Searching for symmetry : 88 results found | RSS Feed for this search

1 2 3 4

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

Description

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

Subjects

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

License

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.324 Relativistic Quantum Field Theory II (MIT) 8.324 Relativistic Quantum Field Theory II (MIT)

Description

This course is the second course of the quantum field theory trimester sequence beginning with Relativistic Quantum Field Theory I (8.323) and ending with Relativistic Quantum Field Theory III (8.325). It develops in depth some of the topics discussed in 8.323 and introduces some advanced material. Topics include functional path integrals, renormalization and renormalization groups, quantization of nonabelian gauge theories, BRST symmetry, renormalization and symmetry breaking, critical exponents and scalar field theory, and perturbation theory anomalies. This course is the second course of the quantum field theory trimester sequence beginning with Relativistic Quantum Field Theory I (8.323) and ending with Relativistic Quantum Field Theory III (8.325). It develops in depth some of the topics discussed in 8.323 and introduces some advanced material. Topics include functional path integrals, renormalization and renormalization groups, quantization of nonabelian gauge theories, BRST symmetry, renormalization and symmetry breaking, critical exponents and scalar field theory, and perturbation theory anomalies.

Subjects

Quantum Field Theory | Quantum Field Theory | nonabelian gauge theories | nonabelian gauge theories | BRST symmetry | BRST symmetry | Perturbation theory anomalies | Perturbation theory anomalies | Renormalization | Renormalization | symmetry breaking | symmetry breaking | Critical exponents | Critical exponents | scalar field theory | scalar field theory | Conformal field theory | Conformal field theory

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

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

Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C

Subjects

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

License

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-3.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.324 Relativistic Quantum Field Theory II (MIT) 8.324 Relativistic Quantum Field Theory II (MIT)

Description

This course is the second course of the quantum field theory trimester sequence beginning with Relativistic Quantum Field Theory I (8.323) and ending with Relativistic Quantum Field Theory III (8.325). It develops in depth some of the topics discussed in 8.323 and introduces some advanced material. This course is the second course of the quantum field theory trimester sequence beginning with Relativistic Quantum Field Theory I (8.323) and ending with Relativistic Quantum Field Theory III (8.325). It develops in depth some of the topics discussed in 8.323 and introduces some advanced material.

Subjects

Quantum Field Theory | Quantum Field Theory | nonabelian gauge theories | nonabelian gauge theories | BRST symmetry | BRST symmetry | Perturbation theory anomalies | Perturbation theory anomalies | Renormalization | Renormalization | symmetry breaking | symmetry breaking | Critical exponents | Critical exponents | scalar field theory | scalar field theory | Conformal field theory | Conformal field theory

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.069 Crystal Structure Analysis (MIT) 5.069 Crystal Structure Analysis (MIT)

Description

This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases. This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases.

Subjects

crystallography | crystallography | inorganic chemistry | inorganic chemistry | physical methods | physical methods | crystal structure determination | crystal structure determination | 3D structure | 3D structure | x-ray crystallagraphy | x-ray crystallagraphy | diffraction | diffraction | x-rays | x-rays | symmetry | symmetry | phasing | phasing | crystal structure | crystal structure | symmetry operations | symmetry operations | crystal lattice | crystal lattice | structure refinement | structure refinement | electron density maps | electron density maps | space group determination | space group determination | anomalous scattering | anomalous scattering

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-5.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.04 Principles of Inorganic Chemistry II (MIT) 5.04 Principles of Inorganic Chemistry II (MIT)

Description

This course provides a systematic presentation of the chemical applications of group theory with emphasis on the formal development of the subject and its applications to the physical methods of inorganic chemical compounds. Against the backdrop of electronic structure, the electronic, vibrational, and magnetic properties of transition metal complexes are presented and their investigation by the appropriate spectroscopy described. This course provides a systematic presentation of the chemical applications of group theory with emphasis on the formal development of the subject and its applications to the physical methods of inorganic chemical compounds. Against the backdrop of electronic structure, the electronic, vibrational, and magnetic properties of transition metal complexes are presented and their investigation by the appropriate spectroscopy described.

