Multivariable

Multivariable refers to anything that involves more than one variable. In mathematics and related fields, it typically describes functions of several variables, such as f(x1, x2, ..., xn). A point in the domain lies in R^n, and the function maps to R or to another space. The study of multivariable phenomena emphasizes how variables interact and how quantities change when multiple inputs vary.

In mathematics, multivariable functions extend the ideas of single-variable calculus to several dimensions. Core tools include partial derivatives, which measure how a function changes with respect to one variable while holding others fixed; gradients, which combine partial derivatives into a vector indicating the direction of steepest ascent; and Hessians, which summarize second-order changes. Integration extends to multiple integrals over regions in higher dimensions. Vector fields, line and surface integrals, and the multivariable version of the chain rule (involving Jacobians) are central components.

Optimization in several variables seeks extrema of functions of multiple inputs, with methods for unconstrained settings and for constrained problems, such as Lagrange multipliers. Coordinate changes and geometry often rely on different coordinate systems (Cartesian, cylindrical, spherical) and on concepts like level sets and manifolds.

Applications are broad, spanning physics, engineering, economics, statistics, and computer science. In statistics, multivariate analysis studies joint distributions, covariance matrices, and correlations. In data science and machine learning, multivariable (high-dimensional) spaces raise considerations such as the curse of dimensionality.