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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

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18.01 Single Variable Calculus (MIT) 18.01 Single Variable Calculus (MIT)

Description

This introductory Calculus course covers differentiation and integration of functions of one variable, with applications. Topics include:Concepts of function, limits, and continuityDifferentiation rules, application to graphing, rates, approximations, and extremum problemsDefinite and indefinite integrationFundamental theorem of calculusApplications of integration to geometry and scienceElementary functionsTechniques of integrationApproximation of definite integrals, improper integrals, and L'Hôpital's rule MATLAB® is a trademark of The MathWorks, Inc. This introductory Calculus course covers differentiation and integration of functions of one variable, with applications. Topics include:Concepts of function, limits, and continuityDifferentiation rules, application to graphing, rates, approximations, and extremum problemsDefinite and indefinite integrationFundamental theorem of calculusApplications of integration to geometry and scienceElementary functionsTechniques of integrationApproximation of definite integrals, improper integrals, and L'Hôpital's rule MATLAB® is a trademark of The MathWorks, Inc.

Subjects

differentiation and integration of functions of one variable | differentiation and integration of functions of one variable | limits | limits | continuity | continuity | differentiation rules | differentiation rules | extremum problems | extremum problems | definite and indefinite integration | definite and indefinite integration | fundamental theorem of calculus | fundamental theorem of calculus | elementary | elementary | techniques of integration | techniques of integration | approximation of definite integrals | approximation of definite integrals | improper integrals | improper integrals | l'H?pital's rule | l'H?pital's rule | single variable calculus | single variable calculus | mathematical applications | mathematical applications | function | function | graphing | graphing | rates | rates | approximations | approximations | definite integration | definite integration | indefinite integration | indefinite integration | geometry | geometry | science | science | elementary functions | elementary functions | definite integrals | definite integrals

License

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18.02SC Multivariable Calculus (MIT) 18.02SC Multivariable Calculus (MIT)

Description

Includes audio/video content: AV lectures. This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics. Includes audio/video content: AV lectures. This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.

Subjects

calculus | calculus | calculus of several variables | calculus of several variables | vector algebra | vector algebra | determinants | determinants | matrix | matrix | matrices | matrices | vector-valued function | vector-valued function | space motion | space motion | scalar function | scalar function | partial differentiation | partial differentiation | gradient | gradient | optimization techniques | optimization techniques | double integrals | double integrals | line integrals | line integrals | exact differential | exact differential | conservative fields | conservative fields | Green's theorem | Green's theorem | triple integrals | triple integrals | surface integrals | surface integrals | divergence theorem Stokes' theorem | divergence theorem Stokes' theorem | applications | applications

License

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18.02 Multivariable Calculus (MIT) 18.02 Multivariable Calculus (MIT)

Description

Includes audio/video content: AV lectures. This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates. Includes audio/video content: AV lectures. This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates.

Subjects

calculus | calculus | calculus of several variables | calculus of several variables | vector algebra | vector algebra | determinants | determinants | matrix | matrix | matrices | matrices | vector-valued function | vector-valued function | space motion | space motion | scalar function | scalar function | partial differentiation | partial differentiation | gradient | gradient | optimization techniques | optimization techniques | double integrals | double integrals | line integrals | line integrals | exact differential | exact differential | conservative fields | conservative fields | Green's theorem | Green's theorem | triple integrals | triple integrals | surface integrals | surface integrals | divergence theorem Stokes' theorem | divergence theorem Stokes' theorem | applications | applications

License

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18.024 Calculus with Theory II (MIT) 18.024 Calculus with Theory II (MIT)

Description

This course is a continuation of 18.014. It covers the same material as 18.02 (Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.Topics include: Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor This course is a continuation of 18.014. It covers the same material as 18.02 (Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.Topics include: Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor

Subjects

linear algebra | linear algebra | vector integral calculus | vector integral calculus | Calculus of several variables | Calculus of several variables | Vector algebra in 3-space | Vector algebra in 3-space | determinants | determinants | matrices | matrices | Vector-valued functions of one variable | Vector-valued functions of one variable | space motion | space motion | Scalar functions of several variables: partial differentiation | Scalar functions of several variables: partial differentiation | gradient | gradient | optimization techniques | optimization techniques | Double integrals and line integrals in the plane | Double integrals and line integrals in the plane | exact differentials and conservative fields | exact differentials and conservative fields | Green's theorem and applications | Green's theorem and applications | triple integrals | triple integrals | line and surface integrals in space | line and surface integrals in space | Divergence theorem | Divergence theorem | Stokes' theorem | Stokes' theorem | applications | applications

License

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18.014 Calculus with Theory I (MIT) 18.014 Calculus with Theory I (MIT)

Description

18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site. 18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site.