Subjects

inorganic chemistry | inorganic chemistry | group theory | group theory | electronic structure of molecules | electronic structure of molecules | transition metal complexes | transition metal complexes | spectroscopy | spectroscopy | symmetry elements | symmetry elements | mathematical groups | mathematical groups | character tables | character tables | molecular point groups | molecular point groups | Huckel Theory | Huckel Theory | N-Dimensional cyclic systems | N-Dimensional cyclic systems | solid state theory | solid state theory | band theory | band theory | frontier molecular orbitals | frontier molecular orbitals | similarity transformations | similarity transformations | complexes | complexes | organometallic complexes | organometallic complexes | two electron bond | two electron bond | vibrational spectroscopy | vibrational spectroscopy | symmetry | symmetry | overtones | overtones | normal coordinat analysis | normal coordinat analysis | AOM | AOM | single electron CFT | single electron CFT | tanabe-sugano diagram | tanabe-sugano diagram | ligand | ligand | crystal field theory | crystal field theory | LCAO | LCAO

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-5.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.321 Quantum Theory I (MIT) 8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement. 8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | eigenstates | uncertainty relation | uncertainty relation | observables | observables | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.322 Quantum Theory II (MIT) 8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation. 8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | uncertainty relation | observables | observables | eigenstates | eigenstates | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | quantization | quantization | EM radiation field | EM radiation field | electromagnetic radiation field | electromagnetic radiation field | adiabatic theorem | adiabatic theorem | Berry?s phase | Berry?s phase | many-particle systems | many-particle systems | Dirac equation | Dirac equation | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.069 Crystal Structure Analysis (MIT) 5.069 Crystal Structure Analysis (MIT)

Description

This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases. This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases.

Subjects

crystallography | crystallography | inorganic chemistry | inorganic chemistry | physical methods | physical methods | crystal structure determination | crystal structure determination | 3D structure | 3D structure | x-ray crystallagraphy | x-ray crystallagraphy | diffraction | diffraction | x-rays | x-rays | symmetry | symmetry | phasing | phasing | crystal structure | crystal structure | symmetry operations | symmetry operations | crystal lattice | crystal lattice | structure refinement | structure refinement | electron density maps | electron density maps | space group determination | space group determination | anomalous scattering | anomalous scattering

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

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

Description

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

Subjects

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

License

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-6.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.068 Physical Methods in Inorganic Chemistry (MIT) 5.068 Physical Methods in Inorganic Chemistry (MIT)

Description

This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases. This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases.

Subjects

crystallography | crystallography | inorganic chemistry | inorganic chemistry | physical methods | physical methods | crystal structure determination | crystal structure determination | 3D structure | 3D structure | x-ray crystallagraphy | x-ray crystallagraphy | diffraction | diffraction | x-rays | x-rays | symmetry | symmetry | phasing | phasing | crystal structure | crystal structure | symmetry operations | symmetry operations | crystal lattice | crystal lattice | structure refinement | structure refinement | electron density maps | electron density maps | space group determination | space group determination | anomalous scattering | anomalous scattering

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

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

Subjects

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

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

Site sourced from

https://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics (MIT) 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics (MIT)

Description

This course covers algebraic approaches to electromagnetism and nano-photonics. Topics include photonic crystals, waveguides, perturbation theory, diffraction, computational methods, applications to integrated optical devices, and fiber-optic systems. Emphasis is placed on abstract algebraic approaches rather than detailed solutions of partial differential equations, the latter being done by computers. This course covers algebraic approaches to electromagnetism and nano-photonics. Topics include photonic crystals, waveguides, perturbation theory, diffraction, computational methods, applications to integrated optical devices, and fiber-optic systems. Emphasis is placed on abstract algebraic approaches rather than detailed solutions of partial differential equations, the latter being done by computers.

