Calculus
The two concepts define inverse operations, in a sense made quite precise by the fundamental theorem of calculus. This means that either may in fact be given priority, but the usual educational approach is to introduce differential calculus first.
The development of calculus is credited to Archimedes, Leibniz and Newton. However, when calculus was first being developed, there was a controversy to who came up with the idea "first" - Leibniz and Newton being the contenders for the crown.
It is thought that Newton had discovered several ideas related to calculus earlier than Leibniz had, however Leibniz was the first to publish. Today, both Leibniz and Newton are considered to have discovered calculus independently.
Lesser credit is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. One of the primary motives for the development of differential calculus was the solution of the so-called "tangent line problem".
Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes of the function's argumentss. This idea lies at the heart of most of the physical sciences. For example basic theory of electrical circuits is formulated in terms of differential equations, to describe the cases where there is oscillation.
The derivative of a function is directly relevant to finding its maxima and minima — because those are points at which the graph is (expected to be) flat. Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the function by its tangents. These are just some of a large number of ways in which calculus is applied in questions that at first sight are not formulated in calculus terms.
de Fermat is sometimes described as the "father" of differential calculus.
Integral calculus studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms, each of which corresponds to a small strip of area under the graph of a function. Considered as such, integration provides effective ways to calculate the area under a curve, and the surface area and volume of solids such as spheres and cones.
The conceptual foundations of calculus include the notions of functions, limits, infinite sequences, infinite series, and continuity. Its tools include the symbol manipulation techniques associated with elementary algebra, and mathematical induction. The modern version of calculus is known as real analysis; this consists of a rigorous derivation of the results of calculus as well as generalisations such as measure theory and functional analysis.
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. It was this realization by Newton and Leibniz that was the key to the explosion of analytic results after their work became known. This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many definite integrals algebraically, without actually performing the limit processes, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivatives. Differential equations are ubiquitous in the sciences.
The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, and especially physics. Almost all modern developments such as building techniques, aviation, and nearly all other technologies make fundamental use of calculus.
Calculus has been extended to differential equations, vector calculus, calculus of variations, complex analysis, time scale calculus and differential topology.
In mathematics and related fields, the term '\calculus' more generally refers to a system of formal rules of inference and axioms that are used for computation.
This usage is particularly common in mathematical logic, where a calculus is applied to compute universally true statements of a certain formal logic. Examples include the calculus of natural deduction, the sequent calculus, as well as many other calculi that are deviced in proof theory.
Derived from the Latin word for "pebble", calculus in its most general sense can mean any method or system of calculation. Other topics where the term calculus is used in this sense include:
History
See main article History of calculusDifferential calculus
Main article derivativeIntegral calculus
Main article integralFoundations
Fundamental theorem of calculus
Applications
See also
Further reading
External link
Other uses of the term
| Topics in mathematics related to change | Edit |
| Arithmetic | Calculus | Vector calculus | Analysis | Differential equations | Dynamical systems and chaos theory | List of functions |