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8.033 Relativity (MIT) 8.033 Relativity (MIT)

Description

Relativity is normally taken by physics majors in their sophomore year. Topics include: Einstein's postulates; consequences for simultaneity, time dilation, length contraction, clock synchronization; Lorentz transformation; relativistic effects and paradoxes; Minkowski diagrams; invariants and four-vectors; momentum, energy and mass; and particle collisions. Also covered is: Relativity and electricity; Coulomb's law; and magnetic fields. Brief introduction to Newtonian cosmology. There is also an introduction to some concepts of General Relativity; principle of equivalence; the Schwarzchild metric; gravitational red shift, particle and light trajectories, geodesics, and Shapiro delay. Relativity is normally taken by physics majors in their sophomore year. Topics include: Einstein's postulates; consequences for simultaneity, time dilation, length contraction, clock synchronization; Lorentz transformation; relativistic effects and paradoxes; Minkowski diagrams; invariants and four-vectors; momentum, energy and mass; and particle collisions. Also covered is: Relativity and electricity; Coulomb's law; and magnetic fields. Brief introduction to Newtonian cosmology. There is also an introduction to some concepts of General Relativity; principle of equivalence; the Schwarzchild metric; gravitational red shift, particle and light trajectories, geodesics, and Shapiro delay.Subjects

Einstein's postulates | Einstein's postulates | consequences for simultaneity | time dilation | length contraction | clock synchronization | consequences for simultaneity | time dilation | length contraction | clock synchronization | Lorentz transformation | Lorentz transformation | relativistic effects and paradoxes | relativistic effects and paradoxes | Minkowski diagrams | Minkowski diagrams | invariants and four-vectors | invariants and four-vectors | momentum | energy and mass | momentum | energy and mass | particle collisions | particle collisions | Relativity and electricity | Relativity and electricity | Coulomb's law | Coulomb's law | magnetic fields | magnetic fields | Newtonian cosmology | Newtonian cosmology | General Relativity | General Relativity | principle of equivalence | principle of equivalence | the Schwarzchild metric | the Schwarzchild metric | gravitational red shift | particle and light trajectories | geodesics | Shapiro delay | gravitational red shift | particle and light trajectories | geodesics | Shapiro delay | gravitational red shift | gravitational red shift | particle trajectories | particle trajectories | light trajectories | light trajectories | invariants | invariants | four-vectors | four-vectors | momentum | momentum | energy | energy | mass | mass | relativistic effects | relativistic effects | paradoxes | paradoxes | electricity | electricity | time dilation | time dilation | length contraction | length contraction | clock synchronization | clock synchronization | Schwarzchild metric | Schwarzchild metric | geodesics | geodesics | Shaprio delay | Shaprio delay | relativistic kinematics | relativistic kinematics | relativistic dynamics | relativistic dynamics | electromagnetism | electromagnetism | hubble expansion | hubble expansion | universe | universe | equivalence principle | equivalence principle | curved space time | curved space time | Ether Theory | Ether Theory | constants | constants | speed of light | speed of light | c | c | graph | graph | pythagorem theorem | pythagorem theorem | triangle | triangle | arrows | arrowsLicense

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See all metadata2.26 Compressible Fluid Dynamics (MIT) 2.26 Compressible Fluid Dynamics (MIT)

Description

2.26 is a 6-unit Honors-level subject serving as the Mechanical Engineering department's sole course in compressible fluid dynamics. The prerequisites for this course are undergraduate courses in thermodynamics, fluid dynamics, and heat transfer. The goal of this course is to lay out the fundamental concepts and results for the compressible flow of gases. Topics to be covered include: appropriate conservation laws; propagation of disturbances; isentropic flows; normal shock wave relations, oblique shock waves, weak and strong shocks, and shock wave structure; compressible flows in ducts with area changes, friction, or heat addition; heat transfer to high speed flows; unsteady compressible flows, Riemann invariants, and piston and shock tube problems; steady 2D supersonic flow, Prandtl-Mey 2.26 is a 6-unit Honors-level subject serving as the Mechanical Engineering department's sole course in compressible fluid dynamics. The prerequisites for this course are undergraduate courses in thermodynamics, fluid dynamics, and heat transfer. The goal of this course is to lay out the fundamental concepts and results for the compressible flow of gases. Topics to be covered include: appropriate conservation laws; propagation of disturbances; isentropic flows; normal shock wave relations, oblique shock waves, weak and strong shocks, and shock wave structure; compressible flows in ducts with area changes, friction, or heat addition; heat transfer to high speed flows; unsteady compressible flows, Riemann invariants, and piston and shock tube problems; steady 2D supersonic flow, Prandtl-MeySubjects

