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2.032 Dynamics (MIT) 2.032 Dynamics (MIT)

Description

This course reviews momentum and energy principles, and then covers the following topics: Hamilton's principle and Lagrange's equations; three-dimensional kinematics and dynamics of rigid bodies; steady motions and small deviations therefrom, gyroscopic effects, and causes of instability; free and forced vibrations of lumped-parameter and continuous systems; nonlinear oscillations and the phase plane; nonholonomic systems; and an introduction to wave propagation in continuous systems. This course was originally developed by Professor T. Akylas. This course reviews momentum and energy principles, and then covers the following topics: Hamilton's principle and Lagrange's equations; three-dimensional kinematics and dynamics of rigid bodies; steady motions and small deviations therefrom, gyroscopic effects, and causes of instability; free and forced vibrations of lumped-parameter and continuous systems; nonlinear oscillations and the phase plane; nonholonomic systems; and an introduction to wave propagation in continuous systems. This course was originally developed by Professor T. Akylas.

Subjects

motion | motion | momentum | momentum | work-energy principle | work-energy principle | degrees of freedom | degrees of freedom | Lagrange's equations | Lagrange's equations | D'Alembert's principle | D'Alembert's principle | Hamilton's principle | Hamilton's principle | gyroscope | gyroscope | gyroscopic effect | gyroscopic effect | steady motions | steady motions | nature of small deviations | nature of small deviations | natural modes | natural modes | natural frequencies for continuous and lumped parameter systems | natural frequencies for continuous and lumped parameter systems | mode shapes | mode shapes | forced vibrations | forced vibrations | dynamic stability theory | dynamic stability theory | instability | instability

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IV (MIT) IV (MIT)

Description

Includes audio/video content: AV selected lectures, AV faculty introductions, AV special element video. The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines. Includes audio/video content: AV selected lectures, AV faculty introductions, AV special element video. The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.

