Multivariable calculus

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Short description: Calculus of functions of several independent variables

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.[1]

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

Typical operations

Limits and continuity

A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.[1]:19–22 For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function.

[math]\displaystyle{ f(x,y) = \frac{x^2y}{x^4+y^2} }[/math]
Plot of the function f(x, y) = (x²y)/(x4 + y2)

approaches zero whenever the point [math]\displaystyle{ (0,0) }[/math] is approached along lines through the origin ([math]\displaystyle{ y=kx }[/math]). However, when the origin is approached along a parabola [math]\displaystyle{ y=\pm x^2 }[/math], the function value has a limit of [math]\displaystyle{ \pm 1/2 }[/math]. Since taking different paths toward the same point yields different limit values, a general limit does not exist there.

Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example.[1]:17–19 In particular, for a real-valued function with two real-valued parameters, [math]\displaystyle{ f(x,y) }[/math], continuity of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ x }[/math] for fixed [math]\displaystyle{ y }[/math] and continuity of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ y }[/math] for fixed [math]\displaystyle{ x }[/math] does not imply continuity of [math]\displaystyle{ f }[/math].


[math]\displaystyle{ f(x,y)= \begin{cases} \frac{y}{x}-y & \text{if}\quad 0 \leq y \lt x \leq 1 \\ \frac{x}{y}-x & \text{if}\quad 0 \leq x \lt y \leq 1 \\ 1-x & \text{if}\quad 0 \lt x=y \\ 0 & \text{everywhere else}. \end{cases} }[/math]

It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle [math]\displaystyle{ (0,1)\times (0,1) }[/math]. Furthermore, the functions defined for constant [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] and [math]\displaystyle{ 0 \le a \le 1 }[/math] by

[math]\displaystyle{ g_a(x) = f(x,a)\quad }[/math] and [math]\displaystyle{ \quad h_a(y) = f(a,y)\quad }[/math]

are continuous. Specifically,

[math]\displaystyle{ g_0(x) = f(x,0) = h_0(0,y) = f(0,y) = 0 }[/math] for all x and y.

However, the sequence [math]\displaystyle{ f \left(\tfrac{1}{n},\tfrac{1}{n}\right) }[/math] (for natural [math]\displaystyle{ n }[/math]) converges to [math]\displaystyle{ \lim_{n\to\infty}f \left(\tfrac{1}{n},\tfrac{1}{n}\right) = 1 }[/math], rendering the function as discontinuous at [math]\displaystyle{ (0,0) }[/math]. Approaching the origin not along parallels to the [math]\displaystyle{ x }[/math]- and [math]\displaystyle{ y }[/math]-axis reveals this discontinuity.

Continuity of function composition

If [math]\displaystyle{ f(x,y) }[/math] is continuous at [math]\displaystyle{ (a,b), }[/math] and [math]\displaystyle{ g }[/math] is a single variable function continuous at [math]\displaystyle{ f(a,b), }[/math] then the composite function [math]\displaystyle{ h=g\circ f }[/math] defined by [math]\displaystyle{ h(x,y)=g(f(x,y)) }[/math] is continuous at [math]\displaystyle{ (a,b). }[/math]

For examples, [math]\displaystyle{ \exp(x-y) }[/math] and [math]\displaystyle{ \ln(1+xy-4x+10y). }[/math]

Properties of continuous functions

If [math]\displaystyle{ f(x,y) }[/math] and [math]\displaystyle{ g(x,y) }[/math] are both continuous at [math]\displaystyle{ (a,b) }[/math] then

(i) [math]\displaystyle{ f(x,y) \pm g(x,y) }[/math] are continuous at [math]\displaystyle{ (a,b). }[/math]

(ii) [math]\displaystyle{ cf(x,y) }[/math] is continuous at [math]\displaystyle{ (a,b) }[/math] for any constant c.

(iii) [math]\displaystyle{ f(x,y) }[/math] [math]\displaystyle{ . }[/math] [math]\displaystyle{ g(x,y) }[/math] is continuous at point [math]\displaystyle{ (a,b). }[/math]

(iv) [math]\displaystyle{ \frac{f(x,y)}{g(x,y)} }[/math] is continuous at [math]\displaystyle{ (a,b), }[/math] if [math]\displaystyle{ g(a,b)\ne 0. }[/math]

(v) [math]\displaystyle{ \mid f(x,y) \mid }[/math] is continuous at [math]\displaystyle{ (a,b). }[/math]

Partial differentiation

The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.[1]:26ff

Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator ([math]\displaystyle{ \nabla }[/math]) is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation which directly varies from point to point in the domain of the function.

Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable.[1]:654ff

Multiple integration

The multiple integral expands the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration.[1]:367ff

The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves.

Fundamental theorem of calculus in multiple dimensions

In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:[1]:543ff

In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.[2]

Applications and uses

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

Type of functions Applicable techniques
Curves Osculating circle.svg [math]\displaystyle{ f: \mathbb{R} \to \mathbb{R}^n }[/math]
for [math]\displaystyle{ n \gt 1 }[/math]
Lengths of curves, line integrals, and curvature.
Surfaces Helicoid.svg [math]\displaystyle{ f: \mathbb{R}^2 \to \mathbb{R}^n }[/math]
for [math]\displaystyle{ n \gt 2 }[/math]
Areas of surfaces, surface integrals, flux through surfaces, and curvature.
Scalar fields Surface-plot.png [math]\displaystyle{ f: \mathbb{R}^n \to \mathbb{R} }[/math] Maxima and minima, Lagrange multipliers, directional derivatives, level sets.
Vector fields Vector field.svg [math]\displaystyle{ 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


  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Richard Courant; Fritz John (14 December 1999). Introduction to Calculus and Analysis Volume II/2. Springer Science & Business Media. ISBN 978-3-540-66570-0. 
  2. Spivak, Michael (1965). Calculus on Manifolds. New York: W. A. Benjamin, Inc.. ISBN 9780805390216. 

External links