Calculus

In its modern sense, calculus includes several mathematical disciplines. Most commonly, it means the detailed analysis of rates of change in functions and is of utmost importance in all sciences. It is usually divided into two (closely related) branches: differential calculus and integral calculus.

The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.

Initially, the derivative is defined via a process involving taking the limit of secant slopes as the two points defining the secant converge and the secant turns into a tangent line. This formula is called the difference quotient, or Newton quotient after Sir Isaac Newton, one of the discoverers of calculus along with Gottfried Wilhelm Leibniz. The difference quotient is:

f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}

where $h$ is the distance between the two secant points.

Since immediately substituting 0 for $h$ would lead to 0/0, which cannot be computed, the numerator must first be simplified until $h$ can be factored out and then canceled against the $h$ of the denominator. The resulting function $f'(x)$ is the derivative of the function, and is the computation of the instantaneous slopes at each point $x$ .

These messy limit calculations can be avoided in concrete cases because of powerful differentiation rules which allow us to find derivatives easily using simple algebraic manipulations. One should not therefore infer that the definition of derivatives in terms of limits is unnecessary. Rather, that definition is the means of proving those "powerful differentiation rules". See derivative for the details.

At a maximal or minimal point (these are collectively referred to as extrema), a function's derivative must be zero, and this yields a very useful optimization method. Another application of differential calculus is Newton's method, an algorithm to find zeros of a function by approximating the function by its tangent.

The second branch of 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.

The fundamental theorem of calculus states that derivatives and integrals are inverse operations. It was this realization by Newton/Leibniz that was the key to the explosion of analytic results after their work became known. Archimedes, and many others, had used what are essentially integral methods well before Newton/Leibniz, and a great many had effectively invented the idea of a derivative (eg, Barrow, Fermat, Pascal, Wallis, ...). The understanding of the 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 integrals algebraically, without actually performing the limit process, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivative. Differential equations are ubiquitous in the sciences.

The conceptual foundations of calculus include the function, limit, infinite sequences, infinite series, and continuity. Its tools include the symbol manipulation techniques associated with elementary algebra, and mathematical induction.

Calculus has been extended to differential equations, vector calculus, calculus of variations, and differential topology. The modern, formally correct version of calculus is known as real analysis.

History

Gottfried Wilhelm Leibniz and Sir Isaac Newton independently invented differential and integral calculus in the late 1600s. Newton was probably somewhat earlier, but he did not publish his discoveries. Newton (who represented derivatives as $f'$ , $f''$ , etc.) provided a host of applications in physics, but Leibniz' more flexible notation ( $df/dx$ , $d^{2}f/dx^{2}$ , etc.) was eventually adopted. (The simpler $f'$ notation is still used in some cases where it is sufficient.)

An ugly controversy erupted about precedence with supporters of Leibniz. The controversy was foolish on all sides. In particular, it set back British analysis (i.e. calculus based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than Leibniz', but was retained in British usage for 100+ years.

The limit definition of the derivative presented above was not evolved until much later, and neither Newton nor Leibniz, nor any of their followers until the mid-1800s, developed calculus with acceptable rigour. Nevertheless, the calculus was widely used, as it was a very powerful mathematical tool, but it was not until the nineteenth century that mathematicians like Augustin Louis Cauchy, Bernhard Bolzano, and Karl Weierstrass were able to provide a mathematically rigorous exposition. This eventually resulted in deep explorations of the concept of infinity by Georg Cantor and others.

Derived from the Latin word for "pebble", calculus in its most general sense can mean a method or system of calculation. Other disciplines called calculus include:

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