Subjects

Unified | Unified | Unified Engineering | Unified Engineering | aerospace | aerospace | CDIO | CDIO | C-D-I-O | C-D-I-O | conceive | conceive | design | design | implement | implement | operate | operate | team | team | team-based | team-based | discipline | discipline | materials | materials | structures | structures | materials and structures | materials and structures | computers | computers | programming | programming | computers and programming | computers and programming | fluids | fluids | fluid mechanics | fluid mechanics | thermodynamics | thermodynamics | propulsion | propulsion | signals | signals | systems | systems | signals and systems | signals and systems | systems problems | systems problems | fundamentals | fundamentals | technical communication | technical communication | graphical communication | graphical communication | communication | communication | reading | reading | research | research | experimentation | experimentation | personal response system | personal response system | prs | prs | active learning | active learning | First law | First law | first law of thermodynamics | first law of thermodynamics | thermo-mechanical | thermo-mechanical | energy | energy | energy conversion | energy conversion | aerospace power systems | aerospace power systems | propulsion systems | propulsion systems | aerospace propulsion systems | aerospace propulsion systems | heat | heat | work | work | thermal efficiency | thermal efficiency | forms of energy | forms of energy | energy exchange | energy exchange | processes | processes | heat engines | heat engines | engines | engines | steady-flow energy equation | steady-flow energy equation | energy flow | energy flow | flows | flows | path-dependence | path-dependence | path-independence | path-independence | reversibility | reversibility | irreversibility | irreversibility | state | state | thermodynamic state | thermodynamic state | performance | performance | ideal cycle | ideal cycle | simple heat engine | simple heat engine | cycles | cycles | thermal pressures | thermal pressures | temperatures | temperatures | linear static networks | linear static networks | loop method | loop method | node method | node method | linear dynamic networks | linear dynamic networks | classical methods | classical methods | state methods | state methods | state concepts | state concepts | dynamic systems | dynamic systems | resistive circuits | resistive circuits | sources | sources | voltages | voltages | currents | currents | Thevinin | Thevinin | Norton | Norton | initial value problems | initial value problems | RLC networks | RLC networks | characteristic values | characteristic values | characteristic vectors | characteristic vectors | transfer function | transfer function | ada | ada | ada programming | ada programming | programming language | programming language | software systems | software systems | programming style | programming style | computer architecture | computer architecture | program language evolution | program language evolution | classification | classification | numerical computation | numerical computation | number representation systems | number representation systems | assembly | assembly | SimpleSIM | SimpleSIM | RISC | RISC | CISC | CISC | operating systems | operating systems | single user | single user | multitasking | multitasking | multiprocessing | multiprocessing | domain-specific classification | domain-specific classification | recursive | recursive | execution time | execution time | fluid dynamics | fluid dynamics | physical properties of a fluid | physical properties of a fluid | fluid flow | fluid flow | mach | mach | reynolds | reynolds | conservation | conservation | conservation principles | conservation principles | conservation of mass | conservation of mass | conservation of momentum | conservation of momentum | conservation of energy | conservation of energy | continuity | continuity | inviscid | inviscid | steady flow | steady flow | simple bodies | simple bodies | airfoils | airfoils | wings | wings | channels | channels | aerodynamics | aerodynamics | forces | forces | moments | moments | equilibrium | equilibrium | freebody diagram | freebody diagram | free-body | free-body | free body | free body | planar force systems | planar force systems | equipollent systems | equipollent systems | equipollence | equipollence | support reactions | support reactions | reactions | reactions | static determinance | static determinance | determinate systems | determinate systems | truss analysis | truss analysis | trusses | trusses | method of joints | method of joints | method of sections | method of sections | statically indeterminate | statically indeterminate | three great principles | three great principles | 3 great principles | 3 great principles | indicial notation | indicial notation | rotation of coordinates | rotation of coordinates | coordinate rotation | coordinate rotation | stress | stress | extensional stress | extensional stress | shear stress | shear stress | notation | notation | plane stress | plane stress | stress equilbrium | stress equilbrium | stress transformation | stress transformation | mohr | mohr | mohr's circle | mohr's circle | principal stress | principal stress | principal stresses | principal stresses | extreme shear stress | extreme shear stress | strain | strain | extensional strain | extensional strain | shear strain | shear strain | strain-displacement | strain-displacement | compatibility | compatibility | strain transformation | strain transformation | transformation of strain | transformation of strain | mohr's circle for strain | mohr's circle for strain | principal strain | principal strain | extreme shear strain | extreme shear strain | uniaxial stress-strain | uniaxial stress-strain | material properties | material properties | classes of materials | classes of materials | bulk material properties | bulk material properties | origin of elastic properties | origin of elastic properties | structures of materials | structures of materials | atomic bonding | atomic bonding | packing of atoms | packing of atoms | atomic packing | atomic packing | crystals | crystals | crystal structures | crystal structures | polymers | polymers | estimate of moduli | estimate of moduli | moduli | moduli | composites | composites | composite materials | composite materials | modulus limited design | modulus limited design | material selection | material selection | materials selection | materials selection | measurement of elastic properties | measurement of elastic properties | stress-strain | stress-strain | stress-strain relations | stress-strain relations | anisotropy | anisotropy | orthotropy | orthotropy | measurements | measurements | engineering notation | engineering notation | Hooke | Hooke | Hooke's law | Hooke's law | general hooke's law | general hooke's law | equations of elasticity | equations of elasticity | boundary conditions | boundary conditions | multi-disciplinary | multi-disciplinary | models | models | engineering systems | engineering systems | experiments | experiments | investigations | investigations | experimental error | experimental error | design evaluation | design evaluation | evaluation | evaluation | trade studies | trade studies | effects of engineering | effects of engineering | social context | social context | engineering drawings | engineering drawings

License

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IV (MIT) IV (MIT)

