Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable: the differentiation and integration of functions involving multiple variables, rather than just one.
Typical operations
Limits and continuity
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by singlevariable functions. For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola. For example, the function

f(x,y) = \frac{x^2y}{x^4+y^2}
approaches zero along any line through the origin. However, when the origin is approached along a parabola y=x^2, it has a limit of 0.5. Since taking different paths toward the same point yields different values for the limit, the limit does not exist.
Continuity in each argument is not sufficient for multivariate continuity: For instance, in the case of a realvalued function with two realvalued parameters, f(x,y), continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f. As an example, consider

f(x,y)= \begin{cases} \frac{y}{x}y & \text{if } 1 \geq x > y \geq 0 \\ \frac{x}{y}x & \text{if } 1 \geq y > x \geq 0 \\ 1x & \text{if } x=y>0 \\ 0 & \text{else}. \end{cases}
It is easy to check that all realvalued functions (with one realvalued argument) that are given by f_y(x):= f(x,y) are continuous in x (for any fixed y). Similarly, all f_x are continuous as f is symmetric with regards to x and y. However, f itself is not continuous as can be seen by considering the sequence f(\frac{1}{n},\frac{1}{n}) (for natural n) which should converge to f(0,0)=0 if f was continuous. However, \lim_{n \to \infty} f(\frac{1}{n},\frac{1}{n}) = 1. Thus, the limit does not exist.
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.
Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator (\nabla) 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.
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.
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 singlevariable 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 famous integral theorems of vector calculus:
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.
Applications and uses
Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,
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.
Multivariable calculus is used in many fields of natural and social science and engineering to model and study highdimensional systems that exhibit deterministic behavior. Nondeterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus. Quantitative analysts in finance also often use multivariate calculus to predict future trends in the stock market.
See also
External links

UC Berkeley video lectures on Multivariable Calculus, Fall 2009, Professor Edward Frenkel

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
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