Online courses directory (418)

This course is a continuation of 18.014. It covers the same material as 18.02 (Multivariable Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis. Assignments require MATLAB^® programming.

Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®.

Acknowledgements

The instructor would like to thank Robert Ashcraft, Sandeep Sharma, David Weingeist, and Nikolay Zaborenko for their work in preparing materials for this course site.

This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods.

This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.

Electrostatics (part 1): Introduction to Charge and Coulomb's Law. Electrostatics (part 2). Proof (Advanced): Field from infinite plate (part 1). Proof (Advanced): Field from infinite plate (part 2). Electric Potential Energy. Electric Potential Energy (part 2-- involves calculus). Voltage. Capacitance. Circuits (part 1). Circuits (part 2). Circuits (part 3). Circuits (part 4). Cross product 1. Cross Product 2. Cross Product and Torque. Introduction to Magnetism. Magnetism 2. Magnetism 3. Magnetism 4. Magnetism 5. Magnetism 6: Magnetic field due to current. Magnetism 7. Magnetism 8. Magnetism 9: Electric Motors. Magnetism 10: Electric Motors. Magnetism 11: Electric Motors. Magnetism 12: Induced Current in a Wire. The dot product. Dot vs. Cross Product. Calculating dot and cross products with unit vector notation. Electrostatics (part 1): Introduction to Charge and Coulomb's Law. Electrostatics (part 2). Proof (Advanced): Field from infinite plate (part 1). Proof (Advanced): Field from infinite plate (part 2). Electric Potential Energy. Electric Potential Energy (part 2-- involves calculus). Voltage. Capacitance. Circuits (part 1). Circuits (part 2). Circuits (part 3). Circuits (part 4). Cross product 1. Cross Product 2. Cross Product and Torque. Introduction to Magnetism. Magnetism 2. Magnetism 3. Magnetism 4. Magnetism 5. Magnetism 6: Magnetic field due to current. Magnetism 7. Magnetism 8. Magnetism 9: Electric Motors. Magnetism 10: Electric Motors. Magnetism 11: Electric Motors. Magnetism 12: Induced Current in a Wire. The dot product. Dot vs. Cross Product. Calculating dot and cross products with unit vector notation.

This tutorial is the meat of much of classical physics. We think about what a force is and how Newton changed the world's (and possibly your) view of how reality works. Newton's First Law of Motion. Newton's First Law of Motion Concepts. Newton's First Law of Motion. Newton's First Law. Newton's Second Law of Motion. Newton's Third Law of Motion. Newton's Third Law of Motion. All of Newton's Laws of Motion. Normal Force and Contact Force. Normal Force in an Elevator. Balanced and Unbalanced Forces. Unbalanced Forces and Motion. Slow Sock on Lubricon VI. Normal Forces on Lubricon VI. Inclined Plane Force Components. Ice Accelerating Down an Incline. Force of Friction Keeping the Block Stationary. Correction to Force of Friction Keeping the Block Stationary. Force of Friction Keeping Velocity Constant. Intuition on Static and Kinetic Friction Comparisons. Static and Kinetic Friction Example. Introduction to Tension. Introduction to Tension (Part 2). Tension in an accelerating system and pie in the face. Newton's First Law of Motion. Newton's First Law of Motion Concepts. Newton's First Law of Motion. Newton's First Law. Newton's Second Law of Motion. Newton's Third Law of Motion. Newton's Third Law of Motion. All of Newton's Laws of Motion. Normal Force and Contact Force. Normal Force in an Elevator. Balanced and Unbalanced Forces. Unbalanced Forces and Motion. Slow Sock on Lubricon VI. Normal Forces on Lubricon VI. Inclined Plane Force Components. Ice Accelerating Down an Incline. Force of Friction Keeping the Block Stationary. Correction to Force of Friction Keeping the Block Stationary. Force of Friction Keeping Velocity Constant. Intuition on Static and Kinetic Friction Comparisons. Static and Kinetic Friction Example. Introduction to Tension. Introduction to Tension (Part 2). Tension in an accelerating system and pie in the face.

