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18.04 Complex Variables with Applications (MIT) 18.04 Complex Variables with Applications (MIT)

Description

This course explored topics such as complex algebra and functions, analyticity, contour integration, Cauchy's theorem, singularities, Taylor and Laurent series, residues, evaluation of integrals, multivalued functions, potential theory in two dimensions, Fourier analysis and Laplace transforms. This course explored topics such as complex algebra and functions, analyticity, contour integration, Cauchy's theorem, singularities, Taylor and Laurent series, residues, evaluation of integrals, multivalued functions, potential theory in two dimensions, Fourier analysis and Laplace transforms.

Subjects

Complex algebra and functions | Complex algebra and functions | analyticity | analyticity | contour integration | Cauchy's theorem | contour integration | Cauchy's theorem | singularities | Taylor and Laurent series | singularities | Taylor and Laurent series | residues | evaluation of integrals | residues | evaluation of integrals | multivalued functions | potential theory in two dimensions | multivalued functions | potential theory in two dimensions | Fourier analysis and Laplace transforms. | Fourier analysis and Laplace transforms. | Fourier analysis and Laplace transforms | Fourier analysis and Laplace transforms

License

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2.04A Systems and Controls (MIT) 2.04A Systems and Controls (MIT)

Description

This course provides an introduction to linear systems, transfer functions, and Laplace transforms. It covers stability and feedback, and provides basic design tools for specifications of transient response. It also briefly covers frequency-domain techniques.   This course provides an introduction to linear systems, transfer functions, and Laplace transforms. It covers stability and feedback, and provides basic design tools for specifications of transient response. It also briefly covers frequency-domain techniques.  

Subjects

systems | systems | controls | controls | ordinary differential equations | ordinary differential equations | ODEs | ODEs | differential equations | differential equations | Laplace | Laplace | transfer function | transfer function | flywheel | flywheel | circuits | circuits | impedance | impedance | feedback | feedback | root locus | root locus | linear systems | linear systems | Laplace transforms | Laplace transforms | stability | stability | frequency-domain | frequency-domain | skyscaper | skyscaper

License

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18.04 Complex Variables with Applications (MIT)

Description

This course explored topics such as complex algebra and functions, analyticity, contour integration, Cauchy's theorem, singularities, Taylor and Laurent series, residues, evaluation of integrals, multivalued functions, potential theory in two dimensions, Fourier analysis and Laplace transforms.

Subjects

Complex algebra and functions | analyticity | contour integration | Cauchy's theorem | singularities | Taylor and Laurent series | residues | evaluation of integrals | multivalued functions | potential theory in two dimensions | Fourier analysis and Laplace transforms. | Fourier analysis and Laplace transforms

License

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12.802 Wave Motions in the Ocean and Atmosphere (MIT) 12.802 Wave Motions in the Ocean and Atmosphere (MIT)

Description

This course is an introduction to basic ideas of geophysical wave motion in rotating, stratified, and rotating-stratified fluids. Subject begins with general wave concepts of phase and group velocity. It also covers the dynamics and kinematics of gravity waves with a focus on dispersion, energy flux, initial value problems, etc. Also addressed are subject foundation used to study internal and inertial waves, Kelvin, Poincare, and Rossby waves in homogeneous and stratified fluids. Laplace tidal equations are applied to equatorial waves. Other topics include: resonant interactions, potential vorticity, wave-mean flow interactions, and instability. This course is an introduction to basic ideas of geophysical wave motion in rotating, stratified, and rotating-stratified fluids. Subject begins with general wave concepts of phase and group velocity. It also covers the dynamics and kinematics of gravity waves with a focus on dispersion, energy flux, initial value problems, etc. Also addressed are subject foundation used to study internal and inertial waves, Kelvin, Poincare, and Rossby waves in homogeneous and stratified fluids. Laplace tidal equations are applied to equatorial waves. Other topics include: resonant interactions, potential vorticity, wave-mean flow interactions, and instability.