Subjects

axioms for the real numbers | axioms for the real numbers | the Riemann integral | the Riemann integral | limits | limits | theorems on continuous functions | theorems on continuous functions | derivatives of functions of one variablethe fundamental theorems of calculus | derivatives of functions of one variablethe fundamental theorems of calculus | Taylor's theorem | Taylor's theorem | infinite series | infinite series | power series | power series | rigorous treatment of the elementary functions | rigorous treatment of the elementary functions

License

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14.30 Introduction to Statistical Method in Economics (MIT) 14.30 Introduction to Statistical Method in Economics (MIT)

Description

This course will provide a solid foundation in probability and statistics for economists and other social scientists. We will emphasize topics needed in the further study of econometrics and provide basic preparation for 14.32. No prior preparation in probability and statistics is required, but familiarity with basic algebra and calculus is assumed. This course will provide a solid foundation in probability and statistics for economists and other social scientists. We will emphasize topics needed in the further study of econometrics and provide basic preparation for 14.32. No prior preparation in probability and statistics is required, but familiarity with basic algebra and calculus is assumed.

Subjects

Economics | Economics | statistics | statistics | methods | methods | probability | probability | economists | economists | social scientists | social scientists | econometrics | econometrics | algebra | algebra | calculus | calculus

License

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16.323 Principles of Optimal Control (MIT) 16.323 Principles of Optimal Control (MIT)

Description

This course studies the principles of deterministic optimal control. It uses variational calculus and Pontryagin's maximum principle. It focuses on applications of the theory, including optimal feedback control, time-optimal control, and others. Dynamic programming and numerical search algorithms are introduced briefly. This course studies the principles of deterministic optimal control. It uses variational calculus and Pontryagin's maximum principle. It focuses on applications of the theory, including optimal feedback control, time-optimal control, and others. Dynamic programming and numerical search algorithms are introduced briefly.

Subjects

nonlinear optimization | nonlinear optimization | linear quadratic regulators | linear quadratic regulators | MATLAB implementation | MATLAB implementation | dynamic programming | dynamic programming | calculus of variations | calculus of variations | LQR | LQR | LQG | LQG | stochastic optimization | stochastic optimization | on-line optimization and control | on-line optimization and control | constrained optimization | constrained optimization | signals | signals | system norms | system norms | Model Predictive Behavior | Model Predictive Behavior | quadratic programming | quadratic programming | mixed-integer linear programming | mixed-integer linear programming | linear programming | linear programming

License

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14.471 Public Economics I (MIT) 14.471 Public Economics I (MIT)

Description

This course is a one-semester introduction to the economic analysis of taxation. It covers both theoretical contributions, such as the theory of optimal income and commodity taxation, as well as empirical work, such as the study of how taxes affect labor supply. The course is designed to acquaint students with key questions in the economics of taxation, and to equip them to carry out their own research in this field. This course is a one-semester introduction to the economic analysis of taxation. It covers both theoretical contributions, such as the theory of optimal income and commodity taxation, as well as empirical work, such as the study of how taxes affect labor supply. The course is designed to acquaint students with key questions in the economics of taxation, and to equip them to carry out their own research in this field.

Subjects

economic analysis | economic analysis | taxation | taxation | wealth | wealth | financial policy | financial policy | income | income | investment | investment | asset | asset | political economy | political economy | labor | labor | capital | capital | public policy | public policy | theory | theory | evidence | evidence | government taxation policy | government taxation policy | tax incidence | tax incidence | optimal tax theory | optimal tax theory | labor supply | labor supply | savings | savings | corrective taxes for externalities | corrective taxes for externalities | corporate behavior | corporate behavior | tax expenditure policy | tax expenditure policy | theory of optimal income | theory of optimal income | commodity taxation | commodity taxation | calculus-based microeconomic analysis | calculus-based microeconomic analysis | duality methods | duality methods | household theory | household theory | firm theory | firm theory | growth theory | growth theory

License

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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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18.013A Calculus with Applications (MIT) 18.013A Calculus with Applications (MIT)