Relationship between angular velocity and speed. Why Distance is Area under Velocity-Time Line. Introduction to Vectors and Scalars. Calculating Average Velocity or Speed. Solving for Time. Displacement from Time and Velocity Example. Acceleration. Balanced and Unbalanced Forces. Unbalanced Forces and Motion. Newton's First Law of Motion. Newton's First Law of Motion Concepts. Newton's First Law of Motion. Newton's Second Law of Motion. Newton's Third Law of Motion. Airbus A380 Take-off Time. Airbus A380 Take-off Distance. Average Velocity for Constant Acceleration. Acceleration of Aircraft Carrier Takeoff. Race Cars with Constant Speed Around Curve. Introduction to Gravity. Mass and Weight Clarification. Gravity for Astronauts in Orbit. Would a Brick or Feather Fall Faster. Deriving Displacement as a Function of Time, Acceleration and Initial Velocity. Plotting Projectile Displacement, Acceleration, and Velocity. Projectile Height Given Time. Deriving Max Projectile Displacement Given Time. Impact Velocity From Given Height. Visualizing Vectors in 2 Dimensions. Projectile at an Angle. Different Way to Determine Time in Air. Launching and Landing on Different Elevations. Total Displacement for Projectile. Total Final Velocity for Projectile. Correction to Total Final Velocity for Projectile. Projectile on an Incline. Unit Vectors and Engineering Notation. Clearing the Green Monster at Fenway. Green Monster at Fenway Part 2. Optimal angle for a projectile part 1. Optimal angle for a projectile part 2 - Hangtime. Optimal angle for a projectile part 3 - Horizontal distance as a function of angle (and speed). Optimal angle for a projectile part 4 Finding the optimal angle and distance with a bit of calculus. Slow Sock on Lubricon VI. Normal Forces on Lubricon VI. Normal Force and Contact Force. Normal Force in an Elevator. Inclined Plane Force Components. Ice Accelerating Down an Incline. Force of Friction Keeping the Block Stationary. Correction to Force of Friction Keeping the Block Stationary. Force of Friction Keeping Velocity Constant. Intuition on Static and Kinetic Friction Comparisons. Static and Kinetic Friction Example. Introduction to Tension. Introduction to Tension (Part 2). Tension in an accelerating system and pie in the face. Introduction to Momentum. Momentum: Ice skater throws a ball. 2-dimensional momentum problem. 2-dimensional momentum problem (part 2). Introduction to work and energy. Work and Energy (part 2). Conservation of Energy. Work/Energy problem with Friction. Introduction to mechanical advantage. Mechanical Advantage (part 2). Mechanical Advantage (part 3). Center of Mass. Introduction to Torque. Moments. Moments (part 2). Unit Vector Notation. Unit Vector Notation (part 2). Projectile Motion with Ordered Set Notation. Projectile motion (part 1). Projectile motion (part 2). Projectile motion (part 3). Projectile motion (part 4). Projectile motion (part 5). Centripetal Force and Acceleration Intuition. Visual Understanding of Centripetal Acceleration Formula. Calculus proof of centripetal acceleration formula. Loop De Loop Question. Loop De Loop Answer part 1. Loop De Loop Answer part 2. Acceleration Due to Gravity at the Space Station. Space Station Speed in Orbit. Conservation of angular momentum. Introduction to Newton's Law of Gravitation. Gravitation (part 2). Viewing g as the value of Earth's Gravitational Field Near the Surface. Intro to springs and Hooke's Law. Potential energy stored in a spring. Spring potential energy example (mistake in math). Introduction to Harmonic Motion. Harmonic Motion Part 2 (calculus). Harmonic Motion Part 3 (no calculus).

The foundations of Pre-Algebra from The Khan Academy.

This free online course in Pre-Algebra mathematics for high school students will guide you through several different areas of mathematics such as integers, one-step equations, inequalities and equations, graphs and functions, percent, probabilities. There is also an introduction into geometry and right triangles. The course is divided into twelve modules and each module is divided into several lessons. Under each lesson you will find theory, examples and video lessons. This course is ideal for learners who want to gain a comprehensive knowledge and understanding of topics in pre-algebra which they can build on in later courses.

This course covers mathematical topics in algebra and trigonometry and is designed to prepare students to enroll for a first semester course in single variable calculus.