Subjects

geophysical wave motion | geophysical wave motion | rotating | stratified | and rotating-stratified fluids | rotating | stratified | and rotating-stratified fluids | general wave concepts | general wave concepts | phase | phase | group velocity | group velocity | dynamics and kinematics of gravity waves | dynamics and kinematics of gravity waves | dispersion | dispersion | energy flux | energy flux | initial value problems | initial value problems | internal and inertial waves | internal and inertial waves | Kelvin | Kelvin | Poincare | Poincare | and Rossby waves | and Rossby waves | homogeneous and stratified fluids | homogeneous and stratified fluids | Laplace tidal equations | Laplace tidal equations | equatorial waves | equatorial waves | resonant interactions | resonant interactions | potential vorticity | potential vorticity | wave-mean flow interactions | wave-mean flow interactions | instability | instability | 12. Kelvin | Poincare | and Rossby waves | 12. Kelvin | Poincare | and Rossby waves | Kelvin | Poincare | and Rossby waves | Kelvin | Poincare | and Rossby waves | internal gravity waves | internal gravity waves | surface gravity waves | surface gravity waves | rotation | rotation | large-scale hydrostatic motions | large-scale hydrostatic motions | vertical structure equation | vertical structure equation | equatorial ?-plane | equatorial ?-plane | Stratified Quasi-Geostrophic Motion | Stratified Quasi-Geostrophic Motion

License

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

Description

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

Subjects

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

License

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18.085 Computational Science and Engineering I (MIT) 18.085 Computational Science and Engineering I (MIT)

Description

Includes audio/video content: AV lectures, AV faculty introductions. This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications. Note: This course was previously called "Mathematical Methods for Engineers I." Includes audio/video content: AV lectures, AV faculty introductions. This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications. Note: This course was previously called "Mathematical Methods for Engineers I."

Subjects

linear algebra | linear algebra | networks | networks | Lagrange multipliers | Lagrange multipliers | differential equations of equilibrium | differential equations of equilibrium | Laplace's equation | Laplace's equation | potential flow | potential flow | boundary-value problems | boundary-value problems | Fourier series | Fourier series | discrete Fourier transform | discrete Fourier transform | convolution | convolution

License

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18.085 Mathematical Methods for Engineers I (MIT) 18.085 Mathematical Methods for Engineers I (MIT)

Description

This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications. This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.

Subjects

linear algebra | linear algebra | networks | networks | Lagrange multipliers | Lagrange multipliers | differential equations of equilibrium | differential equations of equilibrium | Laplace's equation | Laplace's equation | potential flow | potential flow | boundary-value problems | boundary-value problems | Fourier series | Fourier series | discrete Fourier transform | discrete Fourier transform | convolution | convolution

License

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

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18.034 Honors Differential Equations (MIT) 18.034 Honors Differential Equations (MIT)

Description

This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems. This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.

Subjects

Quadrature | Quadrature | Maximum Principle | Maximum Principle | Laplace Transform | Laplace Transform | Existence Theory | Existence Theory | Autonomous System | Autonomous System | Lyapunov | Lyapunov | Limit Cycles | Limit Cycles | Fourier Series | Fourier Series | Boundary Value Problems | Boundary Value Problems

License

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6.003 Signals and Systems (MIT) 6.003 Signals and Systems (MIT)

Description

This course covers fundamentals of signal and system analysis, with applications drawn from filtering, audio and image processing, communications, and automatic control. Topics include convolution, Fourier series and transforms, sampling and discrete-time processing of continuous-time signals, modulation, Laplace and Z-transforms, and feedback systems. This course covers fundamentals of signal and system analysis, with applications drawn from filtering, audio and image processing, communications, and automatic control. Topics include convolution, Fourier series and transforms, sampling and discrete-time processing of continuous-time signals, modulation, Laplace and Z-transforms, and feedback systems.

Subjects

signal and system analysis | signal and system analysis | filtering | filtering | audio | audio | audio processing | audio processing | image processing | image processing | communications | communications | automatic control | automatic control | convolution | convolution | Fourier series | Fourier series | fourier transforms | fourier transforms | sampling | sampling | discrete-time processing | discrete-time processing | modulation | modulation | Laplace transforms | Laplace transforms | Z-transforms | Z-transforms | feedback systems | feedback systems

License

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8.022 Physics II: Electricity and Magnetism (MIT) 8.022 Physics II: Electricity and Magnetism (MIT)

