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MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 21_Boundary_Value_Problems_2.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today we're... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 22_Partial_Differential_Equations_1.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today we'r... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 11_Unconstrained_Optimization_NewtonRaphson_and_Trust_Region_Methods.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JAMES SWAN: OK. Let's go ... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 20_Boundary_Value_Problem_1.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, le... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 6_Singular_Value_Decomposition_Iterative_Solutions_of_Linear_Equations.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JAMES W. SWAN: So this is... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 9_Homotopy_and_Bifurcation.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, let's go a... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 16_ODEIVP_and_Numerical_Integration_4.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, I t... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 12_Constrained_Optimization_Equality_Constraints_and_Lagrange_Multipliers.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JAMES SWAN: OK. Well, eve... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 27_Probability_Theory_2.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. WILLIAM GREEN: All right,... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 26_Partial_Differential_Equations_2.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. WILLIAM GREEN, JR: All ri... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 28_Models_vs_Data_1.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. WILLIAM GREEN: So today w... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 25_Review_Session.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JAMES SWAN: OK, shall we ... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 36_Final_Lecture.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So this is our... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 34_Stochastic_Chemical_Kinetics_1.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. WILLIAM GREEN JR: All rig... |
MIT_1034_Numerical_Methods_Applied_to_Chemical_Engineering_Fall_2015 | 5_Eigenvalues_and_Eigenvectors.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JAMES W. SWAN: Let's go a... |
MIT_561_Physical_Chemistry_Fall_2017 | 31_TimeDependent_Perturbation_Theory_II_H_is_TimeDependent_TwoLevel_Problem.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: My spies tell ... |
MIT_561_Physical_Chemistry_Fall_2017 | 3_TwoSlit_Experiment_Quantum_Weirdness.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: All right. So ... |
MIT_561_Physical_Chemistry_Fall_2017 | 36_Time_Dependence_of_TwoLevel_Systems_Density_Matrix_Rotating_Wave_Approximation.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Today's le... |
MIT_561_Physical_Chemistry_Fall_2017 | 30_TimeDependent_Perturbation_Theory_I_H_is_TimeIndependent_Zewail_Wavepacket.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: This is the... |
MIT_561_Physical_Chemistry_Fall_2017 | 32_Intermolecular_Interactions_by_NonDegenerate_Perturbation_Theory.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseware continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseware at ocw.mit.edu. PROFESSOR: Today, since t... |
MIT_561_Physical_Chemistry_Fall_2017 | 18_Rigid_Rotor_II_Derivation_by_Commutation_Rules.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Now the mai... |
MIT_561_Physical_Chemistry_Fall_2017 | 29_Modern_Electronic_Structure_Theory_Electronic_Correlation.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseware continue to offer high-quality, educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseware at ocw.mit.edu. TROY VAN VOORHIS: All right... |
MIT_561_Physical_Chemistry_Fall_2017 | 1_Quantum_MechanicsHistorical_Background_Photoelectric_Effect_Compton_Scattering.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. BOB FIELD: I'm Bob Field. Th... |
MIT_561_Physical_Chemistry_Fall_2017 | 25_Molecular_Orbital_Theory_II_H2_A2_AB_Diatomics.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: I like this lectu... |
MIT_561_Physical_Chemistry_Fall_2017 | 11_Wavepacket_Dynamics_for_Harmonic_Oscillator_and_PIB.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT Open Courseware at ocw.mit.edu. ROBERT FIELD: OK. So this ... |
MIT_561_Physical_Chemistry_Fall_2017 | 13_From_Hij_Integrals_to_H_Matrices_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: OK. So, today ... |
MIT_561_Physical_Chemistry_Fall_2017 | 27_NonDegenerate_Perturbation_Theory_III.txt | ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Whe... |
MIT_561_Physical_Chemistry_Fall_2017 | 19_Spectroscopy_Probing_Molecules_with_Light.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseware continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseware at ocw.mit.edu. ROBERT FIELD: This lecture i... |
MIT_561_Physical_Chemistry_Fall_2017 | 26_Qualitative_MO_Theory_Hückel.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Last time I... |
MIT_561_Physical_Chemistry_Fall_2017 | 22_Helium_Atom.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: I'm really sic... |
MIT_561_Physical_Chemistry_Fall_2017 | 7_Classical_Mechanical_Harmonic_Oscillator.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality, educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Last time, I ... |
MIT_561_Physical_Chemistry_Fall_2017 | 15_NonDegenerate_Perturbation_Theory_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT open courseware continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Now if you lo... |
