Two dramatically different philosophical approaches to classical mechanics were proposed during the 17th – 18th centuries. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action. These variational formulations now play a pivotal role in science and engineering.

This book introduces variational principles and their application to classical mechanics. The relative merits of the intuitive Newtonian vectorial formulation, and the more powerful variational formulations are compared. Applications to a wide variety of topics illustrate the intellectual beauty, remarkable power, and broad scope provided by use of variational principles in physics.

The growth of blood vessels, a process known as angiogenesis, is one of the earliest events in mammalian development and is regulated by a sensitive interplay of growth factors and other molecules. In this course, we will discuss the key molecular regulators of blood vessel development as well as the techniques and experimental systems that have been utilized by vascular biologists. We will also examine the success of several anti-angiogenic treatments that have been approved by the Food and Drug Administration (FDA), that inhibit the pro-angiogenic vascular endothelial growth factor, VEGF, and that are now being used to treat age-related macular degeneration. Finally, we will explore how during the course of cancer progression, establishment of a blood supply into a tumor can lead to the growth and spread of cancer cells to secondary sites. We will discuss the caveats and potential pitfalls of targeting tumor blood vessels to starve cancer cells and prevent the spread of cancer, which remains one of the leading causes of death in the USA. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. Many instructors of the Advanced Undergraduate Seminars are postdoctoral scientists with a strong interest in teaching.

Online resource focused on budgeting, savings, income and careers, and investing that contains interactive activities and built in pre and post assessments for you to easily track student progress.

Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.

Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.

This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

In this activity, students will use vector analysis to understand the concept of dead reckoning. Students will use vectors to plot their course based on a time and speed. They will then correct the positions with vectors representing winds and currents.

This art history video discussion looks at Diego Rodriguez de Silva y Velazquez's "Las Meninas", c. 1656, oil on canvas (Museo Nacional del Prado, Madrid).

This art history video discussion examines Diego Rodriguez de Silva y Velazquez's "Los Borrachos" (The Drunks) or The Triumph of Bacchus, 1628-1629 (Museo Nacional del Prado, Madrid).

This art history video discussion looks at Diego Velazquez's "Vulcan's Forge", c. 1630, oil on canvas (Museo del Prado, Madrid).

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

This art history video discussion looks at Domenico Veneziano's "Saint Lucy Altarpiece", 1445-47, fresco transfered to canvas (Galleria degli Uffizi, Florence).

Expose middle school students to a first taste of Shakespeare from the angle of the ghost story and launch into the subject of verbs. In this lesson, they learn how Shakespeare uses verbs to move the action of the play. Students then distinguish generic verbs from vivid verbs by working with selected lines in Hamlet's Ghost scene. Finally they test their knowledge of verbs through a crossword interactive puzzle.

Venture uses case studies, interactive business simulations, and personal development activities to teach students how to think entrepreneurially in business and in life. Students will design a simulated business, perfect their business pitch and create an ePortfolio reflecting their personal career goals.

Uses primary sources to explore the impacts World War I had on the struggle for civil rights in Connecticut and America.

A library of primary source documents relating to colonial Connecticut, scaffolded for student use

A social studies resource, curriculum, and lesson plans for study of colonial Connecticut that foregrounds the first person narrative of Venture Smith

This art history video discussion looks at Jan Vermeer's "The Glass of Wine", c. 1661, oil on canvas (Gemaldegalerie, Staatliche Museen zu Berlin).

This art history video discussion looks at Paolo Veronese's "The Dream of Saint Helena", c. 1570, oil on canvas (The National Gallery, London).

This art history video discussion looks at Paolo Veronese's "The Family of Darius before Alexander", 1565-67, oil on canvas (The National Gallery, London).