Calculus

This page is about infinitesmal calculus. For other uses of the word in mathematics and other fields, click here

Calculus usually refers to the elementary study of real-valued functions and their applications to the study of quantities. The central tools of Calculus are the limit, the derivative, and the integral. The subject can be divided into two major branches: Differential Calculus and Integral Calculus, concerned with the study of the derivatives and integrals of functions respectively. The relationship between these two branches of Calculus is encapsulated in the Fundamental Theorem of Calculus. Calculus can be extended to Multivariable Calculus, which studies the properties and applications of functions in multiple variables. Calculus belongs to the more general field of Analysis, which is concerned with the study of functions in a more general setting. The study of real-valued functions is called real analysis and the study of complex-valued functions is called complex analysis.

Calculus vs. Analysis

Strictly speaking, there is virtually no distinction between the topic called Calculus and the topic called Analysis. The distinction is made on historical and pedagogical grounds. Calculus usually refers to the material taught to first and second year university students. It is usually non-rigorous and more concerned with applications and problem solving than theoretical development. Analysis usually refers to the study of functions in a more technical and rigorous setting, usually starting with a first course in the theoretical foundations of elementary calculus. The treatment of Calculus generally follows the historical development pioneered by Isaac Newton and Gottfried Leibniz. The development of introductory Analysis follows the rigorous treatment of the subject that was formulated by mathematicians such as Karl Weierstrass and Augustin Cauchy.