conservation laws | conservation laws | isentropic flows | isentropic flows | normal shock wave relations | normal shock wave relations | oblique shock waves | oblique shock waves | weak shock | weak shock | strong shock | strong shock | ducts | ducts | heat transfer | heat transfer | unsteady flows | unsteady flows | Riemann invariants | Riemann invariants | piston | piston | shock tube | shock tube | steady 2D supersonic flow | steady 2D supersonic flow | Prandtl-Meyer function | Prandtl-Meyer function | self-similar compressible flows | self-similar compressible flowsLicense

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See all metadata8.033 Relativity (MIT) 8.033 Relativity (MIT)

Description

This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. Topics include Einstein's postulates, the Lorentz transformation, relativistic effects and paradoxes, and applications involving electromagnetism and particle physics. This course also provides a brief introduction to some concepts of general relativity, including the principle of equivalence, the Schwartzschild metric and black holes, and the FRW metric and cosmology. This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. Topics include Einstein's postulates, the Lorentz transformation, relativistic effects and paradoxes, and applications involving electromagnetism and particle physics. This course also provides a brief introduction to some concepts of general relativity, including the principle of equivalence, the Schwartzschild metric and black holes, and the FRW metric and cosmology.Subjects

relativity | relativity | special relativity | special relativity | Einstein's postulates | Einstein's postulates | simultaneity | simultaneity | time dilation | time dilation | length contraction | length contraction | clock synchronization | clock synchronization | Lorentz transformation | Lorentz transformation | relativistic effects | relativistic effects | Minkowski diagrams | Minkowski diagrams | relativistic invariants | relativistic invariants | four-vectors | four-vectors | relativitistic particle collisions | relativitistic particle collisions | relativity and electricity | relativity and electricity | Coulomb's law | Coulomb's law | magnetic fields | magnetic fields | Newtonian cosmology | Newtonian cosmology | general relativity | general relativity | Schwarzchild metric | Schwarzchild metric | gravitational | gravitational | red shift | red shift | light trajectories | light trajectories | geodesics | geodesics | Shapiro delay | Shapiro delayLicense

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See all metadata8.20 Introduction to Special Relativity (MIT) 8.20 Introduction to Special Relativity (MIT)

Description

This course introduces the basic ideas and equations of Einstein's Special Theory of Relativity. If you have hoped to understand the physics of Lorentz contraction, time dilation, the "twin paradox", and E=mc2, you're in the right place.AcknowledgementsProf. Knuteson wishes to acknowledge that this course was originally designed and taught by Prof. Robert Jaffe. This course introduces the basic ideas and equations of Einstein's Special Theory of Relativity. If you have hoped to understand the physics of Lorentz contraction, time dilation, the "twin paradox", and E=mc2, you're in the right place.AcknowledgementsProf. Knuteson wishes to acknowledge that this course was originally designed and taught by Prof. Robert Jaffe.Subjects

Einstein's Special Theory of Relativity | Einstein's Special Theory of Relativity | Lorentz transformations | Lorentz transformations | length contraction | length contraction | time dilation | time dilation | four vectors | four vectors | Lorentz invariants | Lorentz invariants | relativistic energy and momentum | relativistic energy and momentum | relativistic kinematics | relativistic kinematics | Doppler shift | Doppler shift | space-time diagrams | space-time diagrams | relativity paradoxes | relativity paradoxes | General Relativity | General RelativityLicense

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The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory. The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.Subjects

intersection theory | intersection theory | moduli spaces | moduli spaces | geometry of homogeneous varieties | geometry of homogeneous varieties | Deligne-Mumford moduli spaces | Deligne-Mumford moduli spaces | stable curves | stable curves | Kontsevich moduli spaces | Kontsevich moduli spaces | stable maps | stable maps | Littlewood-Richardson rules | Littlewood-Richardson rules | Grassmannians | Grassmannians | divisor theory | divisor theory | cohomology | cohomology | Brill-Noether theory | Brill-Noether theory | limit linear series | limit linear series | ample cones | ample cones | effective cones | effective cones | Gromov-Witten invariants | Gromov-Witten invariants | simple homogeneous varieties | simple homogeneous varietiesLicense

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See all metadata6.005 Elements of Software Construction (MIT) 6.005 Elements of Software Construction (MIT)

