calculus

Calculus is a branch of mathematics that studies change and motion. It uses limits, derivatives, and integrals to analyze quantities that vary continuously. The two classical branches are differential calculus, which concerns rates of change and slopes, and integral calculus, which concerns accumulation and area, length, and volume. Together they provide a framework for modeling dynamic processes and solving problems involving continuous change. In higher dimensions, multivariable and vector calculus extend these ideas to functions of several variables and to fields, with tools such as partial derivatives, line and surface integrals, and operators like gradient, divergence, and curl.

The fundamental theorem of calculus connects differentiation and integration, showing that, under appropriate conditions, differentiation and integration are inverse processes. It links local rates to global quantities and underpins many methods in analysis and applied science.

Origins trace to methods for area and motion used by ancient mathematicians and, in the 17th century, Newton and Leibniz formulated the calculus with notation and procedures that are still used today. The subject was later given rigorous foundations in the 19th century by Cauchy and Weierstrass through the theory of limits and epsilon-delta definitions.

Calculus plays a central role in physics, engineering, economics, biology, and computer science. It is used to model motion, optimize systems, compute areas and probabilities, and solve differential equations describing natural phenomena.

Common techniques include differentiation rules, integration methods, and series expansions, as well as numerical approaches for cases lacking closed-form answers. Calculus is taught across science and engineering as a tool for analysis and problem solving.