systems of equations
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Introduction to basic algebraic equations of the form Ax=B.
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Equations with the variable on both sides.
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Using the method of undetermined coefficients to solve nonhomogeneous linear differential equations.
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What a differential equation is and some terminology?
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Introduction to separable differential equations.
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This lecture is all about motion of projectiles (if air drag can be ignored). The objects experience a constant vertical acceleration due to the acceleration of gravity (see also Lecture 12). Professor Lewin reviews the equations for projectile motion, showing that the trajectory is a parabola. He continues with a demonstration that shows how to measure the initial speed of a projectile and how to reach maximum horizontal distance shooting...more
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Professor Shankar introduces the course and answers student questions about the material and the requirements. He gives an overview of Newtonian mechanics and explains its two components: kinematics and dynamics. He then reviews basic concepts in calculus through two key equations: x0 + v0t + ½ at2 and v2 = v02+ 2 a (x-x0), tracing the fate of a particle in one dimension along the x-axis.
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In the absence of a net external torque on an object, angular momentum is conserved. When an object oscillates about an axis of rotation, there is a variable restoring torque acting on the object. A review is given of equations for angular momentum and torque, and the importance of choosing the point of origin. These equations are exercised using an example of a circular orbit.
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The focus of the lecture is simple harmonic motion. Professor Shankar gives several examples of physical systems, such as a mass M attached to a spring, and explains what happens when such systems are disturbed. Amplitude, frequency and period of simple harmonic motion are also defined in the course of the lecture. Several problems are solved in order to demonstrate various cases of oscillation.
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State-action Rewards, Finite Horizon MDPs, The Concept of Dynamical Systems, Examples of Dynamical Models, Linear Quadratic Regulation (LQR), Linearizing a Non-Linear Model, Computing Rewards, Riccati Equation