Description

This course introduces fundamental principles and techniques of software development. Students learn how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; state machines; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. The course includes weekly programming exercises and two substantial group projects. This course introduces fundamental principles and techniques of software development. Students learn how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; state machines; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. The course includes weekly programming exercises and two substantial group projects.Subjects

software development | software development | specifications | specifications | invariants | invariants | state machines | state machines | test-driven development | test-driven development | design patterns | design patterns | object-oriented programming | object-oriented programming | concurrent programming | concurrent programming | functional programming | functional programmingLicense

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See all metadata6.005 Elements of Software Construction (MIT) 6.005 Elements of Software Construction (MIT)

Description

This course provides an introduction to the fundamental principles and techniques of software development that have greatest impact on practice. Topics include capturing the essence of a problem by recognizing and inventing suitable abstractions; key paradigms, including state machines, functional programming, and object-oriented programming; use of design patterns to bridge gap between models and code; the role of interfaces and specification in achieving modularity and decoupling; reasoning about code using invariants; testing, test-case generation and coverage; and essentials of programming with objects, functions, and abstract types. The course includes exercises in modeling, design, implementation and reasoning. This course provides an introduction to the fundamental principles and techniques of software development that have greatest impact on practice. Topics include capturing the essence of a problem by recognizing and inventing suitable abstractions; key paradigms, including state machines, functional programming, and object-oriented programming; use of design patterns to bridge gap between models and code; the role of interfaces and specification in achieving modularity and decoupling; reasoning about code using invariants; testing, test-case generation and coverage; and essentials of programming with objects, functions, and abstract types. The course includes exercises in modeling, design, implementation and reasoning.Subjects

software development | software development | java programming | java programming | java | java | invariants | invariants | decoupling | decoupling | data abstraction | data abstraction | state machine | state machine | module dependency | module dependency | object model | object model | model view controller | model view controller | mvc | mvc | client server | client server | eclipse | eclipse | junit | junit | subversion | subversion | swing | swing | design | design | implement | implement | midi player | midi player | sat solver | sat solver | photo organizer | photo organizer | testing | testing | coverage | coverage | event based programming | event based programming | concurrency | concurrencyLicense

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12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov 12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ovSubjects

classical mechanics | classical mechanics | phase space | phase space | computation | computation | Lagrangian formulation | Lagrangian formulation | action | action | variational principles | variational principles | equations of motion | equations of motion | Hamilton's principle | Hamilton's principle | conserved quantities | conserved quantities | rigid bodies and tops | rigid bodies and tops | Hamiltonian formulation | Hamiltonian formulation | canonical equations | canonical equations | surfaces of section | surfaces of section | chaos | chaos | canonical transformations | canonical transformations | generating functions | generating functions | Liouville's theorem | Liouville's theorem | Poincar? integral invariants | Poincar? integral invariants | Poincar?-Birkhoff | Poincar?-Birkhoff | KAM theorem | KAM theorem | invariant curves | invariant curves | cantori | cantori | nonlinear resonances | nonlinear resonances | resonance overlap | resonance overlap | transition to chaos | transition to chaos | chaotic motion | chaotic motion | 12.620 | 12.620 | 6.946 | 6.946 | 8.351 | 8.351License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata6.005 Elements of Software Construction (MIT) 6.005 Elements of Software Construction (MIT)

Description

This course introduces fundamental principles and techniques of software development. Students learn how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; state machines; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. The course includes weekly programming exercises and two substantial group projects. This course introduces fundamental principles and techniques of software development. Students learn how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; state machines; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. The course includes weekly programming exercises and two substantial group projects.Subjects

software development | software development | specifications | specifications | invariants | invariants | state machines | state machines | test-driven development | test-driven development | design patterns | design patterns | object-oriented programming | object-oriented programming | concurrent programming | concurrent programming | functional programming | functional programmingLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata6.042J Mathematics for Computer Science (MIT) 6.042J Mathematics for Computer Science (MIT)

Description

This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science). This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science).Subjects

mathematical definitions | mathematical definitions | proofs and applicable methods | proofs and applicable methods | formal logic notation | formal logic notation | proof methods | proof methods | induction | induction | well-ordering | well-ordering | sets | sets | relations | relations | elementary graph theory | elementary graph theory | integer congruences | integer congruences | asymptotic notation and growth of functions | asymptotic notation and growth of functions | permutations and combinations | counting principles | permutations and combinations | counting principles | discrete probability | discrete probability | recursive definition | recursive definition | structural induction | structural induction | state machines and invariants | state machines and invariants | recurrences | recurrences | generating functions | generating functions | permutations and combinations | permutations and combinations | counting principles | counting principles | discrete mathematics | discrete mathematics | computer science | computer scienceLicense