Description

Differential calculus in one and several dimensions. Java applets and spreadsheet assignments. Vector algebra in 3D, vector- valued functions, gradient, divergence and curl, Taylor series, numerical methods and applications. Given in the first half of the first term. However, those wishing credit for 18.013A only, must attend the entire semester. Prerequisites: a year of high school calculus or the equivalent, with a score of 4 or 5 on the AB, or the AB portion of the BC, Calculus test, or an equivalent score on a standard international exam, or a passing grade on the first half of the 18.01 Advanced Standing exam.Technical RequirementsThe MathML version of the textbook requires the Internet Explorer 6 browser or above with the MathPlayer plug-in  or Netscape 7.x / Mozill Differential calculus in one and several dimensions. Java applets and spreadsheet assignments. Vector algebra in 3D, vector- valued functions, gradient, divergence and curl, Taylor series, numerical methods and applications. Given in the first half of the first term. However, those wishing credit for 18.013A only, must attend the entire semester. Prerequisites: a year of high school calculus or the equivalent, with a score of 4 or 5 on the AB, or the AB portion of the BC, Calculus test, or an equivalent score on a standard international exam, or a passing grade on the first half of the 18.01 Advanced Standing exam.Technical RequirementsThe MathML version of the textbook requires the Internet Explorer 6 browser or above with the MathPlayer plug-in  or Netscape 7.x / Mozill

Subjects

vector algebra | vector algebra | taylor series | taylor series | numerical methods | numerical methods | differential calculus | differential calculus | 18.013 | 18.013

License

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17.881 Game Theory and Political Theory (MIT) 17.881 Game Theory and Political Theory (MIT)

Description

Increasingly, political scientists are using game theory to analyze strategic interactions across many different settings. Each of the sub-fields, to differing degrees, has seen game theoretic concepts enter its vocabulary, and students entering the profession will need to understand the potential and limits of game theory. This course aims to give students an entry-level understanding of the basic concepts of game theory, and how these concepts have been applied to the study of political phenomena. Because an important component of game theory in political science and political economy is the analysis of substantive political phenomena, we will cover illustrative examples each week in combination with methodological developments. The political and economic phenomena that we will examine Increasingly, political scientists are using game theory to analyze strategic interactions across many different settings. Each of the sub-fields, to differing degrees, has seen game theoretic concepts enter its vocabulary, and students entering the profession will need to understand the potential and limits of game theory. This course aims to give students an entry-level understanding of the basic concepts of game theory, and how these concepts have been applied to the study of political phenomena. Because an important component of game theory in political science and political economy is the analysis of substantive political phenomena, we will cover illustrative examples each week in combination with methodological developments. The political and economic phenomena that we will examine

Subjects

game theory | game theory | game theoretic concepts | game theoretic concepts | games of complete information | games of complete information | games of incomplete information | games of incomplete information | political phenomena | political phenomena | legislative rules | legislative rules | nuclear deterrence | nuclear deterrence | electoral competition | electoral competition | imperfect markets | imperfect markets | probability | probability | calculus | calculus

License

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2.086 Numerical Computation for Mechanical Engineers (MIT) 2.086 Numerical Computation for Mechanical Engineers (MIT)

Description

Includes audio/video content: AV special element video. This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis. Assignments require MATLAB® programming. Includes audio/video content: AV special element video. This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis. Assignments require MATLAB® programming.

Subjects

MATLAB | MATLAB | numerical analysis | numerical analysis | programming | programming | physical modeling | physical modeling | calculus | calculus | linear algebra | linear algebra | Monte Carlo Method | Monte Carlo Method | differential equations | differential equations | nonlinear systems | nonlinear systems

License

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18.01SC Single Variable Calculus (MIT) 18.01SC Single Variable Calculus (MIT)

Description

Includes audio/video content: AV lectures. This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics. Includes audio/video content: AV lectures. This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.

Subjects

differentiation of functions | differentiation of functions | integration of functions | integration of functions | limits | limits | continuity | continuity | differentiation rules | differentiation rules | extremum problems | extremum problems | definite integration | definite integration | indefinite integration | indefinite integration | fundamental theorem of calculus | fundamental theorem of calculus | techniques of integration | techniques of integration | approximation of definite integrals | approximation of definite integrals | improper integrals | improper integrals | l'Hôpital's rule | l'Hôpital's rule

License

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18.01 Single Variable Calculus (MIT) 18.01 Single Variable Calculus (MIT)

Description

Includes audio/video content: AV lectures. This introductory calculus course covers differentiation and integration of functions of one variable, with applications. Includes audio/video content: AV lectures. This introductory calculus course covers differentiation and integration of functions of one variable, with applications.