Description

The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files

Subjects

Unified | Unified | Unified Engineering | Unified Engineering | aerospace | aerospace | CDIO | CDIO | C-D-I-O | C-D-I-O | conceive | conceive | design | design | implement | implement | operate | operate | team | team | team-based | team-based | discipline | discipline | materials | materials | structures | structures | materials and structures | materials and structures | computers | computers | programming | programming | computers and programming | computers and programming | fluids | fluids | fluid mechanics | fluid mechanics | thermodynamics | thermodynamics | propulsion | propulsion | signals | signals | systems | systems | signals and systems | signals and systems | systems problems | systems problems | fundamentals | fundamentals | technical communication | technical communication | graphical communication | graphical communication | communication | communication | reading | reading | research | research | experimentation | experimentation | personal response system | personal response system | prs | prs | active learning | active learning | First law | First law | first law of thermodynamics | first law of thermodynamics | thermo-mechanical | thermo-mechanical | energy | energy | energy conversion | energy conversion | aerospace power systems | aerospace power systems | propulsion systems | propulsion systems | aerospace propulsion systems | aerospace propulsion systems | heat | heat | work | work | thermal efficiency | thermal efficiency | forms of energy | forms of energy | energy exchange | energy exchange | processes | processes | heat engines | heat engines | engines | engines | steady-flow energy equation | steady-flow energy equation | energy flow | energy flow | flows | flows | path-dependence | path-dependence | path-independence | path-independence | reversibility | reversibility | irreversibility | irreversibility | state | state | thermodynamic state | thermodynamic state | performance | performance | ideal cycle | ideal cycle | simple heat engine | simple heat engine | cycles | cycles | thermal pressures | thermal pressures | temperatures | temperatures | linear static networks | linear static networks | loop method | loop method | node method | node method | linear dynamic networks | linear dynamic networks | classical methods | classical methods | state methods | state methods | state concepts | state concepts | dynamic systems | dynamic systems | resistive circuits | resistive circuits | sources | sources | voltages | voltages | currents | currents | Thevinin | Thevinin | Norton | Norton | initial value problems | initial value problems | RLC networks | RLC networks | characteristic values | characteristic values | characteristic vectors | characteristic vectors | transfer function | transfer function | ada | ada | ada programming | ada programming | programming language | programming language | software systems | software systems | programming style | programming style | computer architecture | computer architecture | program language evolution | program language evolution | classification | classification | numerical computation | numerical computation | number representation systems | number representation systems | assembly | assembly | SimpleSIM | SimpleSIM | RISC | RISC | CISC | CISC | operating systems | operating systems | single user | single user | multitasking | multitasking | multiprocessing | multiprocessing | domain-specific classification | domain-specific classification | recursive | recursive | execution time | execution time | fluid dynamics | fluid dynamics | physical properties of a fluid | physical properties of a fluid | fluid flow | fluid flow | mach | mach | reynolds | reynolds | conservation | conservation | conservation principles | conservation principles | conservation of mass | conservation of mass | conservation of momentum | conservation of momentum | conservation of energy | conservation of energy | continuity | continuity | inviscid | inviscid | steady flow | steady flow | simple bodies | simple bodies | airfoils | airfoils | wings | wings | channels | channels | aerodynamics | aerodynamics | forces | forces | moments | moments | equilibrium | equilibrium | freebody diagram | freebody diagram | free-body | free-body | free body | free body | planar force systems | planar force systems | equipollent systems | equipollent systems | equipollence | equipollence | support reactions | support reactions | reactions | reactions | static determinance | static determinance | determinate systems | determinate systems | truss analysis | truss analysis | trusses | trusses | method of joints | method of joints | method of sections | method of sections | statically indeterminate | statically indeterminate | three great principles | three great principles | 3 great principles | 3 great principles | indicial notation | indicial notation | rotation of coordinates | rotation of coordinates | coordinate rotation | coordinate rotation | stress | stress | extensional stress | extensional stress | shear stress | shear stress | notation | notation | plane stress | plane stress | stress equilbrium | stress equilbrium | stress transformation | stress transformation | mohr | mohr | mohr's circle | mohr's circle | principal stress | principal stress | principal stresses | principal stresses | extreme shear stress | extreme shear stress | strain | strain | extensional strain | extensional strain | shear strain | shear strain | strain-displacement | strain-displacement | compatibility | compatibility | strain transformation | strain transformation | transformation of strain | transformation of strain | mohr's circle for strain | mohr's circle for strain | principal strain | principal strain | extreme shear strain | extreme shear strain | uniaxial stress-strain | uniaxial stress-strain | material properties | material properties | classes of materials | classes of materials | bulk material properties | bulk material properties | origin of elastic properties | origin of elastic properties | structures of materials | structures of materials | atomic bonding | atomic bonding | packing of atoms | packing of atoms | atomic packing | atomic packing | crystals | crystals | crystal structures | crystal structures | polymers | polymers | estimate of moduli | estimate of moduli | moduli | moduli | composites | composites | composite materials | composite materials | modulus limited design | modulus limited design | material selection | material selection | materials selection | materials selection | measurement of elastic properties | measurement of elastic properties | stress-strain | stress-strain | stress-strain relations | stress-strain relations | anisotropy | anisotropy | orthotropy | orthotropy | measurements | measurements | engineering notation | engineering notation | Hooke | Hooke | Hooke's law | Hooke's law | general hooke's law | general hooke's law | equations of elasticity | equations of elasticity | boundary conditions | boundary conditions | multi-disciplinary | multi-disciplinary | models | models | engineering systems | engineering systems | experiments | experiments | investigations | investigations | experimental error | experimental error | design evaluation | design evaluation | evaluation | evaluation | trade studies | trade studies | effects of engineering | effects of engineering | social context | social context | engineering drawings | engineering drawings | 16.01 | 16.01 | 16.02 | 16.02 | 16.03 | 16.03 | 16.04 | 16.04

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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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

License

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5.73 Introductory Quantum Mechanics I (MIT) 5.73 Introductory Quantum Mechanics I (MIT)

