Calculus

Calculus is a branch of mathematics, developed from algebra and geometry. It involves two major complementary ideas. The first, called differential calculus, involving the method of differentiation, focuses on rates of change (within functions), such as accelerations, curves, and slopes. The second, called integral calculus, involving the idea of integration, focuses on a generalization of the idea of area under a function, and similar concepts such as volume. These two ideas are inverse operations in a sense shown by the fundamental theorem of calculus.

Table of contents

1 Main branches

2 Foundations

3 Fundamental theorem of calculus

4 History

5 Applications

6 Notations for differentiation

7 See also

8 Further reading

9 External Link

Main branches

There are two main branches of calculus:

Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes within the function's arguments. Another application of differential calculus is Newton's method, an algorithm to find zeros of a function by approximating the function by its tangent. 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 which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.

Foundations

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.

Fundamental theorem of calculus

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.

History

See main article History of calculus

The development of calculus is credited to Archimedes, Leibniz and Newton; 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".

Applications

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.

Notations for differentiation

See Leibniz notation for a fuller discussion; also separation of variables for certain advantages of this notation. See also [[differential operator

Lagrange notation for differentiation is

f '(x) for the first derivative,

f ''(x) for the second derivative,

f '''(x) for the third derivative and then

f⁽ⁿ⁾(x) for the nth derivative (n > 3).

This is usually most convenient for contexts where derivatives are treated as functions.

In the Leibniz notation for differentiation, the derivative of the function f(x) is written:

If we have a variable representing a function, for example if we set:

then we can write the derivative as:

Using Lagrange's notation , we could write:

Higher derivatives are expressed as

for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

which we can loosely write as:

Now drop the brackets and we have:

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear symbolically to cancel:

etc.

and:

Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation.

Newton notation for differentiation involves placing a dot over the function name:

and so on.

Newton termed the derivative the fluxion. Newton notation is mainly used in mechanics, normally for time derivatives suhc as velocity and acceleration, and ODE theory. It is not so convenient for high derivatives.

Newton used a number of different notations for integrals (fluents). The widely adopted notation is Leibniz notation for integration.

External Link

The Role of Calculus in College Mathematics

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:

Lambda calculus (a formulation of the theory of reflexive functions with deep connections to computational theory; due in final form to Alonzo Church of Princeton)
Predicate calculus (the rules governing the logic of predicates in symbolic logic)

Topics in mathematics related to change
Arithmetic \| Calculus \| Vector calculus \| Analysis \| Differential equations \| Dynamical systems and chaos theory \| List of functions