Subjects

differentiation and integration of functions of one variable | differentiation and integration of functions of one variable | limits | limits | continuity | continuity | differentiation rules | differentiation rules | extremum problems | extremum problems | definite and indefinite integration | definite and indefinite integration | fundamental theorem of calculus | fundamental theorem of calculus | elementary | elementary | techniques of integration | techniques of integration | approximation of definite integrals | approximation of definite integrals | improper integrals | improper integrals | l'H?pital's rule | l'H?pital's rule

License

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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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1.060 Engineering Mechanics II (MIT) 1.060 Engineering Mechanics II (MIT)

Description

This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from calculus and differential equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics. This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from calculus and differential equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics.

Subjects

fluid mechanics | fluid mechanics | fluids | fluids | civil and environmental engineering | civil and environmental engineering | differential equations | differential equations | calculus | calculus | flow | flow | movement | movement | wave forms | wave forms | Bernoulli's theorem | Bernoulli's theorem | wavelets | wavelets | mechanics | mechanics | solids | solids | hydrostatics | hydrostatics | mass | mass | momentum | momentum | energy | energy | flow nets | flow nets | velocity | velocity | laminar flow | laminar flow | turbulent flow | turbulent flow | groundwater | groundwater | hydraulics | hydraulics | backwater curves | backwater curves

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2.035 Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations (MIT) 2.035 Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations (MIT)

Description

This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. App This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. App

Subjects

calculus of variations | calculus of variations | linear algebra | linear algebra | solid mechanics | solid mechanics | fluid mechanics | fluid mechanics | energy methods | energy methods | microstructures of crystalline | microstructures of crystalline

License

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

Description

This course covers the 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 the materials science and engineering core courses (3.012 and 3.014) 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, and fourier analysis. Users may find additional or updated materials at Professor C This course covers the 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 the materials science and engineering core courses (3.012 and 3.014) 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, and fourier analysis. Users may find additional or updated materials at Professor C

Subjects

energetics | energetics | visualization | visualization | graph | graph | plot | plot | chart | chart | materials science | materials science | DMSE | DMSE | structure | structure | symmetry | symmetry | mechanics | mechanics | physicss | physicss | solids and soft materials | solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalue | eigenvalue | eigenvector | eigenvector | quadratic form | quadratic form | tensor operation | tensor operation | symmetry operation | symmetry operation | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | ODE | ODE | solution | solution | vector | vector | matrix | matrix | determinant | determinant | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walk | random walk | Mathematica | Mathematica | simulation | simulation

License

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6.055J The Art of Approximation in Science and Engineering (MIT) 6.055J The Art of Approximation in Science and Engineering (MIT)

Description

This course teaches simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications are drawn from the physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis is on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world. This course teaches simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications are drawn from the physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis is on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world.

Subjects

approximation | approximation | science | science | engineering | engineering | managing complexity | managing complexity | divide and conquer | divide and conquer | heterogeneous hierarchies | heterogeneous hierarchies | homogeneous hierarchies | homogeneous hierarchies | proportional reasoning | proportional reasoning | conservation/box models | conservation/box models | dimensional analysis | dimensional analysis | special cases | special cases | extreme cases | extreme cases | discretization | discretization | spring models | spring models | symmetry | symmetry | invariance | invariance | discarding information | discarding information | oil imports | oil imports | tree representations | tree representations | gold | gold | random walks | random walks | UNIX | UNIX | triangle bisection | triangle bisection | pentagonal heat flow | pentagonal heat flow | jump heights | jump heights | simple calculus | simple calculus | drag | drag | cycling | cycling | swimming | swimming | flying | flying | flight | flight | algebraic symmetry | algebraic symmetry | densities | densities | hydrogen size | hydrogen size | bending of light | bending of light | Buckingham Pi Theorem | Buckingham Pi Theorem | pulley acceleration | pulley acceleration | waves | waves

License

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6.827 Multithreaded Parallelism: Languages and Compilers (MIT) 6.827 Multithreaded Parallelism: Languages and Compilers (MIT)