Description

This course runs parallel to 8.02, but assumes that students have some knowledge of vector calculus. The class introduces Maxwell's equations, in both differential and integral form, along with electrostatic and magnetic vector potential, and the properties of dielectrics and magnetic materials. This class was taught by an undergraduate in the Experimental Study Group (ESG). Student instructors are paired with ESG faculty members, who advise and oversee the students' teaching efforts. This course runs parallel to 8.02, but assumes that students have some knowledge of vector calculus. The class introduces Maxwell's equations, in both differential and integral form, along with electrostatic and magnetic vector potential, and the properties of dielectrics and magnetic materials. This class was taught by an undergraduate in the Experimental Study Group (ESG). Student instructors are paired with ESG faculty members, who advise and oversee the students' teaching efforts.

Subjects

Electricity | Electricity | Magnetism | Magnetism | Maxwell's equations | Maxwell's equations | electrostatic potential | electrostatic potential | vector potential | vector potential | dielectrics | dielectrics | Coulomb's Law | Coulomb's Law | Electric Field | Electric Field | Electric Flux | Electric Flux | Gauss's Law | Gauss's Law | Electric Potential Gradient | Electric Potential Gradient | Poisson Equations | Poisson Equations | Laplace Equations | Laplace Equations | Curl | Curl | Conductors | Conductors | Capacitance | Capacitance | Resistance | Resistance | Kirchhoff's Rules | Kirchhoff's Rules | EMF | EMF | RC Circuits | RC Circuits | Th?venin Equivalence | Th?venin Equivalence | Magnetic Force | Magnetic Force | Magnetic Field | Magnetic Field | Ampere's Law | Ampere's Law | Special Relativity | Special Relativity | Spacetime | Spacetime | Biot-Savart Law | Biot-Savart Law | Faraday's Law | Faraday's Law | Lenz's Law | Lenz's Law | RL Circuits | RL Circuits | AC Circuits | AC Circuits | Electromagnetic Radiation | Electromagnetic Radiation | Poynting Vector | Poynting Vector

License

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18.085 Mathematical Methods for Engineers I (MIT) 18.085 Mathematical Methods for Engineers I (MIT)

Description

Review of linear algebra, applications to networks, structures, and estimation, Lagrange multipliers, differential equations of equilibrium, Laplace's equation and potential flow, boundary-value problems, minimum principles and calculus of variations, Fourier series, discrete Fourier transform, convolution, applications.Technical RequirementsRealOne Player software is required to run the .rm files found on this course site. MATLAB® is a trademark of The MathWorks, Inc. Review of linear algebra, applications to networks, structures, and estimation, Lagrange multipliers, differential equations of equilibrium, Laplace's equation and potential flow, boundary-value problems, minimum principles and calculus of variations, Fourier series, discrete Fourier transform, convolution, applications.Technical RequirementsRealOne Player software is required to run the .rm files found on this course site. MATLAB® is a trademark of The MathWorks, Inc.

Subjects

linear algebra | linear algebra | networks | networks | Lagrange multipliers | Lagrange multipliers | differential equations of equilibrium | differential equations of equilibrium | Laplace's equation | Laplace's equation | potential flow | potential flow | boundary-value problems | boundary-value problems | Fourier series | Fourier series | discrete Fourier transform | discrete Fourier transform | convolution | convolution

License

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18.303 Linear Partial Differential Equations (MIT) 18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions. This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functions | Bessel functions

License

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2.003 Modeling Dynamics and Control I (MIT) 2.003 Modeling Dynamics and Control I (MIT)

Description

Includes audio/video content: AV special element video. This course is the first of a two term sequence in modeling, analysis and control of dynamic systems. The various topics covered are as follows: mechanical translation, uniaxial rotation, electrical circuits and their coupling via levers, gears and electro-mechanical devices, analytical and computational solution of linear differential equations, state-determined systems, Laplace transforms, transfer functions, frequency response, Bode plots, vibrations, modal analysis, open- and closed-loop control, instability, time-domain controller design, and introduction to frequency-domain control design techniques. Case studies of engineering applications are also covered. Includes audio/video content: AV special element video. This course is the first of a two term sequence in modeling, analysis and control of dynamic systems. The various topics covered are as follows: mechanical translation, uniaxial rotation, electrical circuits and their coupling via levers, gears and electro-mechanical devices, analytical and computational solution of linear differential equations, state-determined systems, Laplace transforms, transfer functions, frequency response, Bode plots, vibrations, modal analysis, open- and closed-loop control, instability, time-domain controller design, and introduction to frequency-domain control design techniques. Case studies of engineering applications are also covered.