MIT_561_Physical_Chemistry_Fall_2017 | 12_Catch_Up_and_Review_Postulates.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at osu.mit.edu. ROBERT FIELD: You know th... |
MIT_561_Physical_Chemistry_Fall_2017 | 6_3D_Box_and_QM_Separation_of_Variables.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Well, we're... |
MIT_561_Physical_Chemistry_Fall_2017 | 4_Classical_Wave_Equation_and_Separation_of_Variables.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT Open Courseware at ocw.mit.edu. ROBERT FIELD: That's th... |
MIT_561_Physical_Chemistry_Fall_2017 | 2_Wave_Nature_of_the_Electron_and_the_Internal_Structure_of_an_Atom.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Last time,... |
MIT_561_Physical_Chemistry_Fall_2017 | 14_From_Hij_Integrals_to_H_Matrices_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: So as I sa... |
MIT_561_Physical_Chemistry_Fall_2017 | 35_DeltaFunctions_EigenFunctions_of_X_Discrete_Variable_Representation.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Last time, we... |
MIT_561_Physical_Chemistry_Fall_2017 | 33_Electronic_Spectroscopy_FranckCondon.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at OCW.MIT.edu. PROFESSOR: Today's lecture i... |
MIT_561_Physical_Chemistry_Fall_2017 | 9_The_Harmonic_Oscillator_Creation_and_Annihilation_Operators.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Last lecture, ... |
MIT_561_Physical_Chemistry_Fall_2017 | 17_Rigid_Rotor_I_Orbital_Angular_Momentum.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT Open Courseware at ocw.mit.edu. ROBERT FIELD: Today is the... |
MIT_561_Physical_Chemistry_Fall_2017 | 8_Quantum_Mechanical_Harmonic_Oscillator.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: So today we... |
MIT_561_Physical_Chemistry_Fall_2017 | 10_The_TimeDependent_Schrödinger_Equation.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: The outline fo... |
MIT_561_Physical_Chemistry_Fall_2017 | 16_NonDegenerate_Perturbation_Theory_II_HO_using_aa.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Today, I'm ... |
MIT_561_Physical_Chemistry_Fall_2017 | 34_Electronic_Spectroscopy_and_Photochemistry.txt | The following content is provided under a Creative Commons license. Your support will help MIT Open Courseware continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseware at ocw.mit.edu. PROFESSOR ROBERT FIELD: ... |
MIT_561_Physical_Chemistry_Fall_2017 | 28_Modern_Electronic_Structure_Theory_Basis_Sets.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseware at ocw.mit.edu. TROY VAN VOORHIS: All rig... |
MIT_561_Physical_Chemistry_Fall_2017 | 24_Molecular_Orbital_Theory_I_Variational_Principle_and_Matrix_Mechanics.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: So in the p... |
MIT_561_Physical_Chemistry_Fall_2017 | 21_Hydrogen_Atom_II_Rydberg_States.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: This is one of... |
MIT_561_Physical_Chemistry_Fall_2017 | 5_Quantum_Mechanics_Free_Particle_and_Particle_in_1D_Box.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: OK, let's get ... |
MIT_561_Physical_Chemistry_Fall_2017 | 23_ManyElectron_Atoms.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Let's get star... |
MIT_561_Physical_Chemistry_Fall_2017 | 20_Hydrogen_Atom_I.txt | FEMALE SPEAKER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR... |
MIT_8962_General_Relativity_Spring_2020 | 20_Spherical_compact_sources_I.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So we're switching gears. We're done with the large scale structure of the universe. Now we're done with cosmology. And I'm going to move into the topic that will dominate the last several lectures of 8.962. Let's look at how to construct the spacetime of a compact body. ... |
MIT_8962_General_Relativity_Spring_2020 | 19_Cosmology_II.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So in the lectures I'm going to record today, we're going to conclude cosmology. And then we're going to begin talking about a another system in which we saw the Einstein field equations using asymmetry. So this is one where we will again consider systems that are spheric... |
MIT_8962_General_Relativity_Spring_2020 | 2_Introduction_to_tensors.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: Good afternoon. So we spent our last lecture laying out some of the basic foundations, making a couple of definitions. I want to quickly recap the most important concepts and definitions. And then, let me be blunt, I kind of want to get through these definitions, which I ... |
MIT_8962_General_Relativity_Spring_2020 | 23_Black_holes_II.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So in this final lecture, I want to think a bit sort of with an eye towards thinking about how one might actually make measurements that prove the nature of the black hole spacetime that was discussed the previous lecture. I'm going to discuss motion in a black hole space... |
MIT_8962_General_Relativity_Spring_2020 | 1_Introduction_and_the_geometric_viewpoint_on_physics.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: The key textbook for this class is Sean Carroll's textbook on general relativity. It's now almost 20 years old. I would say-- I think it's listed on the website as required. I would actually call it sort of semi-required. It is where I will tend to post most of the readin... |
MIT_8962_General_Relativity_Spring_2020 | 11_More_on_spacetime_curvature.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So this is unfortunately a rather sad lecture, [LAUGHS] since this will be the last one that I'm giving to a live audience here, although any of you who are grad students, if you want to come and keep me company while I'm recording my next couple of lectures-- I'm actuall... |