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See all metadata6.042J Mathematics for Computer Science (MIT) 6.042J Mathematics for Computer Science (MIT)

Description

This course is offered to undergraduates and is an elementary discrete mathematics course oriented towards applications in computer science and engineering. Topics covered include: formal logic notation, induction, sets and relations, permutations and combinations, counting principles, and discrete probability. This course is offered to undergraduates and is an elementary discrete mathematics course oriented towards applications in computer science and engineering. Topics covered include: formal logic notation, induction, sets and relations, permutations and combinations, counting principles, and discrete probability.Subjects

Elementary discrete mathematics for computer science and engineering | Elementary discrete mathematics for computer science and engineering | mathematical definitions | mathematical definitions | proofs and applicable methods | proofs and applicable methods | formal logic notation | formal logic notation | proof methods | proof methods | induction | induction | well-ordering | well-ordering | sets | sets | relations | relations | elementary graph theory | elementary graph theory | integer congruences | integer congruences | asymptotic notation and growth of functions | asymptotic notation and growth of functions | permutations and combinations | permutations and combinations | counting principles | counting principles | discrete probability | discrete probability | recursive definition | recursive definition | structural induction | structural induction | state machines and invariants | state machines and invariants | recurrences | recurrences | generating functions | generating functions | 6.042 | 6.042 | 18.062 | 18.062License

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See all metadata16.55 Ionized Gases (MIT) 16.55 Ionized Gases (MIT)

Description

This course highlights the properties and behavior of low-temperature plasmas in relation to energy conversion, plasma propulsion, and gas lasers. The course includes material on the equilibrium (energy states, statistical mechanics, and relationship to thermodynamics) and kinetic theory of ionized gases (motion of charged particles, distribution function, collisions, characteristic lengths and times, cross sections, and transport properties). In addition, the course discusses gas surface interactions (thermionic emission, sheaths, and probe theory) and radiation in plasmas and diagnostics. This course highlights the properties and behavior of low-temperature plasmas in relation to energy conversion, plasma propulsion, and gas lasers. The course includes material on the equilibrium (energy states, statistical mechanics, and relationship to thermodynamics) and kinetic theory of ionized gases (motion of charged particles, distribution function, collisions, characteristic lengths and times, cross sections, and transport properties). In addition, the course discusses gas surface interactions (thermionic emission, sheaths, and probe theory) and radiation in plasmas and diagnostics.Subjects

Ionized gases | Ionized gases | plasma physics | plasma physics | motion of charges | motion of charges | drift | drift | adiabatic invariants | adiabatic invariants | collision theory | collision theory | kinetic theory | kinetic theory | H theorem | H theorem | entropy | entropy | Maxwellian distribution | Maxwellian distribution | Boltzmann equation | Boltzmann equation | plasma sheath | plasma sheath | electrostatic probe | electrostatic probe | orbital motion limit | orbital motion limit | equilibrium statistical mechanics | equilibrium statistical mechanics | radiation transport | radiation transportLicense

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Relativity is normally taken by physics majors in their sophomore year. Topics include: Einstein's postulates; consequences for simultaneity, time dilation, length contraction, clock synchronization; Lorentz transformation; relativistic effects and paradoxes; Minkowski diagrams; invariants and four-vectors; momentum, energy and mass; and particle collisions. Also covered is: Relativity and electricity; Coulomb's law; and magnetic fields. Brief introduction to Newtonian cosmology. There is also an introduction to some concepts of General Relativity; principle of equivalence; the Schwarzchild metric; gravitational red shift, particle and light trajectories, geodesics, and Shapiro delay.Subjects

Einstein's postulates | consequences for simultaneity | time dilation | length contraction | clock synchronization | Lorentz transformation | relativistic effects and paradoxes | Minkowski diagrams | invariants and four-vectors | momentum | energy and mass | particle collisions | Relativity and electricity | Coulomb's law | magnetic fields | Newtonian cosmology | General Relativity | principle of equivalence | the Schwarzchild metric | gravitational red shift | particle and light trajectories | geodesics | Shapiro delay | gravitational red shift | particle trajectories | light trajectories | invariants | four-vectors | momentum | energy | mass | relativistic effects | paradoxes | electricity | time dilation | length contraction | clock synchronization | Schwarzchild metric | geodesics | Shaprio delay | relativistic kinematics | relativistic dynamics | electromagnetism | hubble expansion | universe | equivalence principle | curved space time | Ether Theory | constants | speed of light | c | graph | pythagorem theorem | triangle | arrowsLicense