Description

The topics covered in this course include: Languages and compilers to exploit multithreaded parallelism Implicit parallel programming using functional languages and their extensions Higher-order functions, non-strictness, and polymorphism Explicit parallel programming and nondeterminism The lambda calculus and its variants Term rewriting and operational semantics Compiling multithreaded code for symmetric multiprocessors and clusters Static analysis and compiler optimizations This course is worth 4 Engineering Design Points. The topics covered in this course include: Languages and compilers to exploit multithreaded parallelism Implicit parallel programming using functional languages and their extensions Higher-order functions, non-strictness, and polymorphism Explicit parallel programming and nondeterminism The lambda calculus and its variants Term rewriting and operational semantics Compiling multithreaded code for symmetric multiprocessors and clusters Static analysis and compiler optimizations This course is worth 4 Engineering Design Points.

Subjects

languages | languages | compilers | compilers | multithreaded parallelism | multithreaded parallelism | implicit parallel programming | implicit parallel programming | higher order functions | higher order functions | non-strictness | non-strictness | polymorphism | polymorphism | explicit parallel programming | explicit parallel programming | nondeterminism | nondeterminism | lambda calculus | lambda calculus | term rewriting | term rewriting | symmetric multiprocessors | symmetric multiprocessors | clusters | clusters | static analysis | static analysis | compiler optimizations | compiler optimizations

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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15.450 Analytics of Finance (MIT) 15.450 Analytics of Finance (MIT)

Description

This course covers the key quantitative methods of finance: financial econometrics and statistical inference for financial applications; dynamic optimization; Monte Carlo simulation; stochastic (Itô) calculus. These techniques, along with their computer implementation, are covered in depth. Application areas include portfolio management, risk management, derivatives, and proprietary trading. This course covers the key quantitative methods of finance: financial econometrics and statistical inference for financial applications; dynamic optimization; Monte Carlo simulation; stochastic (Itô) calculus. These techniques, along with their computer implementation, are covered in depth. Application areas include portfolio management, risk management, derivatives, and proprietary trading.

Subjects

financial econometrics | financial econometrics | statistical inference | statistical inference | dynamic optimization | dynamic optimization | Monte Carlo simulation | Monte Carlo simulation | stochastic (Itô) calculus | stochastic (Itô) calculus | portfolio management | portfolio management | risk management | risk management | proprietary trading | proprietary trading | derivative pricing | derivative pricing | generalized method of moments | generalized method of moments | Black-Scholes model | Black-Scholes model

License

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16.323 Principles of Optimal Control (MIT) 16.323 Principles of Optimal Control (MIT)

Description

This course studies basic optimization and the principles of optimal control. It considers deterministic and stochastic problems for both discrete and continuous systems. The course covers solution methods including numerical search algorithms, model predictive control, dynamic programming, variational calculus, and approaches based on Pontryagin's maximum principle, and it includes many examples and applications of the theory. This course studies basic optimization and the principles of optimal control. It considers deterministic and stochastic problems for both discrete and continuous systems. The course covers solution methods including numerical search algorithms, model predictive control, dynamic programming, variational calculus, and approaches based on Pontryagin's maximum principle, and it includes many examples and applications of the theory.

Subjects

nonlinear optimization | nonlinear optimization | dynamic programming | dynamic programming | HJB Equation | HJB Equation | calculus of variations | calculus of variations | constrained optimal control | constrained optimal control | singular arcs | singular arcs | stochastic optimal control | stochastic optimal control | LQG robustness | LQG robustness | feedback control systems | feedback control systems | model predictive control | model predictive control | line search methods | line search methods | Lagrange multipliers | Lagrange multipliers | discrete LQR | discrete LQR

License

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18.100A Introduction to Analysis (MIT) 18.100A Introduction to Analysis (MIT)

Description

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Subjects

mathematical analysis | mathematical analysis | estimations | estimations | limit of a sequence | limit of a sequence | limit theorems | limit theorems | subsequences | subsequences | cluster points | cluster points | infinite series | infinite series | power series | power series | local and global properties | local and global properties | continuity | continuity | intermediate-value theorem | intermediate-value theorem | convexity | convexity | integrability | integrability | Riemann integral | Riemann integral | calculus | calculus | convergence | convergence | Gamma function | Gamma function | Stirling | Stirling | quantifiers and negation | quantifiers and negation | Leibniz | Leibniz | Fubini | Fubini | improper integrals | improper integrals | Lebesgue integral | Lebesgue integral | mathematical proofs | mathematical proofs | differentiation | differentiation | integration | integration

License

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