MIT_8962_General_Relativity_Spring_2020 | 8_Lie_transport_Killing_vectors_tensor_densities.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: Last Thursday, we began the work of moving from special relativity to general relativity, and we spent a lot of time unpacking two formulations of the principle of equivalence. So one, which goes under the name "weak equivalence principle"-- a simpler way of saying that i... |
MIT_8962_General_Relativity_Spring_2020 | 6_The_principle_of_equivalence.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So we're just picking up where we stopped last time. So we are beginning to discuss how we are going to sort of do a geometrical approach to physics, using a more general set of coordinates now. So we began talking about how things change when I discuss special relativity... |
MIT_8962_General_Relativity_Spring_2020 | 3_Tensors_continued.txt | [SQUEAKING][RUSTLING][CLICKING] SCOTT HUGHES: All right, welcome to Tuesday. So hopefully, you've all saw the brief announcement I send to the class. I have to introduce a colloquium speaker over in Astrophysics, basically at the second this class officially ends, so I will be wrapping things up a little bit early so t... |
MIT_8962_General_Relativity_Spring_2020 | 15_Linearized_gravity_II_Dynamic_sources.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: In the previous lecture, I began with the Einstein field equations in the most general form. And then I specialized to linearized gravity, to taking the spacetime metric, assuming that I can find a coordinate system in which it is close to the eta alpha beta form that we ... |
MIT_8962_General_Relativity_Spring_2020 | 16_Gravitational_radiation_I.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: OK. So let's just dive in. And I'll just quickly recap where we were last time. So what I did last time was, we are now beginning the adventure, so to speak, of looking at how one examines-- how one finds and then studies the properties of solutions to the Einstein field ... |
MIT_8962_General_Relativity_Spring_2020 | 7_Principle_of_equivalence_continued_parallel_transport.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right, good afternoon. So today's lecture is a particularly important one. We are going to introduce the main physical principle that underlies general relativity. We'll be spending a bunch of time connecting that principle to the mathematics as the rest of the term u... |
MIT_8962_General_Relativity_Spring_2020 | 13_The_Einstein_field_equation_variant_derivation.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right, so we are ready to move to the next lecture. I'm going to record today. So if we just quickly recap the highlights of what we did in our previous lecture, the goal of this lecture was to take all the tools that we have built and develop a theory of gravity. So ... |
MIT_8962_General_Relativity_Spring_2020 | 9_Geodesics.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So let me just do a quick recap of what we did last time. So today, we're going to move into things that are a little bit more physics. Last time we were really doing some things that allows us to establish some of the critical mathematical concepts we need to study the t... |
MIT_8962_General_Relativity_Spring_2020 | 4_Volumes_and_volume_elements_conservation_laws.txt | [SQUEAKING] [RUSTLING] [CLICKING] PROFESSOR: OK. Great. So at this point, we basically have all of the most important-- well, we now have the full understanding of how to work with tensors. I haven't really done too much physics with them at this point, but we've very carefully-- one might even argue excessively carefu... |
MIT_8962_General_Relativity_Spring_2020 | 10_Spacetime_curvature.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: We're in for a uncomfortable couple of weeks, but we will do our damnedest to make sure that-- I'm not going to say life won't be disrupted. Life is going to be bloody well disrupted. No question of that. But number one goal is making sure everyone remains healthy, both p... |
MIT_8962_General_Relativity_Spring_2020 | 12_The_Einstein_field_equation.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right. Good morning, 8.962. This is a very weird experience. I am standing in here talking to an empty classroom. I have some experience talking to myself, because like many of us, I am probably a little weirder than the average. But that does not change the fact that... |
MIT_8962_General_Relativity_Spring_2020 | 18_Cosmology_I.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right. So at this point we're going to switch gears. Everything that we have done over the past several lectures has been in service of the approach to solving the Einstein field equations in which we assume a small perturbation around an exact background. Most of it ... |
MIT_8962_General_Relativity_Spring_2020 | 17_Gravitational_radiation_II.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: OK. Let's move on to the next lecture. Might have gotten a little bit of me getting prepped here. I really have to give a lot of credit to the people in the Office of Digital Learning for making this entire thing possible. So I'm really grateful particularly to Elaine Mel... |
MIT_8962_General_Relativity_Spring_2020 | 14_Linearized_gravity_I_Principles_and_static_limit.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right, so in today's recorded lecture, I would like to pick up where we started-- excuse me. I'd like to pick up where we stopped last time. So I discussed the Einstein field equations in the previous two lectures. I derived them first from the method that was used by... |
MIT_8962_General_Relativity_Spring_2020 | 21_Spherical_compact_sources_II.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: So we'll pick up where we ended last time. We're looking at the spacetime of a compact spherical body, working in what we call Schwarzschild coordinates. We deduce that the line element describing this body is of the form ds squared equals negative e to the 2 phi dt squar... |