Subjects

modeling | modeling | analysis | analysis | dynamic | dynamic | systems | systems | mechanical | mechanical | translation | translation | uniaxial | uniaxial | rotation | rotation | electrical | electrical | circuits | circuits | coupling | coupling | levers | levers | gears | gears | electro-mechanical | electro-mechanical | devices | devices | linear | linear | differential | differential | equations | equations | state-determined | state-determined | Laplace | Laplace | transforms | transforms | transfer | transfer | functions | functions | frequency | frequency | response | response | Bode | Bode | vibrations | vibrations | modal | modal | open-loop | open-loop | closed-loop | closed-loop | control | control | instability | instability | time-domain | time-domain | controller | controller | frequency-domain | frequency-domain

License

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8.022 Physics II: Electricity and Magnetism (MIT) 8.022 Physics II: Electricity and Magnetism (MIT)

Description

Course 8.022 is one of several second-term freshman physics courses offered at MIT. It is geared towards students who are looking for a thorough and challenging introduction to electricity and magnetism. Topics covered include: Electric and magnetic field and potential; introduction to special relativity; Maxwell's equations, in both differential and integral form; and properties of dielectrics and magnetic materials. In addition to the theoretical subject matter, several experiments in electricity and magnetism are performed by the students in the laboratory. Acknowledgments Prof. Sciolla would like to acknowledge the contributions of MIT Professors Scott Hughes and Peter Fisher to the development of this course. She would also like to acknowledge that these course materials include cont Course 8.022 is one of several second-term freshman physics courses offered at MIT. It is geared towards students who are looking for a thorough and challenging introduction to electricity and magnetism. Topics covered include: Electric and magnetic field and potential; introduction to special relativity; Maxwell's equations, in both differential and integral form; and properties of dielectrics and magnetic materials. In addition to the theoretical subject matter, several experiments in electricity and magnetism are performed by the students in the laboratory. Acknowledgments Prof. Sciolla would like to acknowledge the contributions of MIT Professors Scott Hughes and Peter Fisher to the development of this course. She would also like to acknowledge that these course materials include cont

Subjects

Electricity | Electricity | Magnetism | Magnetism | Maxwell's equations | Maxwell's equations | electrostatic potential | electrostatic potential | vector potential | vector potential | dielectrics | dielectrics | Coulomb's Law | Coulomb's Law | Electric Field | Electric Field | Electric Flux | Electric Flux | Gauss's Law | Gauss's Law | Electric Potential Gradient | Electric Potential Gradient | Poisson Equations | Poisson Equations | Laplace Equations | Laplace Equations | Curl | Curl | Conductors | Conductors | Capacitance | Capacitance | Resistance | Resistance | Kirchhoff's Rules | Kirchhoff's Rules | EMF | EMF | RC Circuits | RC Circuits | Th?venin Equivalence | Th?venin Equivalence | Magnetic Force | Magnetic Force | Magnetic Field | Magnetic Field | Ampere's Law | Ampere's Law | Special Relativity | Special Relativity | Spacetime | Spacetime | Biot-Savart Law | Biot-Savart Law | Faraday's Law | Faraday's Law | Lenz's Law | Lenz's Law | RL Circuits | RL Circuits | AC Circuits | AC Circuits | Electromagnetic Radiation | Electromagnetic Radiation | Poynting Vector | Poynting Vector | Magnetism | Maxwell's equations; | Magnetism | Maxwell's equations;

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18.034 Honors Differential Equations (MIT) 18.034 Honors Differential Equations (MIT)

Description

This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems. This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.

Subjects

Quadrature | Quadrature | Maximum Principle | Maximum Principle | Laplace Transform | Laplace Transform | Existence Theory | Existence Theory | Autonomous System | Autonomous System | Lyapunov | Lyapunov | Limit Cycles | Limit Cycles | Fourier Series | Fourier Series | Boundary Value Problems | Boundary Value Problems

License

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18.04 Complex Variables with Applications (MIT)

Description

This course explored topics such as complex algebra and functions, analyticity, contour integration, Cauchy's theorem, singularities, Taylor and Laurent series, residues, evaluation of integrals, multivalued functions, potential theory in two dimensions, Fourier analysis and Laplace transforms.

