What is "differential calculus"

differential \dif`fer*en"tial\, a. [Cf. F. diff['e]rentiel.]

1. Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.

   For whom he produced differential favors.
   --Motley.

2. (Math.) Of or pertaining to a differential, or to differentials.

3. (Mech.) Relating to differences of motion or leverage; producing effects by such differences; said of mechanism. Differential calculus. (Math.) See under Calculus. Differential coefficient, the limit of the ratio of the increment of a function of a variable to the increment of the variable itself, when these increments are made indefinitely small. Differential coupling, a form of slip coupling used in light machinery to regulate at pleasure the velocity of the connected shaft. Differential duties (Polit. Econ.), duties which are not imposed equally upon the same products imported from different countries. Differential galvanometer (Elec.), a galvanometer having two coils or circuits, usually equal, through which currents passing in opposite directions are measured by the difference of their effect upon the needle. Differential gearing, a train of toothed wheels, usually an epicyclic train, so arranged as to constitute a differential motion. Differential motion, a mechanism in which a simple differential combination produces such a change of motion or force as would, with ordinary compound arrangements, require a considerable train of parts. It is used for overcoming great resistance or producing very slow or very rapid motion. Differential pulley. (Mach.)

   1. A portable hoisting apparatus, the same in principle as the differential windlass.

   2. A hoisting pulley to which power is applied through a differential gearing.

      Differential screw, a compound screw by which a motion is produced equal to the difference of the motions of the component screws.

      Differential thermometer, a thermometer usually with a U-shaped tube terminating in two air bulbs, and containing a colored liquid, used for indicating the difference between the temperatures to which the two bulbs are exposed, by the change of position of the colored fluid, in consequence of the different expansions of the air in the bulbs. A graduated scale is attached to one leg of the tube.

      Differential windlass, or Chinese windlass, a windlass whose barrel has two parts of different diameters. The hoisting rope winds upon one part as it unwinds from the other, and a pulley sustaining the weight to be lifted hangs in the bight of the rope. It is an ancient example of a differential motion.

Calculus \Cal"cu*lus\, n.; pl. Calculi. [L, calculus. See Calculate, and Calcule.]

1. (Med.) Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc.

2. (Math.) A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation.

   Barycentric calculus, a method of treating geometry by defining a point as the center of gravity of certain other points to which co["e]fficients or weights are ascribed.

   Calculus of functions, that branch of mathematics which treats of the forms of functions that shall satisfy given conditions.

   Calculus of operations, that branch of mathematical logic that treats of all operations that satisfy given conditions.

   Calculus of probabilities, the science that treats of the computation of the probabilities of events, or the application of numbers to chance.

   Calculus of variations, a branch of mathematics in which the laws of dependence which bind the variable quantities together are themselves subject to change.

   Differential calculus, a method of investigating mathematical questions by using the ratio of certain indefinitely small quantities called differentials. The problems are primarily of this form: to find how the change in some variable quantity alters at each instant the value of a quantity dependent upon it.

   Exponential calculus, that part of algebra which treats of exponents.

   Imaginary calculus, a method of investigating the relations of real or imaginary quantities by the use of the imaginary symbols and quantities of algebra.

   Integral calculus, a method which in the reverse of the differential, the primary object of which is to learn from the known ratio of the indefinitely small changes of two or more magnitudes, the relation of the magnitudes themselves, or, in other words, from having the differential of an algebraic expression to find the expression itself.

n. (context calculus English) The calculus that deals with instantaneous rate of change.

n. the part of calculus that deals with the variation of a function with respect to changes in the independent variable (or variables) by means of the concepts of derivative and differential [syn: method of fluxions]

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

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. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.

Differentiation has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the force applied to the body; rearranging this derivative statement leads to the famous equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.