MIT_8962_General_Relativity_Spring_2020 | 22_Black_holes_I.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: I'm going to record two lectures today. These are the final two lectures I will be recording for 8.962. I actually have notes on an additional two lectures. I have notes for an additional two lectures. But those additional two lectures are somewhat advanced material. It's... |
MIT_8962_General_Relativity_Spring_2020 | 5_The_stress_energy_tensor_and_the_Christoffel_symbol.txt | [SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: All right. So welcome back. I had a little bit of a break. Before I get into-- I go over my quick recap, you hopefully all have seen the announcements that we're going to delay the due date of the next problem set until Tuesday. That's in part because some of the material... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 253_Digraphs_Connected_Vertices_Video.txt | ALBERT MEYER: The topic for this video lecture is Connected Vertices, the kind of connections we can have between different vertices through edges. We're going to start by showing that the shortest walk between two vertices is a path. We're going to prove it by contradiction. Suppose we have some path from u to v and i... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 343_Two_Pair_Poker_Hands_Video.txt | PROFESSOR: So let's do a basic example of counting that illustrates the use of these new generalized rules of the Division Rule and the Generalized Product Rule. And let's count some particular kind of poker hands called a 2-pair. So poker is a game where each player is dealt five cards from a deck of 52 cards. And the... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 444_Random_Variables_Uniform_Binomial_Video.txt | PROFESSOR: Certain kinds of random variables keep coming up, so let's look at two basic examples now, namely uniform random variables and binomial random variables. Let's begin with uniform, because we've seen those already. So a uniform random variable means that all the values that it takes, it takes with equal proba... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 113_Intro_to_Proofs_Part_2.txt | PROFESSOR: There's other kinds of bogus proofs that come up. Let's just run through this one quickly. Here's a fact that you about, roots of polynomials. Every polynomial has two roots, at least over the complex numbers, over c. And how do you prove that? Well, you just write down the formulas for the roots. You know t... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 261_DAGs_Video.txt | PROFESSOR: Directed acyclic graphs are a special class of graphs that really have and warrant a theory of their own. Of course, "directed acyclic graphs" is lot of syllables, so they're called "DAGs" for short. OK. So here's why they come up all the time. Let's look at a diagram that may be familiar to you. This shows ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 431_Independence_Video.txt | PROFESSOR: Independent events are events that have nothing to do with each other. And needless to say, it's a lot easier to work with them because you don't have to figure out some weird interaction between two events that do have something to do with each other. Typical case where independent events come up is, for ex... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 123_Proof_by_Cases.txt | PROFESSOR: Another basic proof technique is called Proof By Cases, which we prove something by breaking it up into pieces that are easy to prove but that together cover all possibilities. Let's look at an explicit, simple example from computer science. Here's a Java logical expression. The way to decipher this is that ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 353_InclusionExclusion_Example_Video.txt | PROFESSOR: The final general counting rules that we'll examine is called inclusion-exclusion, and it is a straightforward generalization of the sum rule-- at least in the simple case of two sets that we'll look at first. So we're going to look at 6042 example of applying inclusion-exclusion, but let's begin by stating ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 459_Linearity_of_Expectation_Video.txt | PROFESSOR: So we've been saving for the last the property that makes expectation calculating really easy and short circuits a lot of the ingenious methods that we've used up until now-- namely, expectation is linear. So what that means is that if you have two random variables R and S and two constants a and b, the expe... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 457_Mean_Time_to_Failure_Video.txt | PROFESSOR: We're constantly asking how long we have to wait for things. And in the context of probability theory, it turns into the technical question called, the expected time to failure-- or the mean time to failure. Some examples might be that an insurance company wants to know, for a given policy holder, the expect... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 184_Strong_Induction_Video.txt | PROFESSOR: So now we come to an interesting variant of ordinary induction called strong induction, and here's how it works. With strong induction, just as with ordinary induction, you prove the base case P of 0. You're trying to prove for all nP of n, so you prove P of 0, but now in order to prove P of n plus 1 in the ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 152_Predicate_Logic_2_Video.txt | PROFESSOR: So now we're going to talk about the concepts of validity and satisfiability, which take on some extra interest in the context of predicate calculus. So let's remember for propositional validity, if you have a propositional formula with variables taking the truth values ranging over true and false, then a fo... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 2115_Optimal_Stable_Matching_Video.txt | PROFESSOR: So stability has some value, but it doesn't mean that everybody's happy. In fact, it just means that nobody can find anybody else who's equally unhappy that they would want to run off with. So let's examine the question of how well people do using the mating ritual, or in other possible ways of finding stabl... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 274_Equivalence_Relations_Video.txt | Equivalence relations are another kind of binary relation on a set which play a crucial role in mathematics and in computer science in particular. And they can also be explained both in terms of digraphs and in terms of axioms. So let's begin with a digraph explanation of an equivalence relation. And the kind of relati... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 161_Sets_Definitions_Video.txt | PROFESSOR: We're going to look at the most fundamental of all mathematical data types, namely sets, and let's begin with the definitions. So informally, a set is a collection of mathematical objects, and the idea is that you treat the collection of objects as one new object. And that's a working definition. But of cour... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 317_Integral_Method_Video.txt | PROFESSOR: So we figured out that you can get the book stack, the overhang of the books, to be half the harmonic sum. With n books, you can get out Hn over 2 where Hn is this harmonic sum, or harmonic number. The question is, how are we going to estimate or calculate what this sum is? Now it turns out there is no simpl... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 191_State_Machines_Invariants_Video.txt | PROFESSOR: So we're going to talk about state machines, which is a topic that you're going to see in many further courses, because state machines model step by step processes. And of course, if you think about a computation, normally the way you think about it is that it's doing instructions, step by step, one after an... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 1101_Recursive_Data_Video.txt | PROFESSOR: Recursive data play a key role in programming, so let's take a mathematical look at what goes on. So the basic idea of recursive data is roughly that you're going to define a class of objects in terms of the simpler versions of the same object. With a little more precision, the idea is that you [? can ?] bui... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 331_Sum_And_Product_Rules_Video.txt | PROFESSOR: The topic of counting, or combinatorics, is an important one in a number of different disciplines, and notably in computer science. So the origins of the combinatorics and counting are a little bit disreputable. They come, historically, out of people studying gambling and trying to calculate odds and the fra... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 121_Proof_by_Contradiction.txt | PROFESSOR: It's doubtful if you really understand something if you can explain why it's true. That's what proofs are about in mathematics and in computer science. So we're going to be talking about proofs of lots of things that we're trying to understand. And in particular, we're going to look at a proof technique now ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 1111_Cardinality_Video.txt | PROFESSOR: Cardinality is the word that's used to refer to the size of infinite sets. And before we go further, let's take a quick look at why we're interested in infinite sets in a course that's Mathematics for Computer Scientists. Why does computer science care about infinite sets? Well, like every data structure tha... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 415_Sample_Spaces_Video.txt | PROFESSOR: So let's look now at the mathematical foundations of probability theory, the basic definitions of which we've just been doing examples up until now. So a key concept is a probability space. And that's what we're going to talk about in this segment. So the abstract setting of a probability space is the first ... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 144_Truth_Tables_Video.txt | PROFESSOR: Just as truth tables provide a simple, direct way to define the meanings of the individual propositional connectives or propositional operators, they also provide a methodical way to understand the behavior of formulas, and in particular whether two formulas are equivalent or whether a formula is valid, mean... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 323_Asymptotic_Properties_Video.txt | PROFESSOR: An advantage of expressing the asymptotic notations, in terms of limits, is that a bunch of their properties then become immediately obvious. Here's one. If f is little o of g, or f is asymptotically equal to g, then, in fact, f is big o of g. Or we can reason about this informally by saying that the first o... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 2119_Halls_Theorem.txt | PROFESSOR: So let's get set to state Hall's theorem in a way that doesn't mention boys and girls. But let's remember the girl-boy setup to start. So the general setup is a bipartite graph H. And a bipartite graph has two sets of vertices-- the girl vertices and the boy vertices. Formally, there's a set L of H, called t... |
MIT_6042J_Mathematics_for_Computer_Science_Spring_2015 | 1119_Russells_Paradox_Video.txt | PROFESSOR: So in this final segment today, we're going to talk about set theory just a little bit. Because if you're going to take a math class, if you're going to be exposed to math for computer science, it's useful to have at least a glimmering of what the foundations of math looks like, how it starts and how it gets... |
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