Description

5.73 covers fundamental concepts of quantum mechanics: wave properties, uncertainty principles, Schrodinger equation, and operator and matrix methods. Basic applications of the following are discussed: one-dimensional potentials (harmonic oscillator), three-dimensional centrosymetric potentials (hydrogen atom), and angular momentum and spin. The course also examines approximation methods: WKB method, variational principle, and perturbation theory. Acknowledgement The instructor would like to acknowledge Peter Giunta for preparing the original version of the materials for 5.73. 5.73 covers fundamental concepts of quantum mechanics: wave properties, uncertainty principles, Schrodinger equation, and operator and matrix methods. Basic applications of the following are discussed: one-dimensional potentials (harmonic oscillator), three-dimensional centrosymetric potentials (hydrogen atom), and angular momentum and spin. The course also examines approximation methods: WKB method, variational principle, and perturbation theory. Acknowledgement The instructor would like to acknowledge Peter Giunta for preparing the original version of the materials for 5.73.

Subjects

quantum mechanics | quantum mechanics | wave properties | wave properties | uncertainty principles | uncertainty principles | Schrodinger | Schrodinger | operator method | operator method | matrix method | matrix method | one-dimensional potentials | one-dimensional potentials | harmonic oscillator | harmonic oscillator | three- dimensional centrosymetric potentials | three- dimensional centrosymetric potentials | angular momentum | angular momentum | spin | spin | approximation methods | approximation methods | WKB method | WKB method | variational principle | variational principle | perturbation theory | perturbation theory

License

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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 | arrows

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J 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 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 ov

Subjects

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 | 8.351

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6.042J 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 science

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1.033 Mechanics of Material Systems: An Energy Approach (MIT) 1.033 Mechanics of Material Systems: An Energy Approach (MIT)

Description

1.033 provides an introduction to continuum mechanics and material modeling of engineering materials based on first energy principles: deformation and strain; momentum balance, stress and stress states; elasticity and elasticity bounds; plasticity and yield design. The overarching theme is a unified mechanistic language using thermodynamics, which allows understanding, modeling and design of a large range of engineering materials. This course is offered both to undergraduate (1.033) and graduate (1.57) students. 1.033 provides an introduction to continuum mechanics and material modeling of engineering materials based on first energy principles: deformation and strain; momentum balance, stress and stress states; elasticity and elasticity bounds; plasticity and yield design. The overarching theme is a unified mechanistic language using thermodynamics, which allows understanding, modeling and design of a large range of engineering materials. This course is offered both to undergraduate (1.033) and graduate (1.57) students.

Subjects

continuum mechanics | continuum mechanics | material modeling | material modeling | engineering materials | engineering materials | energy principles: deformation and strain | energy principles: deformation and strain | momentum balance | momentum balance | stress | stress | stress states | stress states | elasticity and elasticity bounds | elasticity and elasticity bounds | plasticity | plasticity | yield design | yield design | first energy principles | first energy principles | deformation | deformation | strain | strain | elasticity bounds | elasticity bounds | unified mechanistic language | unified mechanistic language | thermodynamics | thermodynamics | engineering structures | engineering structures | unified framework | unified framework | irreversible processes | irreversible processes | structural engineering | structural engineering | soil mechanics | soil mechanics | mechanical engineering | mechanical engineering | materials science | materials science | solids | solids | durability mechanics | durability mechanics

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J Classical Mechanics: A Computational Approach (MIT)

Description

We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: 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 overl We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: 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 overl

Subjects

classical mechanics | classical mechanics | computational classical mechanics | computational classical mechanics | structure and interpretation of classical mechanics | structure and interpretation of classical mechanics | phase space | phase space | lagrangian | lagrangian | action | action | variational principles | variational principles | equation of motion | equation of motion | hamilton principle | hamilton principle | rigid bodies | rigid bodies | Hamiltonian | Hamiltonian | canonical equations | canonical equations | surfaces of section | surfaces of section | canonical transformations | canonical transformations | liouville | liouville | Poincare | Poincare | birkhoff | birkhoff | kam theorem | kam theorem | invariant curves | invariant curves | resonance | resonance | chaos | chaos

License

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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

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6.632 Electromagnetic Wave Theory (MIT) 6.632 Electromagnetic Wave Theory (MIT)

Description

6.632 is a graduate subject on electromagnetic wave theory, emphasizing mathematical approaches, problem solving, and physical interpretation. Topics covered include: waves in media, equivalence principle, duality and complementarity, Huygens' principle, Fresnel and Fraunhofer diffraction, dyadic Green's functions, Lorentz transformation, and Maxwell-Minkowski theory. Examples deal with limiting cases of Maxwell's theory and diffraction and scattering of electromagnetic waves. 6.632 is a graduate subject on electromagnetic wave theory, emphasizing mathematical approaches, problem solving, and physical interpretation. Topics covered include: waves in media, equivalence principle, duality and complementarity, Huygens' principle, Fresnel and Fraunhofer diffraction, dyadic Green's functions, Lorentz transformation, and Maxwell-Minkowski theory. Examples deal with limiting cases of Maxwell's theory and diffraction and scattering of electromagnetic waves.

