Time derivative

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as .

Notation

A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,

A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E.

(This is called Newton's notation)

Higher time derivatives are also used: the second derivative with respect to time is written as

with the corresponding shorthand of .

As a generalization, the time derivative of a vector, say:

is defined as the vector whose components are the derivatives of the components of the original vector. That is,

Use in physics

Time derivatives are a key concept in physics. For example, for a changing position , its time derivative is its velocity, and its second derivative with respect to time, , is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk. See motion graphs and derivatives.

Khan Academy

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with the aim of providing a free, world-class education for anyone, anywhere. The organization produces short lectures in the form of YouTube videos. In addition to micro lectures, the organization's website features practice exercises and tools for educators. All resources are available for free to anyone around the world. The main language of the website is English, but the content is also available in other languages.

History

The founder of the organization, Salman Khan, was born in New Orleans, Louisiana, United States to immigrant parents from Bangladesh and India. After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MEng in electrical engineering and computer science), he pursued an MBA from Harvard Business School.

In late 2004, Khan began tutoring his cousin Nadia who needed help with math using Yahoo!'s Doodle notepad.When other relatives and friends sought similar help, he decided that it would be more practical to distribute the tutorials on YouTube. The videos' popularity and the testimonials of appreciative students prompted Khan to quit his job in finance as a hedge fund analyst at Connective Capital Management in 2009, and focus on the tutorials (then released under the moniker "Khan Academy") full-time.

The Material

For the New York City band of the same name, see Material (band)

The Material is an American rock quintet from San Diego, California. Colleen D'Agostino (vocals) moved to San Diego to pursue a music degree at San Diego State University. In her third year, she began playing with Jon Moreaux (guitar) and Noah Vowles (drums). The three added bassist Kevin Falk, formerly of Every Time I Die and Between the Buried and Me, and started writing songs for their first demo. Kevin was replaced by Brian Miller (bass), and Roi Elam (guitar) joined shortly after. With the permanent line up, The Material went into the studio to record their 6-song debut EP "Tomorrow", which was co-produced by Brian Grider and was released on September 1, 2007. They placed in the top three of the Dew Circuit Breakout of 2007, losing to Seattle band The Myriad.

The Material has been featured on MTV and MTV2 and their song "Moving to Seattle" is also available to download for the video game, Rock Band. They have toured the entire United States and in September 2009, headlined the coast-to-coast Everlasting Sound Tour with the bands Blameshift and And Then There Was You.

Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable: the differentiation and integration of functions involving multiple variables, rather than just one.

Typical operations

Limits and continuity

A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola. For example, the function

approaches zero along any line through the origin. However, when the origin is approached along a parabola , it has a limit of 0.5. Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

Continuity in each argument is not sufficient for multivariate continuity: For instance, in the case of a real-valued function with two real-valued parameters, , continuity of in for fixed and continuity of in for fixed does not imply continuity of . As an example, consider