Subjects

Complex algebra and functions | analyticity | contour integration | Cauchy's theorem | singularities | Taylor and Laurent series | residues | evaluation of integrals | multivalued functions | potential theory in two dimensions | Fourier analysis and Laplace transforms. | Fourier analysis and Laplace transforms

License

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6.003 Signals and Systems (MIT) 6.003 Signals and Systems (MIT)

Description

6.003 covers the fundamentals of signal and system analysis, focusing on representations of discrete-time and continuous-time signals (singularity functions, complex exponentials and geometrics, Fourier representations, Laplace and Z transforms, sampling) and representations of linear, time-invariant systems (difference and differential equations, block diagrams, system functions, poles and zeros, convolution, impulse and step responses, frequency responses). Applications are drawn broadly from engineering and physics, including feedback and control, communications, and signal processing. 6.003 covers the fundamentals of signal and system analysis, focusing on representations of discrete-time and continuous-time signals (singularity functions, complex exponentials and geometrics, Fourier representations, Laplace and Z transforms, sampling) and representations of linear, time-invariant systems (difference and differential equations, block diagrams, system functions, poles and zeros, convolution, impulse and step responses, frequency responses). Applications are drawn broadly from engineering and physics, including feedback and control, communications, and signal processing.

Subjects

signal and system analysis | signal and system analysis | representations of discrete-time and continuous-time signals | representations of discrete-time and continuous-time signals | representations of linear time-invariant systems | representations of linear time-invariant systems | Fourier representations | Fourier representations | Laplace and Z transforms | Laplace and Z transforms | sampling | sampling | difference and differential equations | difference and differential equations | feedback and control | feedback and control | communications | communications | signal processing | signal processing

License

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18.303 Linear Partial Differential Equations (MIT) 18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsSpecial software is required to use some of the files in this course: .m. This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsSpecial software is required to use some of the files in this course: .m.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functions | Bessel functions

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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18.311 Principles of Applied Mathematics (MIT) 18.311 Principles of Applied Mathematics (MIT)

Description

18.311 Principles of Continuum Applied Mathematics covers fundamental concepts in continuous applied mathematics, including applications from traffic flow, fluids, elasticity, granular flows, etc. The class also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion, and group velocity. 18.311 Principles of Continuum Applied Mathematics covers fundamental concepts in continuous applied mathematics, including applications from traffic flow, fluids, elasticity, granular flows, etc. The class also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion, and group velocity.

Subjects

partial differential equation | partial differential equation | hyperbolic equations | hyperbolic equations | dimensional analysis | dimensional analysis | perturbation methods | perturbation methods | hyperbolic systems | hyperbolic systems | diffusion and reaction processes | diffusion and reaction processes | continuum models | continuum models | equilibrium models | equilibrium models | continuous applied mathematics | continuous applied mathematics | traffic flow | traffic flow | fluids | fluids | elasticity | elasticity | granular flows | granular flows | continuum limit | continuum limit | conservation laws | conservation laws | quasi-equilibrium | quasi-equilibrium | kinematic waves | kinematic waves | characteristics | characteristics | simple waves | simple waves | shocks | shocks | diffusion (linear and nonlinear) | diffusion (linear and nonlinear) | numerical solution of wave equations | numerical solution of wave equations | finite differences | finite differences | consistency | consistency | stability | stability | discrete and fast Fourier transforms | discrete and fast Fourier transforms | spectral methods | spectral methods | transforms and series (Fourier | Laplace) | transforms and series (Fourier | Laplace) | sonic booms | sonic booms | Mach cone | Mach cone | caustics | caustics | lattices | lattices | dispersion | dispersion | group velocity | group velocity

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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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

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2.14 Analysis and Design of Feedback Control Systems (MIT) 2.14 Analysis and Design of Feedback Control Systems (MIT)