Subjects

electromagnetic wave theory | electromagnetic wave theory | waves in media | waves in media | equivalence principle | equivalence principle | duality | duality | complementarity | complementarity | Huygens' principle | Huygens' principle | Fresnel diffraction | Fresnel diffraction | Fraunhofer diffraction | Fraunhofer diffraction | dyadic Green's functions | dyadic Green's functions | Lorentz transformation | Lorentz transformation | Maxwell-Minkowski theory | Maxwell-Minkowski theory | Maxwell | Maxwell | diffraction | diffraction | scattering | scattering

License

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2.032 Dynamics (MIT)

Description

This course reviews momentum and energy principles, and then covers the following topics: Hamilton's principle and Lagrange's equations; three-dimensional kinematics and dynamics of rigid bodies; steady motions and small deviations therefrom, gyroscopic effects, and causes of instability; free and forced vibrations of lumped-parameter and continuous systems; nonlinear oscillations and the phase plane; nonholonomic systems; and an introduction to wave propagation in continuous systems. This course was originally developed by Professor T. Akylas.

Subjects

motion | momentum | work-energy principle | degrees of freedom | Lagrange's equations | D'Alembert's principle | Hamilton's principle | gyroscope | gyroscopic effect | steady motions | nature of small deviations | natural modes | natural frequencies for continuous and lumped parameter systems | mode shapes | forced vibrations | dynamic stability theory | instability

License

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IV (MIT)

Description

The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.

Subjects

Unified | Unified Engineering | aerospace | CDIO | C-D-I-O | conceive | design | implement | operate | team | team-based | discipline | materials | structures | materials and structures | computers | programming | computers and programming | fluids | fluid mechanics | thermodynamics | propulsion | signals | systems | signals and systems | systems problems | fundamentals | technical communication | graphical communication | communication | reading | research | experimentation | personal response system | prs | active learning | First law | first law of thermodynamics | thermo-mechanical | energy | energy conversion | aerospace power systems | propulsion systems | aerospace propulsion systems | heat | work | thermal efficiency | forms of energy | energy exchange | processes | heat engines | engines | steady-flow energy equation | energy flow | flows | path-dependence | path-independence | reversibility | irreversibility | state | thermodynamic state | performance | ideal cycle | simple heat engine | cycles | thermal pressures | temperatures | linear static networks | loop method | node method | linear dynamic networks | classical methods | state methods | state concepts | dynamic systems | resistive circuits | sources | voltages | currents | Thevinin | Norton | initial value problems | RLC networks | characteristic values | characteristic vectors | transfer function | ada | ada programming | programming language | software systems | programming style | computer architecture | program language evolution | classification | numerical computation | number representation systems | assembly | SimpleSIM | RISC | CISC | operating systems | single user | multitasking | multiprocessing | domain-specific classification | recursive | execution time | fluid dynamics | physical properties of a fluid | fluid flow | mach | reynolds | conservation | conservation principles | conservation of mass | conservation of momentum | conservation of energy | continuity | inviscid | steady flow | simple bodies | airfoils | wings | channels | aerodynamics | forces | moments | equilibrium | freebody diagram | free-body | free body | planar force systems | equipollent systems | equipollence | support reactions | reactions | static determinance | determinate systems | truss analysis | trusses | method of joints | method of sections | statically indeterminate | three great principles | 3 great principles | indicial notation | rotation of coordinates | coordinate rotation | stress | extensional stress | shear stress | notation | plane stress | stress equilbrium | stress transformation | mohr | mohr's circle | principal stress | principal stresses | extreme shear stress | strain | extensional strain | shear strain | strain-displacement | compatibility | strain transformation | transformation of strain | mohr's circle for strain | principal strain | extreme shear strain | uniaxial stress-strain | material properties | classes of materials | bulk material properties | origin of elastic properties | structures of materials | atomic bonding | packing of atoms | atomic packing | crystals | crystal structures | polymers | estimate of moduli | moduli | composites | composite materials | modulus limited design | material selection | materials selection | measurement of elastic properties | stress-strain | stress-strain relations | anisotropy | orthotropy | measurements | engineering notation | Hooke | Hooke's law | general hooke's law | equations of elasticity | boundary conditions | multi-disciplinary | models | engineering systems | experiments | investigations | experimental error | design evaluation | evaluation | trade studies | effects of engineering | social context | engineering drawings

License

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.htm

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IV (MIT)

Description

The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.