Subjects

linear algebra | linear algebra | eigensystems for Maxwell's equations | eigensystems for Maxwell's equations | symmetry groups | symmetry groups | representation theory | representation theory | Bloch's theorem | Bloch's theorem | numerical eigensolver methods | numerical eigensolver methods | time and frequency-domain computation | time and frequency-domain computation | perturbation theory | perturbation theory | coupled-mode theories | coupled-mode theories | waveguide theory | waveguide theory | adiabatic transitions | adiabatic transitions | Optical phenomena | Optical phenomena | photonic crystals | photonic crystals | band gaps | band gaps | anomalous diffraction | anomalous diffraction | mechanisms for optical confinement | mechanisms for optical confinement | optical fibers | optical fibers | integrated optical devices | integrated optical devices

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.841 Crystal Structure Refinement (MIT) 5.841 Crystal Structure Refinement (MIT)

Description

This course in crystal structure refinement examines the practical aspects of crystal structure determination from data collection strategies to data reduction and basic and advanced refinement problems of organic and inorganic molecules. This course in crystal structure refinement examines the practical aspects of crystal structure determination from data collection strategies to data reduction and basic and advanced refinement problems of organic and inorganic molecules.

Subjects

chemistry | chemistry | crystal structure refinement | crystal structure refinement | practical aspects | practical aspects | crystal structure determination | crystal structure determination | data collection | data collection | strategies | strategies | data reduction | data reduction | refinement problems | refinement problems | organic | organic | inorganic | inorganic | molecules | molecules | SHELXL | SHELXL | hydrogen atoms | hydrogen atoms | disorder | disorder | pseudo symmetry | pseudo symmetry | merohedral twins | merohedral twins | pseudo-merohedral twins | pseudo-merohedral twins | twinning | twinning | non-merohedral twins | non-merohedral twins | PLATON | PLATON

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

RES.6-011 The Art of Insight in Science and Engineering: Mastering Complexity (MIT) RES.6-011 The Art of Insight in Science and Engineering: Mastering Complexity (MIT)

Description

In this book, Sanjoy Mahajan shows us that the way to master complexity is through insight rather than precision. Precision can overwhelm us with information, whereas insight connects seemingly disparate pieces of information into a simple picture. Unlike computers, humans depend on insight. Based on the author's fifteen years of teaching at MIT, Cambridge University, and Olin College, The Art of Insight in Science and Engineering shows us how to build insight and find understanding, giving readers tools to help them solve any problem in science and engineering. (Description courtesy of MIT Press.) In this book, Sanjoy Mahajan shows us that the way to master complexity is through insight rather than precision. Precision can overwhelm us with information, whereas insight connects seemingly disparate pieces of information into a simple picture. Unlike computers, humans depend on insight. Based on the author's fifteen years of teaching at MIT, Cambridge University, and Olin College, The Art of Insight in Science and Engineering shows us how to build insight and find understanding, giving readers tools to help them solve any problem in science and engineering. (Description courtesy of MIT Press.)

Subjects

approximation | approximation | science | science | engineering | engineering | complexity | complexity | divide and conquer | divide and conquer | abstraction | abstraction | symmetry | symmetry | proportion | proportion | dimension | dimension | lumping | lumping | probabalistic reasoning | probabalistic reasoning

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-RES.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

5.069 Crystal Structure Analysis (MIT)

Description

This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases.

Subjects

crystallography | inorganic chemistry | physical methods | crystal structure determination | 3D structure | x-ray crystallagraphy | diffraction | x-rays | symmetry | phasing | crystal structure | symmetry operations | crystal lattice | structure refinement | electron density maps | space group determination | anomalous scattering

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

Site sourced from

https://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

18.701 Algebra I (MIT) 18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups. This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Group Theory | Linear Algebra | and Geometry | Linear Algebra | and Geometry | groups | groups | vector spaces | vector spaces | linear transformations | linear transformations | symmetry groups | symmetry groups | bilinear | bilinear | bilinear forms | and linear groups | bilinear forms | and linear groups

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C

Subjects

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

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

Site sourced from

https://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

7.340 Nano-life: An Introduction to Virus Structure and Assembly (MIT) 7.340 Nano-life: An Introduction to Virus Structure and Assembly (MIT)

