Laplace Equation
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Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix...more
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This Stanford Continuing Studies course is a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics, quantum mechanics, the general and special theories of relativity, electromagnatism, cosmology, black holes and statistical mechanics. While these courses build upon one another, each section of the course also stands on its own, and...more
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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's,...more
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This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.
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In order for Social Security to work, people have to believe there's some possibility that the world will last forever, so that each old generation will have a young generation to support it. The overlapping generations model, invented by Allais and Samuelson but here augmented with land, represents such a situation. Financial equilibrium can again be reduced to general equilibrium. At first glance it would seem that the model requires a...more
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After showing how a double-minimum potential generates one-dimensional bonding, Professor McBride moves on to multi-dimensional wave functions. Solving Schrödinger's three-dimensional differential equation might have been daunting, but it was not, because the necessary formulas had been worked out more than a century earlier in connection with acoustics. Acoustical "Chladni" figures show how nodal patterns relate to frequencies. The...more
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Structure of the Atom: A Conundrum. The work of E. Rutherford, 1911, lead to the discovery of the nucleus. In this lecture, Professor Sylvia Ceyer begins by explaining the backscattering experiment that lead to this key discovery in the early 20th century. She then moves on to a classical description of the atom, including coulombic interaction and the classical equation of motion (Newton's Second Law). The lecture ends with discussion...more
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This lecture brings experiment to bear on the previous theoretical discussion of bonding by focusing on hybridization of the central atom in three XH3 molecules. Because independent electron pairs must not overlap, hybridization can be related to molecular structure by a simple equation. The "Umbrella Vibration" and the associated rehybridization of the central atom is used to illustrate how a competition between strong bonds and stable...more
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Government and Entrepreneurship: The Evolution of Entrepreneurship in China
Stanford / Entrepreneurship
Tarun Khanna, Professor at Harvard Business School, argues that the old equation that government = inefficient does not hold unilaterally but depends on the context. To illustrate, Khanna describes the evolution of the Communist Party of China and its efforts to co-opt entrepreneurs so that now the government and entrepreneurship is very closely integrated. In contrast, Khanna suggests that in India entrepreneurs keep their distance from...more
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After pointing out several discrepancies between electron difference density results and Lewis bonding theory, the course proceeds to quantum mechanics in search of a fundamental understanding of chemical bonding. The wave function ψ, which beginning students find confusing, was equally confusing to the physicists who created quantum mechanics. The Schrödinger equation reckons kinetic energy through the shape of ψ. When ψ curves toward...more
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Continuing Convolution: Review Of The Formula, Situiation In Which It Arose, Example Of Convolution: Filtering, The Ideas Behind Filtering, Terminology, Interpreting Convolution In The Time Domain, General Properties Of Convolution In The Time Domain, Derivative Theorem For Fourier Transforms, Heat Equation On An Infinite Rod
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Concepts covered in this lecture begin with the restoring force of a spring (Hooke's Law) which leads to an equation of motion that is characteristic of a simple harmonic oscillator (SHO). Using the small angle approximation, a similar expression is reached for a pendulum. To demonstrate that the period is independent of the mass of the bob, Professor Lewin places himself at the end of the 5 meter long cable and measures the period.