Subjects

Unified | Unified Engineering | aerospace | CDIO | C-D-I-O | conceive | design | implement | operate | team | team-based | discipline | materials | structures | materials and structures | computers | programming | computers and programming | fluids | fluid mechanics | thermodynamics | propulsion | signals | systems | signals and systems | systems problems | fundamentals | technical communication | graphical communication | communication | reading | research | experimentation | personal response system | prs | active learning | First law | first law of thermodynamics | thermo-mechanical | energy | energy conversion | aerospace power systems | propulsion systems | aerospace propulsion systems | heat | work | thermal efficiency | forms of energy | energy exchange | processes | heat engines | engines | steady-flow energy equation | energy flow | flows | path-dependence | path-independence | reversibility | irreversibility | state | thermodynamic state | performance | ideal cycle | simple heat engine | cycles | thermal pressures | temperatures | linear static networks | loop method | node method | linear dynamic networks | classical methods | state methods | state concepts | dynamic systems | resistive circuits | sources | voltages | currents | Thevinin | Norton | initial value problems | RLC networks | characteristic values | characteristic vectors | transfer function | ada | ada programming | programming language | software systems | programming style | computer architecture | program language evolution | classification | numerical computation | number representation systems | assembly | SimpleSIM | RISC | CISC | operating systems | single user | multitasking | multiprocessing | domain-specific classification | recursive | execution time | fluid dynamics | physical properties of a fluid | fluid flow | mach | reynolds | conservation | conservation principles | conservation of mass | conservation of momentum | conservation of energy | continuity | inviscid | steady flow | simple bodies | airfoils | wings | channels | aerodynamics | forces | moments | equilibrium | freebody diagram | free-body | free body | planar force systems | equipollent systems | equipollence | support reactions | reactions | static determinance | determinate systems | truss analysis | trusses | method of joints | method of sections | statically indeterminate | three great principles | 3 great principles | indicial notation | rotation of coordinates | coordinate rotation | stress | extensional stress | shear stress | notation | plane stress | stress equilbrium | stress transformation | mohr | mohr's circle | principal stress | principal stresses | extreme shear stress | strain | extensional strain | shear strain | strain-displacement | compatibility | strain transformation | transformation of strain | mohr's circle for strain | principal strain | extreme shear strain | uniaxial stress-strain | material properties | classes of materials | bulk material properties | origin of elastic properties | structures of materials | atomic bonding | packing of atoms | atomic packing | crystals | crystal structures | polymers | estimate of moduli | moduli | composites | composite materials | modulus limited design | material selection | materials selection | measurement of elastic properties | stress-strain | stress-strain relations | anisotropy | orthotropy | measurements | engineering notation | Hooke | Hooke's law | general hooke's law | equations of elasticity | boundary conditions | multi-disciplinary | models | engineering systems | experiments | investigations | experimental error | design evaluation | evaluation | trade studies | effects of engineering | social context | engineering drawings

License

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IV (MIT)

Description

The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files

Subjects

Unified | Unified Engineering | aerospace | CDIO | C-D-I-O | conceive | design | implement | operate | team | team-based | discipline | materials | structures | materials and structures | computers | programming | computers and programming | fluids | fluid mechanics | thermodynamics | propulsion | signals | systems | signals and systems | systems problems | fundamentals | technical communication | graphical communication | communication | reading | research | experimentation | personal response system | prs | active learning | First law | first law of thermodynamics | thermo-mechanical | energy | energy conversion | aerospace power systems | propulsion systems | aerospace propulsion systems | heat | work | thermal efficiency | forms of energy | energy exchange | processes | heat engines | engines | steady-flow energy equation | energy flow | flows | path-dependence | path-independence | reversibility | irreversibility | state | thermodynamic state | performance | ideal cycle | simple heat engine | cycles | thermal pressures | temperatures | linear static networks | loop method | node method | linear dynamic networks | classical methods | state methods | state concepts | dynamic systems | resistive circuits | sources | voltages | currents | Thevinin | Norton | initial value problems | RLC networks | characteristic values | characteristic vectors | transfer function | ada | ada programming | programming language | software systems | programming style | computer architecture | program language evolution | classification | numerical computation | number representation systems | assembly | SimpleSIM | RISC | CISC | operating systems | single user | multitasking | multiprocessing | domain-specific classification | recursive | execution time | fluid dynamics | physical properties of a fluid | fluid flow | mach | reynolds | conservation | conservation principles | conservation of mass | conservation of momentum | conservation of energy | continuity | inviscid | steady flow | simple bodies | airfoils | wings | channels | aerodynamics | forces | moments | equilibrium | freebody diagram | free-body | free body | planar force systems | equipollent systems | equipollence | support reactions | reactions | static determinance | determinate systems | truss analysis | trusses | method of joints | method of sections | statically indeterminate | three great principles | 3 great principles | indicial notation | rotation of coordinates | coordinate rotation | stress | extensional stress | shear stress | notation | plane stress | stress equilbrium | stress transformation | mohr | mohr's circle | principal stress | principal stresses | extreme shear stress | strain | extensional strain | shear strain | strain-displacement | compatibility | strain transformation | transformation of strain | mohr's circle for strain | principal strain | extreme shear strain | uniaxial stress-strain | material properties | classes of materials | bulk material properties | origin of elastic properties | structures of materials | atomic bonding | packing of atoms | atomic packing | crystals | crystal structures | polymers | estimate of moduli | moduli | composites | composite materials | modulus limited design | material selection | materials selection | measurement of elastic properties | stress-strain | stress-strain relations | anisotropy | orthotropy | measurements | engineering notation | Hooke | Hooke's law | general hooke's law | equations of elasticity | boundary conditions | multi-disciplinary | models | engineering systems | experiments | investigations | experimental error | design evaluation | evaluation | trade studies | effects of engineering | social context | engineering drawings | 16.01 | 16.02 | 16.03 | 16.04