Description

This course develops the fundamentals of feedback control using linear transfer function system models. Topics covered include analysis in time and frequency domains; design in the s-plane (root locus) and in the frequency domain (loop shaping); describing functions for stability of certain non-linear systems; extension to state variable systems and multivariable control with observers; discrete and digital hybrid systems and use of z-plane design. Students will complete an extended design case study. Students taking the graduate version (2.140) will attend the recitation sessions and complete additional assignments. This course develops the fundamentals of feedback control using linear transfer function system models. Topics covered include analysis in time and frequency domains; design in the s-plane (root locus) and in the frequency domain (loop shaping); describing functions for stability of certain non-linear systems; extension to state variable systems and multivariable control with observers; discrete and digital hybrid systems and use of z-plane design. Students will complete an extended design case study. Students taking the graduate version (2.140) will attend the recitation sessions and complete additional assignments.

Subjects

feedback loops | feedback loops | control systems | control systems | compensation | compensation | Bode plots | Bode plots | Nyquist plots | Nyquist plots | state space | state space | frequency domain | frequency domain | time domain | time domain | transfer functions | transfer functions | Laplace transform | Laplace transform | root locus | root locus | op-amps | op-amps | gears | gears | motors | motors | actuators | actuators | nonlinear systems | nonlinear systems | stability theory | stability theory | dynamic feedback | dynamic feedback | mechanical engineering problem archive | mechanical engineering problem archive

License

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2.14 Analysis and Design of Feedback Control Systems (MIT) 2.14 Analysis and Design of Feedback Control Systems (MIT)

Description

This course develops the fundamentals of feedback control using linear transfer function system models. It covers analysis in time and frequency domains; design in the s-plane (root locus) and in the frequency domain (loop shaping); describing functions for stability of certain non-linear systems; extension to state variable systems and multivariable control with observers; discrete and digital hybrid systems and the use of z-plane design. Assignments include extended design case studies and capstone group projects. Graduate students are expected to complete additional assignments. This course develops the fundamentals of feedback control using linear transfer function system models. It covers analysis in time and frequency domains; design in the s-plane (root locus) and in the frequency domain (loop shaping); describing functions for stability of certain non-linear systems; extension to state variable systems and multivariable control with observers; discrete and digital hybrid systems and the use of z-plane design. Assignments include extended design case studies and capstone group projects. Graduate students are expected to complete additional assignments.

Subjects

feedback loops | feedback loops | compensation | compensation | Bode plots | Bode plots | Nyquist plots | Nyquist plots | state space | state space | frequency domain | frequency domain | time domain | time domain | transfer functions | transfer functions | Laplace transform | Laplace transform | root locus | root locus | op-amps | op-amps | gears | gears | motors | motors | actuators | actuators | nonlinear systems | nonlinear systems | stability theory | stability theory | control systems | control systems | dynamic feedback | dynamic feedback | mechanical engineering problem archive | mechanical engineering problem archive

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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2.087 Engineering Math: Differential Equations and Linear Algebra (MIT) 2.087 Engineering Math: Differential Equations and Linear Algebra (MIT)

Description

Includes audio/video content: AV selected lectures. This course is about the mathematics that is most widely used in the mechanical engineering core subjects: An introduction to linear algebra and ordinary differential equations (ODEs), including general numerical approaches to solving systems of equations. Includes audio/video content: AV selected lectures. This course is about the mathematics that is most widely used in the mechanical engineering core subjects: An introduction to linear algebra and ordinary differential equations (ODEs), including general numerical approaches to solving systems of equations.

Subjects

differential equations | differential equations | linear algebra | linear algebra | linear differential equations | linear differential equations | ordinary | ordinary | partial | partial | vector space | vector space | first order | first order | second order | second order | Heaviside | Heaviside | delta | delta | Dirac | Dirac | exponential | exponential | sinusoid | sinusoid | real | real | complex | complex | forced oscillations | forced oscillations | Laplace transform | Laplace transform | graph | graph | nonlinear | nonlinear | source | source | sink | sink | saddle | saddle | spiral | spiral | Euler | Euler | linearization | linearization | Guassian | Guassian | matrix | matrix | mechanical engineer | mechanical engineer | eigenvector | eigenvector | eigenvalue | eigenvalue | exponentiation | exponentiation | least squares | least squares

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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