Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.

Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.

Introduction

In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:

  1. There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D;
  2. There are multiple extended objects associated with the dimension; for example, a 1D function is represented as a curve on the 2D Cartesian plane, but a scalar-valued function of two variables is a surface in 3D, while curves can also live in 3D space.

The consequence of the first difference is the difference in the definition of the limits and continuity. Directional limits and derivatives define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators.

The consequence of the second difference is the existence of multiple types of integration, including line integrals, surface integrals and volume integrals. Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot be properly defined.

Limits

A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.

A limit along a path may be defined by considering a parametrised path <math>s(t): \mathbb{R} \to \mathbb{R}^n</math> in n-dimensional Euclidean space. Any function <math>f(\overrightarrow{x}): \mathbb{R}^n \to \mathbb{R}^m</math> can then be projected on the path as a 1D function <math>f(s(t))</math>. The limit of <math>f</math> to the point <math>s(t_0)</math> along the path <math>s(t)</math> can hence be defined as

Note that the value of this limit can be dependent on the form of <math>s(t)</math>, i.e. the path chosen, not just the point which the limit approaches.

Applications and uses

Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,

{| class="wikitable" style="text-align:center"

|-

! !! !!Type of functions!! Applicable techniques

|-

! Curves

| 120px || <math>f: \mathbb{R} \to \mathbb{R}^n</math> <br> for <math>n > 1</math> || Lengths of curves, line integrals, and curvature.

|-

! Surfaces

| 120px || <math>f: \mathbb{R}^2 \to \mathbb{R}^n</math> <br> for <math>n > 2</math> || Areas of surfaces, surface integrals, flux through surfaces, and curvature.

|-

! Scalar fields

| 120px || <math>f: \mathbb{R}^n \to \mathbb{R}</math> || Maxima and minima, Lagrange multipliers, directional derivatives, level sets.

|-

! Vector fields

| 120px || <math>f: \mathbb{R}^m \to \mathbb{R}^n</math> || Any of the operations of vector calculus including gradient, divergence, and curl.

|}

Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.

Multivariate calculus is used in the optimal control of continuous time dynamic systems. It is used in regression analysis to derive formulas for estimating relationships among various sets of empirical data.

Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. In economics, for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate calculus.

Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.

See also

  • List of multivariable calculus topics
  • Multivariate statistics

References

  • MIT video lectures on Multivariable Calculus, Fall 2007
  • Multivariable Calculus: A free online textbook by George Cain and James Herod
  • Multivariable Calculus Online: A free online textbook by Jeff Knisley
  • Multivariable Calculus – A Very Quick Review, Prof. Blair Perot, University of Massachusetts Amherst
  • Multivariable Calculus, Online text by Dr. Jerry Shurman