Description

Watson and Crick noted that the size of a viral genome was insufficient to encode a protein large enough to encapsidate it and reasoned, therefore that a virus shell must be composed of multiple, but identical subunits. Today, high resolution structures of virus capsids reveal the basis of this genetic economy as a highly symmetrical structure, much like a geodesic dome composed of protein subunits. Crystallographic structures and cryo-electron microscopy reconstructions combined with molecular data are beginning to reveal how these nano-structures are built. Topics covered in the course will include basic principles of virus structure and symmetry, capsid assembly, strategies for enclosing nucleic acid, proteins involved in entry and exit, and the life cycles of well understood pathogens Watson and Crick noted that the size of a viral genome was insufficient to encode a protein large enough to encapsidate it and reasoned, therefore that a virus shell must be composed of multiple, but identical subunits. Today, high resolution structures of virus capsids reveal the basis of this genetic economy as a highly symmetrical structure, much like a geodesic dome composed of protein subunits. Crystallographic structures and cryo-electron microscopy reconstructions combined with molecular data are beginning to reveal how these nano-structures are built. Topics covered in the course will include basic principles of virus structure and symmetry, capsid assembly, strategies for enclosing nucleic acid, proteins involved in entry and exit, and the life cycles of well understood pathogens

Subjects

viruses | viruses | virus structure | virus structure | virus assembly | virus assembly | virus shell | virus shell | virus genome | virus genome | capsids | capsids | capsid assembly | capsid assembly | TEM | TEM | transmission electron microscopy | transmission electron microscopy | nano-life | nano-life | nano-structures | nano-structures | virus symmetry | virus symmetry | icosahedral virus | icosahedral virus | electron cryotomography | electron cryotomography | nucleic acid packaging | nucleic acid packaging

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-7.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

18.701 Algebra I (MIT) 18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups. This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Group Theory | Linear Algebra | Linear Algebra | Geometry | Geometry | groups | groups | vector spaces | vector spaces | linear transformations | linear transformations | symmetry groups | symmetry groups | bilinear forms | bilinear forms | linear groups | linear groups

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

3.21 Kinetic Processes in Materials (MIT) 3.21 Kinetic Processes in Materials (MIT)

Description

This course presents a unified treatment of phenomenological and atomistic kinetic processes in materials. It provides the foundation for the advanced understanding of processing, microstructural evolution, and behavior for a broad spectrum of materials. The course emphasizes analysis and development of rigorous comprehension of fundamentals. Topics include: irreversible thermodynamics; diffusion; nucleation; phase transformations; fluid and heat transport; morphological instabilities; gas-solid, liquid-solid, and solid-solid reactions. This course presents a unified treatment of phenomenological and atomistic kinetic processes in materials. It provides the foundation for the advanced understanding of processing, microstructural evolution, and behavior for a broad spectrum of materials. The course emphasizes analysis and development of rigorous comprehension of fundamentals. Topics include: irreversible thermodynamics; diffusion; nucleation; phase transformations; fluid and heat transport; morphological instabilities; gas-solid, liquid-solid, and solid-solid reactions.

Subjects

Thermodynamics | Thermodynamics | field | field | gradient | gradient | continuity equation | continuity equation | irreversible thermodynamics | irreversible thermodynamics | entropy | entropy | Onsager's symmetry principle | Onsager's symmetry principle | diffusion | diffusion | capillarity | capillarity | stress | stress | diffusion equation | diffusion equation | crystal | crystal | jump process | jump process | jump rate | jump rate | diffusivity | diffusivity | interstitial | interstitial | Kroger-Vink | Kroger-Vink | grain boundary | grain boundary | isotropic | isotropic | Rayleigh instability | Rayleigh instability | Gibbs-Thomson | Gibbs-Thomson | particle coarsening | particle coarsening | growth kinetics | growth kinetics | phase transformation | phase transformation | nucleation | nucleation | spinoldal decomposition | spinoldal decomposition

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses-3.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.231 Physics of Solids I (MIT) 8.231 Physics of Solids I (MIT)

