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| {{Calculus}}
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| '''Multivariable calculus''' (also known as '''multivariate calculus''') is the extension of [[calculus]] in one [[Variable (mathematics)|variable]] to calculus in more than one variable: the [[Differentiation (mathematics)|differentiation]] and [[integral|integration]] of functions involving multiple variables, rather than just one.
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| == Typical operations==
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| ===Limits and continuity===
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| A study of [[limit of a function|limits]] and [[continuous function|continuity]] in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable 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
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| :<math>f(x,y) = \frac{x^2y}{x^4+y^2}</math> | |
| approaches zero along any line through the origin. However, when the origin is approached along a parabola <math>y=x^2</math>, 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.
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| '''Continuity in each argument is not sufficient for multivariate continuity:'''
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| For instance, in the case of a real-valued function with two real-valued parameters, <math>f(x,y)</math>, continuity
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| of <math>f</math> in <math>x</math> for fixed <math>y</math> and continuity of <math>f</math> in <math>y</math> for fixed <math>x</math> does not imply continuity of <math>f</math>. As an example, consider
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| :<math>
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| f(x,y)=
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| \begin{cases}
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| \frac{y}{x}-y & \text{if } 1 \geq x > y \geq 0 \\
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| \frac{x}{y}-x & \text{if } 1 \geq y > x \geq 0 \\
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| 1-x & \text{if } x=y>0 \\
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| 0 & \text{else}.
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| \end{cases}
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| </math>
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| It is easy to check that all real-valued functions (with one real-valued argument) that are
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| given by <math>f_y(x):= f(x,y)</math> are continuous in <math>x</math> (for any fixed <math>y</math>). Similarly, all <math>f_x</math>
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| are continuous as <math>f</math> is symmetric with regards to <math>x</math> and <math>y</math>. However, <math>f</math> itself is not continuous as can be seen by
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| considering the sequence <math>f(\frac{1}{n},\frac{1}{n})</math> (for natural <math>n</math>) which should converge to <math>f(0,0)=0</math> if <math>f</math> was continuous. However,
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| <math>\lim_{n \to \infty} f(\frac{1}{n},\frac{1}{n}) = 1.</math> Thus, the limit does not exist.
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| ===Partial differentiation===
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| {{main|Partial derivative}}
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| 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.
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| Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In [[vector calculus]], the [[del]] operator (<math>\nabla</math>) is used to define the concepts of [[gradient]], [[divergence]], and [[Curl (mathematics)|curl]] in terms of partial derivatives. A matrix of partial derivatives, the '''[[Jacobian matrix and determinant|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.
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| [[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.
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| ===Multiple integration===
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| {{main|Multiple integral}}
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| 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.
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| The [[surface integral]] and the [[line integral]] are used to integrate over curved [[manifolds]] such as [[surfaces]] and [[curves]].
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| ===Fundamental theorem of calculus in multiple dimensions===
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| 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 famous integral theorems of vector calculus:
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| * [[Gradient theorem]]
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| * [[Stokes' theorem#Special cases|Stokes' theorem]]
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| * [[Divergence theorem]]
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| * [[Green's theorem]].
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| 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 [[Differentiable manifold|manifolds]].
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| ==Applications and uses==
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| Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,
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| {| class="wikitable" style="text-align:center"
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| |-
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| ! !! !! Domain/Codomain !! Applicable techniques
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| |-
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| ! [[Curve]]s
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| | [[Image:Osculating circle.svg|120px]] || <math>f: \mathbb{R} \to \mathbb{R}^n</math> || Lengths of curves, [[line integral]]s, and [[curvature]].
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| |-
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| ! [[Surfaces]]
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| | [[Image:Helicoid.PNG|120px]] || <math>f: \mathbb{R}^{2} \to \mathbb{R}^n</math> || [[Area]]s of surfaces, [[surface integral]]s, [[flux]] through surfaces, and curvature.
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| |-
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| ! [[Scalar fields]]
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| | [[Image:Surface-plot.png|120px]] || <math>f: \mathbb{R}^n \to \mathbb{R}</math> || Maxima and minima, [[Lagrange multipliers]], [[directional derivative]]s.
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| |-
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| ! [[Vector fields]]
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| | [[Image: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]].
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| |}
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| 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]].
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| 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. Non-deterministic, or [[stochastic process|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.
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| ==See also==
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| * [[List of multivariable calculus topics]]
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| * [[Multivariate statistics]]
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| ==External links==
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| * [http://www.youtube.com/watch?v=cw6pHhjhKmk&feature=list_related&playnext=1&list=SP07CF868151394FE3 UC Berkeley video lectures on Multivariable Calculus, Fall 2009, Professor Edward Frenkel]
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| * [http://www.youtube.com/user/MIT#g/c/4C4C8A7D06566F38 MIT video lectures on Multivariable Calculus, Fall 2007]
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| * [http://www.math.gatech.edu/~cain/notes/calculus.html ''Multivariable Calculus'']: A free online textbook by George Cain and James Herod
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| * [http://math.etsu.edu/Multicalc/ ''Multivariable Calculus Online'']: A free online textbook by Jeff Knisley
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| * [http://www.ecs.umass.edu/mie/faculty/perot/mie440/Multivariable%20Calculus.pdf ''Multivariable Calculus – A Very Quick Review''], Prof Blair Perot, University of Massachusetts Amherst
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| [[Category:Multivariable calculus|*]]
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