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See all metadata8.20 Introduction to Special Relativity (MIT) 8.20 Introduction to Special Relativity (MIT)

Description

Introduces the basic ideas and equations of Einstein's Special Theory of Relativity. Topics include: Lorentz transformations, length contraction and time dilation, four vectors, Lorentz invariants, relativistic energy and momentum, relativistic kinematics, Doppler shift, space-time diagrams, relativity paradoxes, and some concepts of General Relativity. Introduces the basic ideas and equations of Einstein's Special Theory of Relativity. Topics include: Lorentz transformations, length contraction and time dilation, four vectors, Lorentz invariants, relativistic energy and momentum, relativistic kinematics, Doppler shift, space-time diagrams, relativity paradoxes, and some concepts of General Relativity.Subjects

Einstein's Special Theory of Relativity | Einstein's Special Theory of Relativity | Lorentz transformations | Lorentz transformations | length contraction | length contraction | ime dilation | ime dilation | time dilation | time dilation | four vectors | four vectors | Lorentz invariants | Lorentz invariants | relativistic energy and momentum | relativistic energy and momentum | relativistic kinematics | relativistic kinematics | Doppler shift | Doppler shift | space-time diagrams | space-time diagrams | relativity paradoxes | relativity paradoxes | General Relativity | General RelativityLicense

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See all metadata6.042J Mathematics for Computer Science (MIT) 6.042J Mathematics for Computer Science (MIT)

Description

This course is offered to undergraduates and is an elementary discrete mathematics course oriented towards applications in computer science and engineering. Topics covered include: formal logic notation, induction, sets and relations, permutations and combinations, counting principles, and discrete probability. This course is offered to undergraduates and is an elementary discrete mathematics course oriented towards applications in computer science and engineering. Topics covered include: formal logic notation, induction, sets and relations, permutations and combinations, counting principles, and discrete probability.Subjects

Elementary discrete mathematics for computer science and engineering | Elementary discrete mathematics for computer science and engineering | mathematical definitions | mathematical definitions | proofs and applicable methods | proofs and applicable methods | formal logic notation | formal logic notation | proof methods | proof methods | induction | induction | well-ordering | well-ordering | sets | sets | relations | relations | elementary graph theory | elementary graph theory | integer congruences | integer congruences | asymptotic notation and growth of functions | asymptotic notation and growth of functions | permutations and combinations | permutations and combinations | counting principles | counting principles | discrete probability | discrete probability | recursive definition | recursive definition | structural induction | structural induction | state machines and invariants | state machines and invariants | recurrences | recurrences | generating functions | generating functions | 6.042 | 6.042 | 18.062 | 18.062License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata6.042J Mathematics for Computer Science (MIT)

Description

This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science).Subjects

mathematical definitions | proofs and applicable methods | formal logic notation | proof methods | induction | well-ordering | sets | relations | elementary graph theory | integer congruences | asymptotic notation and growth of functions | permutations and combinations | counting principles | discrete probability | recursive definition | structural induction | state machines and invariants | recurrences | generating functions | permutations and combinations | counting principles | discrete mathematics | computer scienceLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata12.620J Classical Mechanics: A Computational Approach (MIT)

Description

12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ovSubjects

classical mechanics | phase space | computation | Lagrangian formulation | action | variational principles | equations of motion | Hamilton's principle | conserved quantities | rigid bodies and tops | Hamiltonian formulation | canonical equations | surfaces of section | chaos | canonical transformations | generating functions | Liouville's theorem | Poincar? integral invariants | Poincar?-Birkhoff | KAM theorem | invariant curves | cantori | nonlinear resonances | resonance overlap | transition to chaos | chaotic motion | 12.620 | 6.946 | 8.351License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata6.042J Mathematics for Computer Science (MIT)

Description

This course is offered to undergraduates and is an elementary discrete mathematics course oriented towards applications in computer science and engineering. Topics covered include: formal logic notation, induction, sets and relations, permutations and combinations, counting principles, and discrete probability.Subjects

Elementary discrete mathematics for computer science and engineering | mathematical definitions | proofs and applicable methods | formal logic notation | proof methods | induction | well-ordering | sets | relations | elementary graph theory | integer congruences | asymptotic notation and growth of functions | permutations and combinations | counting principles | discrete probability | recursive definition | structural induction | state machines and invariants | recurrences | generating functions | 6.042 | 18.062License