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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.

Subjects

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

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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).

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 science

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8.04 Quantum Physics I (MIT) 8.04 Quantum Physics I (MIT)

Description

Includes audio/video content: AV lectures. This course covers the experimental basis of quantum physics. It introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.It is the first course in the undergraduate Quantum Physics sequence, followed by 8.05 Quantum Physics II and 8.06 Quantum Physics III.  Includes audio/video content: AV lectures. This course covers the experimental basis of quantum physics. It introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.It is the first course in the undergraduate Quantum Physics sequence, followed by 8.05 Quantum Physics II and 8.06 Quantum Physics III. 

Subjects

quantum physics: photoelectric effect | quantum physics: photoelectric effect | Compton scattering | Compton scattering | photons | photons | Franck-Hertz experiment | Franck-Hertz experiment | the Bohr atom | the Bohr atom | electron diffraction | electron diffraction | deBroglie waves | deBroglie waves | wave-particle duality of matter and light | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave mechanics: Schroedinger's equation | wave functions | wave functions | wave packets | wave packets | probability amplitudes | probability amplitudes | stationary states | stationary states | the Heisenberg uncertainty principle | the Heisenberg uncertainty principle | zero-point energies | zero-point energies | transmission and reflection at a barrier | transmission and reflection at a barrier | barrier penetration | barrier penetration | potential wells | potential wells | simple harmonic oscillator | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic systems | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic systems

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2.76 Multi-Scale System Design (MIT) 2.76 Multi-Scale System Design (MIT)

Description

Multi-scale systems (MuSS) consist of components from two or more length scales (nano, micro, meso, or macro-scales). In MuSS, the engineering modeling, design principles, and fabrication processes of the components are fundamentally different. The challenge is to make these components so they are conceptually and model-wise compatible with other-scale components with which they interface. This course covers the fundamental properties of scales, design theories, modeling methods and manufacturing issues which must be addressed in these systems. Examples of MuSS include precision instruments, nanomanipulators, fiber optics, micro/nano-photonics, nanorobotics, MEMS (piezoelectric driven manipulators and optics), X-Ray telescopes and carbon nano-tube assemblies. Students master the materials Multi-scale systems (MuSS) consist of components from two or more length scales (nano, micro, meso, or macro-scales). In MuSS, the engineering modeling, design principles, and fabrication processes of the components are fundamentally different. The challenge is to make these components so they are conceptually and model-wise compatible with other-scale components with which they interface. This course covers the fundamental properties of scales, design theories, modeling methods and manufacturing issues which must be addressed in these systems. Examples of MuSS include precision instruments, nanomanipulators, fiber optics, micro/nano-photonics, nanorobotics, MEMS (piezoelectric driven manipulators and optics), X-Ray telescopes and carbon nano-tube assemblies. Students master the materials

Subjects

scale | scale | complexity | complexity | nano | micro | meso | or macro-scale | nano | micro | meso | or macro-scale | kinematics | kinematics | metrology | metrology | engineering modeling | motion | engineering modeling | motion | modeling | modeling | design | design | manufacture | manufacture | design principles | design principles | fabrication process | fabrication process | functional requirements | functional requirements | precision instruments | precision instruments | nanomanipulators | fiber optics | micro- photonics | nano-photonics | nanorobotics | MEMS | nanomanipulators | fiber optics | micro- photonics | nano-photonics | nanorobotics | MEMS | piezoelectric | transducer | actuator | sensor | piezoelectric | transducer | actuator | sensor | constraint | rigid constraint | flexible constraint | ride-flexible constraint | constraint | rigid constraint | flexible constraint | ride-flexible constraint | constaint-based design | constaint-based design | carbon nanotube | carbon nanotube | nanowire | nanowire | scanning tunneling microscope | scanning tunneling microscope | flexure | flexure | protein structure | protein structure | polymer structure | polymer structure | nanopelleting | nanopipette | nanowire | nanopelleting | nanopipette | nanowire | TMA pixel array | TMA pixel array | error modeling | error modeling | repeatability | repeatability

