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==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
|-
! [[Curve]]s
| [[File:Osculating circle.svg|120px]] || <math>f: \mathbb{R} \to \mathbb{R}^n</math> <br> for <math>n > 1</math> || Lengths of curves, [[line integral]]s, and [[curvature]].
|-
! [[Surface (mathematics)|Surface]]s
| [[Image:Helicoid.svg|120px]] || <math>f: \mathbb{R}^2 \to \mathbb{R}^n</math> <br> for <math>n > 2</math> || [[Area]]s of surfaces, [[surface integral]]s, [[flux]] through surfaces, and curvature.
|-
! [[Scalar fields]]
| [[Image:Surface-plot.png|120px]] || <math>f: \mathbb{R}^n \to \mathbb{R}</math> || Maxima and minima, [[Lagrange multipliers]], [[directional derivative]]s, [[level set]]s.
|-
! [[Vector fields]]
| [[File:Vector field.svg|120px]] || <math>f: \mathbb{R}^m \to \mathbb{R}^n</math> || Any of the operations of [[vector calculus]] including [[gradient]], [[divergence]], and [[Curl (mathematics)|curl]].
|}
Multivariable calculus can be applied to analyze [[deterministic system]]s that have multiple [[degrees of freedom (physics and chemistry)|degrees of freedom]].  Functions with [[independent variable]]s 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 science|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 [[profit maximization|producer choice]] over various inputs to use and outputs to produce, are modeled with multivariate calculus. 

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