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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 .
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:
The consequence of the first difference is the difference in the definition of the limit and differentiation. 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.
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 in n-dimensional Euclidean space. Any function can then be projected on the path as a 1D function . The limit of to the point along the path can hence be defined as
(1) |
Note that the value of this limit can be dependent on the form of , i.e. the path chosen, not just the point which the limit approaches. [1] : 19–22 For example, consider the function
If the point is approached through the line , or in parametric form:
(2) |
Then the limit along the path will be:
(3) |
On the other hand, if the path (or parametrically, ) is chosen, then the limit becomes:
(4) |
Since taking different paths towards the same point yields different values, a general limit at the point cannot be defined for the function.
A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a function that the limit of to some point is L, if and only if
(5) |
for all continuous functions such that .
From the concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a function that is continuous at the point , if and only if
(5) |
for all continuous functions such that .
As with limits, being continuous along one path does not imply multivariate continuity.
Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example. [1] : 17–19 For example, for a real-valued function with two real-valued parameters, , continuity of in for fixed and continuity of in for fixed does not imply continuity of .
Consider
It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle . Furthermore, the functions defined for constant and and by
are continuous. Specifically,
However, consider the parametric path . The parametric function becomes
(6) |
Therefore,
(7) |
It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates.
Proof | |||
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From the Lipschitz continuity condition for we have
where is the Lipschitz constant. Note also that, as is continuous at , for every there exists a such that . Hence, for every , choose ; there exists an such that for all satisfying , , and . Hence converges to regardless of the precise form of . |
The derivative of a single-variable function is defined as
(9) |
Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function along some path :
(10) |
Unlike limits, for which the value depends on the exact form of the path , it can be shown that the derivative along the path depends only on the tangent vector of the path at , i.e. , provided that is Lipschitz continuous at , and that the limit exits for at least one such path.
Proof | |||||||||
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For continuous up to the first derivative (this statement is well defined as is a function of one variable), we can write the Taylor expansion of around using Taylor's theorem to construct the remainder:
where . Substituting this into 10 ,
where . Lipschitz continuity gives us for some finite , . It follows that . Note also that given the continuity of , as . Substituting these two conditions into 12 ,
whose limit depends only on as the dominant term. |
It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function along the unit vector at some point is
(14) |
or, when expressed in terms of ordinary differentiation,
(15) |
which is a well defined expression because is a scalar function with one variable in .
It is not possible to define a unique scalar derivative without a direction; it is clear for example that . It is also possible for directional derivatives to exist for some directions but not for others.
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
A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.
Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator () 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
The multiple integral extends 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.
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]
Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,
Type of functions | Applicable techniques | ||
---|---|---|---|
Curves | ![]() | for | Lengths of curves, line integrals, and curvature. |
Surfaces | ![]() | for | Areas of surfaces, surface integrals, flux through surfaces, and curvature. |
Scalar fields | ![]() | Maxima and minima, Lagrange multipliers, directional derivatives, level sets. | |
Vector fields | ![]() | 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.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line:
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function.
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth and does not contain any break, angle, or cusp.
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion. It has important applications in mathematical finance and stochastic differential equations.
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space rather than just the real line.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. Roughly speaking, the two operations can be thought of as inverses of each other.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article.
Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.
In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.