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See all metadata6.005 Elements of Software Construction (MIT)

Description

This course provides an introduction to the fundamental principles and techniques of software development that have greatest impact on practice. Topics include capturing the essence of a problem by recognizing and inventing suitable abstractions; key paradigms, including state machines, functional programming, and object-oriented programming; use of design patterns to bridge gap between models and code; the role of interfaces and specification in achieving modularity and decoupling; reasoning about code using invariants; testing, test-case generation and coverage; and essentials of programming with objects, functions, and abstract types. The course includes exercises in modeling, design, implementation and reasoning.Subjects

software development | java programming | java | invariants | decoupling | data abstraction | state machine | module dependency | object model | model view controller | mvc | client server | eclipse | junit | subversion | swing | design | implement | midi player | sat solver | photo organizer | testing | coverage | event based programming | concurrencyLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata2.26 Compressible Fluid Dynamics (MIT)

Description

2.26 is a 6-unit Honors-level subject serving as the Mechanical Engineering department's sole course in compressible fluid dynamics. The prerequisites for this course are undergraduate courses in thermodynamics, fluid dynamics, and heat transfer. The goal of this course is to lay out the fundamental concepts and results for the compressible flow of gases. Topics to be covered include: appropriate conservation laws; propagation of disturbances; isentropic flows; normal shock wave relations, oblique shock waves, weak and strong shocks, and shock wave structure; compressible flows in ducts with area changes, friction, or heat addition; heat transfer to high speed flows; unsteady compressible flows, Riemann invariants, and piston and shock tube problems; steady 2D supersonic flow, Prandtl-MeySubjects

conservation laws | isentropic flows | normal shock wave relations | oblique shock waves | weak shock | strong shock | ducts | heat transfer | unsteady flows | Riemann invariants | piston | shock tube | steady 2D supersonic flow | Prandtl-Meyer function | self-similar compressible flowsLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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Description

This course introduces fundamental principles and techniques of software development. Students learn how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; state machines; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. The course includes weekly programming exercises and two substantial group projects.Subjects

software development | specifications | invariants | state machines | test-driven development | design patterns | object-oriented programming | concurrent programming | functional programmingLicense

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See all metadata8.20 Introduction to Special Relativity (MIT)

Description

Introduces the basic ideas and equations of Einstein's Special Theory of Relativity. Topics include: Lorentz transformations, length contraction and time dilation, four vectors, Lorentz invariants, relativistic energy and momentum, relativistic kinematics, Doppler shift, space-time diagrams, relativity paradoxes, and some concepts of General Relativity.Subjects

Einstein's Special Theory of Relativity | Lorentz transformations | length contraction | ime dilation | time dilation | four vectors | Lorentz invariants | relativistic energy and momentum | relativistic kinematics | Doppler shift | space-time diagrams | relativity paradoxes | General RelativityLicense

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See all metadata18.727 Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces (MIT)

Description

The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.Subjects

intersection theory | moduli spaces | geometry of homogeneous varieties | Deligne-Mumford moduli spaces | stable curves | Kontsevich moduli spaces | stable maps | Littlewood-Richardson rules | Grassmannians | divisor theory | cohomology | Brill-Noether theory | limit linear series | ample cones | effective cones | Gromov-Witten invariants | simple homogeneous varietiesLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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Description

This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science).Subjects

mathematical definitions | proofs and applicable methods | formal logic notation | proof methods | induction | well-ordering | sets | relations | elementary graph theory | integer congruences | asymptotic notation and growth of functions | permutations and combinations | counting principles | discrete probability | recursive definition | structural induction | state machines and invariants | recurrences | generating functions | permutations and combinations | counting principles | discrete mathematics | computer scienceLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. Topics include Einstein's postulates, the Lorentz transformation, relativistic effects and paradoxes, and applications involving electromagnetism and particle physics. This course also provides a brief introduction to some concepts of general relativity, including the principle of equivalence, the Schwartzschild metric and black holes, and the FRW metric and cosmology.Subjects

relativity | special relativity | Einstein's postulates | simultaneity | time dilation | length contraction | clock synchronization | Lorentz transformation | relativistic effects | Minkowski diagrams | relativistic invariants | four-vectors | relativitistic particle collisions | relativity and electricity | Coulomb's law | magnetic fields | Newtonian cosmology | general relativity | Schwarzchild metric | gravitational | red shift | light trajectories | geodesics | Shapiro delayLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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