Calculus
Calculus is the branch of mathematical analysis that explores how functions behave locally, that is, how they change in the immediate neighborhood of a given point.
What is it for? What does it mean?
At its heart, calculus is the mathematics of change. It provides a precise language for describing variation, growth, and motion, and it rests on two central ideas: limits and continuity for real-valued functions.
Together with geometry and algebra, calculus offers powerful methods for solving equations and for modeling phenomena across mathematics, physics, engineering, and many other fields.
What are the main branches of calculus?
Calculus is traditionally divided into two main branches:
- Differential calculus
It studies derivatives and explains how a function changes at a specific point, capturing concepts such as slope, velocity, and instantaneous rate of change. - Integral calculus
It focuses on the accumulation of quantities and makes it possible to compute areas, volumes, and other global measures derived from local information.
What are the tools of calculus?
The core tools of calculus are the limit, the derivative, and integrals.
- Limit
The concept of a limit provides the rigorous foundation for continuity and is essential for defining derivatives; - Derivatives
A derivative measures the rate at which a function changes, capturing how one quantity responds to variations in another; - Integrals
An integral allows one to compute the accumulation of quantities, linking local behavior to global results; - Sequences and series
A sequence is an ordered list of numbers defined by a specific rule. A series is the sum of the terms of a sequence and plays a key role in approximation and convergence.
These tools are widely used in mathematical analysis to represent functions and to study their behavior, both qualitatively and quantitatively.
A brief history of calculus
Calculus emerged in the seventeenth century, primarily through the work of Isaac Newton and Gottfried Wilhelm Leibniz, who independently developed the fundamental ideas of derivatives and integrals.
Their work made it possible to give a precise mathematical description of processes involving change and accumulation.
During the eighteenth and nineteenth centuries, mathematicians such as Euler, Cauchy, and Weierstrass placed calculus on a firm logical footing by introducing precise definitions of limits and continuity, laying the foundations of modern mathematical analysis.
Many of the most important theorems in analysis still bear their names today.
And so on.