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12.620J 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 ov

Subjects

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 | 8.351

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4.605 Introduction to the History and Theory of Architecture (MIT) 4.605 Introduction to the History and Theory of Architecture (MIT)

Description

This course is a global-oriented survey of the history of architecture, from the prehistoric to the sixteenth century. It treats buildings and environments, including cities, in the context of the cultural and civilizational history. It offers an introduction to design principles and analysis. Being global, it aims to give the student perspective on the larger pushes and pulls that influence architecture and its meanings, whether these be economic, political, religious or climatic. This course is a global-oriented survey of the history of architecture, from the prehistoric to the sixteenth century. It treats buildings and environments, including cities, in the context of the cultural and civilizational history. It offers an introduction to design principles and analysis. Being global, it aims to give the student perspective on the larger pushes and pulls that influence architecture and its meanings, whether these be economic, political, religious or climatic.

Subjects

vernacular architecture | vernacular architecture | ancient civilizations | ancient civilizations | urbanism | urbanism | cities | cities | buildings | buildings | design principles | design principles | architecture analysis | architecture analysis | classical civilizations | classical civilizations | Greece | Greece | Rome | Rome | Asia | Asia | Islam | Islam | cathedrals | cathedrals

License

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2.971 2nd Summer Introduction to Design (MIT) 2.971 2nd Summer Introduction to Design (MIT)

Description

Introduce students to the creative design process, based on the scientific method and peer review, by application of fundamental principles and learning to complete projects according to schedule and within budget. Subject relies on active learning through a major team-based design-and-build project focused on the need for a new consumer product identified by each team. Topics to be learned while teams create, design, build, and test their product ideas include formulating strategies, concepts and modules, and estimation, concept selection, machine elements, design for manufacturing, visual thinking, communication, teamwork, and professional responsibilities. Introduce students to the creative design process, based on the scientific method and peer review, by application of fundamental principles and learning to complete projects according to schedule and within budget. Subject relies on active learning through a major team-based design-and-build project focused on the need for a new consumer product identified by each team. Topics to be learned while teams create, design, build, and test their product ideas include formulating strategies, concepts and modules, and estimation, concept selection, machine elements, design for manufacturing, visual thinking, communication, teamwork, and professional responsibilities.

Subjects

creative design process | creative design process | scientific method | scientific method | peer review | peer review | fundamental principles | fundamental principles | team-based | team-based | design-and-build project focused on the need for a new consumer product | design-and-build project focused on the need for a new consumer product | concept selection | concept selection | machine elements | machine elements | manufacturing design | manufacturing design | visual thinking | visual thinking

License

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.htm

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2.004 Modeling Dynamics and Control II (MIT) 2.004 Modeling Dynamics and Control II (MIT)

Description

This course is the second subject of a two-term sequence on modeling, analysis and control of dynamic systems. Topics covered include: kinematics and dynamics of mechanical systems, including rigid bodies in plane motion linear and angular momentum principles impact and collision problems linearization about equilibrium free and forced vibrations sensors and actuators control of mechanical systems integral and derivative action, lead and lag compensators root-locus design methods frequency-domain design methods applications to case-studies of multi-domain systems This course is the second subject of a two-term sequence on modeling, analysis and control of dynamic systems. Topics covered include: kinematics and dynamics of mechanical systems, including rigid bodies in plane motion linear and angular momentum principles impact and collision problems linearization about equilibrium free and forced vibrations sensors and actuators control of mechanical systems integral and derivative action, lead and lag compensators root-locus design methods frequency-domain design methods applications to case-studies of multi-domain systems

Subjects

Kinematics | | Kinematics | | dynamics of mechanical systems | | dynamics of mechanical systems | | Linear and angular momentum principles | | Linear and angular momentum principles | | Linearization about equilibrium | | Linearization about equilibrium | | Integral and derivative action | | Integral and derivative action | | lead and lag compensators | | lead and lag compensators | | Root-locus design methods | | Root-locus design methods | | Frequency-domain design methods | | Frequency-domain design methods | | multi-domain systems. | multi-domain systems.

License

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.htm

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