Description

The topics covered in this course include:Periodic Structure and Symmetry of CrystalsDiffraction, Reciprocal LatticeChemical BondingLattice DynamicsPhononsThermal PropertiesFree Electron GasModel of MetalsBloch Theorem and Band StructureNearly Free Electron ApproximationTight Binding MethodFermi SurfaceSemiconductorsElectronsHolesImpuritiesOptical PropertiesExcitons andMagnetism The topics covered in this course include:Periodic Structure and Symmetry of CrystalsDiffraction, Reciprocal LatticeChemical BondingLattice DynamicsPhononsThermal PropertiesFree Electron GasModel of MetalsBloch Theorem and Band StructureNearly Free Electron ApproximationTight Binding MethodFermi SurfaceSemiconductorsElectronsHolesImpuritiesOptical PropertiesExcitons andMagnetism

Subjects

periodic structure and symmetry of crystals | periodic structure and symmetry of crystals | diffraction | diffraction | reciprocal lattice | reciprocal lattice | chemical bonding | chemical bonding | phonons | phonons | thermal properties | thermal properties | free electron gas | free electron gas | model of metals | model of metals | Bloch theorem and band structure | Bloch theorem and band structure | nearly free electron approximation | nearly free electron approximation | tight binding method | tight binding method | Fermi surface | Fermi surface | semiconductors | semiconductors | electrons | electrons | holes | holes | impurities | impurities | optical properties | optical properties | excitons | excitons | magnetism | magnetism

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allarchivedcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.231 Physics of Solids I (MIT) 8.231 Physics of Solids I (MIT)

Description

This course offers an introduction to the basic concepts of the quantum theory of solids. This course offers an introduction to the basic concepts of the quantum theory of solids.

Subjects

periodic structure | periodic structure | symmetry of crystals | symmetry of crystals | diffraction | diffraction | reciprocal lattice | reciprocal lattice | chemical bonding | chemical bonding | lattice dynamics | lattice dynamics | phonons | phonons | thermal properties | thermal properties | free electron gas | free electron gas | model of metals | model of metals | Bloch theorem | Bloch theorem | band structure | band structure | nearly free electron approximation | nearly free electron approximation | tight binding method | tight binding method | Fermi surface | Fermi surface | semiconductors | semiconductors | electrons | electrons | holes | holes | impurities | impurities | optical properties | optical properties | excitons | excitons | magnetism. | magnetism.

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

8.325 Relativistic Quantum Field Theory III (MIT) 8.325 Relativistic Quantum Field Theory III (MIT)

Description

This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry. This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry.

Subjects

gauge symmetry | gauge symmetry | confinement | confinement | renormalization | renormalization | asymptotic freedom | asymptotic freedom | anomalies | anomalies | instantons | instantons | zero modes | zero modes | gauge boson and Higgs spectrum | gauge boson and Higgs spectrum | fermion multiplets | fermion multiplets | CKM matrix | CKM matrix | unification in SU(5) and SO(10) | unification in SU(5) and SO(10) | phenomenology of Higgs sector | phenomenology of Higgs sector | lepton and baryon number violation | lepton and baryon number violation | nonperturbative (lattice) formulation | nonperturbative (lattice) formulation

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata

18.969 Topics in Geometry: Mirror Symmetry (MIT) 18.969 Topics in Geometry: Mirror Symmetry (MIT)

Description

This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor. This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.

Subjects

mirror symmetry | mirror symmetry | deformation | deformation | hodge theory | hodge theory | pseudoholomorphic | pseudoholomorphic | gromov-witten | gromov-witten | cohomology | cohomology | yukawa | yukawa | monodromy | monodromy | picard-fuchs | picard-fuchs | lagrangian floer theory | lagrangian floer theory | homology | homology | SYZ conjecture | SYZ conjecture | submanifolds | submanifolds | K3 surfaces | K3 surfaces | matrices | matrices

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

Site sourced from

http://ocw.mit.edu/rss/all/mit-allcourses.xml

Attribution

Click to get HTML | Click to get attribution | Click to get URL

All metadata

See all metadata