | scientific computing | solving sets of nonlinear algebraic equations | solving sets of nonlinear algebraic equations | solving ordinary differential equations | solving ordinary differential equations | solving differential-algebraic (DAE) systems | solving differential-algebraic (DAE) systems | probability theory | probability theory | use of probability theory in physical modeling | use of probability theory in physical modeling | statistical analysis of data estimation | statistical analysis of data estimation | statistical analysis of parameter estimation | statistical analysis of parameter estimation | finite difference techniques | finite difference techniques | finite element techniques | finite element techniques | converting partial differential equations | converting partial differential equations

License

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6.641 Electromagnetic Fields, Forces, and Motion (MIT) 6.641 Electromagnetic Fields, Forces, and Motion (MIT)

Description

Includes audio/video content: AV faculty introductions. This course examines electric and magnetic quasistatic forms of Maxwell's equations applied to dielectric, conduction, and magnetization boundary value problems. Topics covered include: electromagnetic forces, force densities, and stress tensors, including magnetization and polarization; thermodynamics of electromagnetic fields, equations of motion, and energy conservation; applications to synchronous, induction, and commutator machines; sensors and transducers; microelectromechanical systems; propagation and stability of electromechanical waves; and charge transport phenomena. Acknowledgments The instructor would like to thank Thomas Larsen and Matthew Pegler for transcribing into LaTeX the homework problems, homework solutions, and Includes audio/video content: AV faculty introductions. This course examines electric and magnetic quasistatic forms of Maxwell's equations applied to dielectric, conduction, and magnetization boundary value problems. Topics covered include: electromagnetic forces, force densities, and stress tensors, including magnetization and polarization; thermodynamics of electromagnetic fields, equations of motion, and energy conservation; applications to synchronous, induction, and commutator machines; sensors and transducers; microelectromechanical systems; propagation and stability of electromechanical waves; and charge transport phenomena. Acknowledgments The instructor would like to thank Thomas Larsen and Matthew Pegler for transcribing into LaTeX the homework problems, homework solutions, and

Subjects

electromagnetic | electromagnetic | electromagnetic field | electromagnetic field | forces | forces | motion | motion | electric | electric | magnetic | magnetic | quasistatic | quasistatic | Maxwell's equations | Maxwell's equations | dielectric | dielectric | conduction | conduction | magnetization | magnetization | boundary value problems | boundary value problems | force densities | force densities | stress tensors | stress tensors | polarization | polarization | thermodynamics | thermodynamics | equations of motion | equations of motion | energy conservation | energy conservation | synchronous | synchronous | induction | induction | commutator machines | commutator machines | sensors | sensors | transducers | transducers | microelectromechanical systems | microelectromechanical systems | electromechanical waves | electromechanical waves | charge transport phenomena | charge transport phenomena

License

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12.950 Atmospheric and Oceanic Modeling (MIT) 12.950 Atmospheric and Oceanic Modeling (MIT)

Description

The numerical methods, formulation and parameterizations used in models of the circulation of the atmosphere and ocean will be described in detail. Widely used numerical methods will be the focus but we will also review emerging concepts and new methods. The numerics underlying a hierarchy of models will be discussed, ranging from simple GFD models to the high-end GCMs. In the context of ocean GCMs, we will describe parameterization of geostrophic eddies, mixing and the surface and bottom boundary layers. In the atmosphere, we will review parameterizations of convection and large scale condensation, the planetary boundary layer and radiative transfer. The numerical methods, formulation and parameterizations used in models of the circulation of the atmosphere and ocean will be described in detail. Widely used numerical methods will be the focus but we will also review emerging concepts and new methods. The numerics underlying a hierarchy of models will be discussed, ranging from simple GFD models to the high-end GCMs. In the context of ocean GCMs, we will describe parameterization of geostrophic eddies, mixing and the surface and bottom boundary layers. In the atmosphere, we will review parameterizations of convection and large scale condensation, the planetary boundary layer and radiative transfer.

