Time derivativeA 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 . 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 .
Differential of a functionIn calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation holds, where the derivative is represented in the Leibniz notation , and this is consistent with regarding the derivative as the quotient of the differentials.
Second derivativeIn calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.
Leibniz's notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively. Consider y as a function of a variable x, or y = f(x).
∂The character ∂ (Unicode: U+2202) is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as (read as "the partial derivative of z with respect to x"). It is also used for boundary of a set, the boundary operator in a chain complex, and the conjugate of the Dolbeault operator on smooth differential forms over a complex manifold. It should be distinguished from other similar-looking symbols such as lowercase Greek letter delta (δ) or the lowercase Latin letter eth (ð).
Fundamental theorem of calculusThe fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
DelDel, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Differential calculusIn mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value.
FluxionA fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time). Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions. Fluxions and fluents made up Newton's early calculus. Fluxions were central to the Leibniz–Newton calculus controversy, when Newton sent a letter to Gottfried Wilhelm Leibniz explaining them, but concealing his words in code due to his suspicion.
Method of FluxionsMethod of Fluxions (De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671 and published in 1736. Fluxion is Newton's term for a derivative. He originally developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known (similarly, his findings which eventually became the Philosophiae Naturalis Principia Mathematica were developed at this time and hidden from the world in Newton's notes for many years).