Subjects

numerical methods | numerical methods | formulation | formulation | parameterizations | parameterizations | models of the circulation of the atmosphere and ocean | models of the circulation of the atmosphere and ocean | numerics underlying a hierarchy of models | numerics underlying a hierarchy of models | simple GFD models | simple GFD models | high-end GCMs | high-end GCMs | ocean GCMs | ocean GCMs | parameterization of geostrophic eddies | parameterization of geostrophic eddies | mixing | mixing | surface and bottom boundary layers | surface and bottom boundary layers | atmosphere | atmosphere | parameterizations of convection | parameterizations of convection | large scale condensation | large scale condensation | planetary boundary layer | planetary boundary layer | radiative transfer | radiative transfer | finite difference method | finite difference method | Spatial discretization | Spatial discretization | numerical dispersion | numerical dispersion | Series expansion | Series expansion | Time-stepping | Time-stepping | Space-time discretization | Space-time discretization | Shallow water dynamics | Shallow water dynamics | Barotropic models | Barotropic models | Quasi-geostrophic equations | Quasi-geostrophic equations | Quasi-geostrophic models | Quasi-geostrophic models | Eddy parameterization | Eddy parameterization | Vertical coordinates | Vertical coordinates | primitive equations | primitive equations | Boundary layer parameterizations | Boundary layer parameterizations

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc. Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations

License

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18.152 Introduction to Partial Differential Equations (MIT) 18.152 Introduction to Partial Differential Equations (MIT)

Description

This course introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. It includes mathematical tools, real-world examples and applications. This course introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. It includes mathematical tools, real-world examples and applications.

Subjects

diffusion | diffusion | elliptic | elliptic | hyperbolic | hyperbolic | partial differential equation | partial differential equation | Initial and boundary value problems for ordinary differential equations | Initial and boundary value problems for ordinary differential equations | Sturm-Liouville theory and eigenfunction expansions | Sturm-Liouville theory and eigenfunction expansions | initial value problems | initial value problems | wave equation;heat equation | wave equation;heat equation | Dirichlet problem | Dirichlet problem | Laplace operator and potential theory | Laplace operator and potential theory | Black-Scholes equation | Black-Scholes equation | water waves | water waves | scalar conservation laws | scalar conservation laws | first order equations | first order equations

License

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14.451 Dynamic Optimization Methods with Applications (MIT) 14.451 Dynamic Optimization Methods with Applications (MIT)

Description

This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from a dynamic programming and optimal control perspective. We also study the dynamic systems that come from the solutions to these problems. The course will illustrate how these techniques are useful in various applications, drawing on many economic examples. However, the focus will remain on gaining a general command of the tools so that they can be applied later in other classes. This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from a dynamic programming and optimal control perspective. We also study the dynamic systems that come from the solutions to these problems. The course will illustrate how these techniques are useful in various applications, drawing on many economic examples. However, the focus will remain on gaining a general command of the tools so that they can be applied later in other classes.

Subjects

vector spaces | vector spaces | principle of optimality | principle of optimality | concavity of the value function | concavity of the value function | differentiability of the value function | differentiability of the value function | Euler equations | Euler equations | deterministic dynamics | deterministic dynamics | models with constant returns to scale | models with constant returns to scale | nonstationary models | nonstationary models | stochastic dynamic programming | stochastic dynamic programming | stochastic Euler equations | stochastic Euler equations | stochastic dynamics | stochastic dynamics | calculus of variations | calculus of variations | the maximum principle | the maximum principle | discounted infinite-horizon optimal control | discounted infinite-horizon optimal control | saddle-path stability | saddle-path stability

License

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13.002J Introduction to Numeric Analysis for Engineering (MIT)

Description

An introduction to the formulation, methodology, and techniques for numerical solution of engineering problems. Fundamental principles of digital computing and the implications for algorithm accuracy and stability. Error propagation and stability. The solution of systems of linear equations, including direct and iterative techniques. Roots of equations and systems of equations. Numerical interpolation, differentiation and integration. Fundamentals of finite-difference solutions to ordinary differential equations. Error and convergence analysis. Subject taught first half of term.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

digital computing | algorithm accuracy | error propagation | linear equations | iterative techniques | roots of equations | systems of equations | numerical interpolation | differentiation | integration | finite-difference solutions | differential equations | 10.002J | 13.002 | 10.002

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