View source for Multivariable calculus

{{Short description|Calculus of functions of several variables}}
{{Calculus}}


'''Multivariable calculus''' (also known as '''multivariate calculus''') is the extension of [[Calculus|calculus]] in one [[Variable (mathematics)|variable]] to calculus with [[Function of several real variables|functions of several variables]]: the [[Differential calculus|differentiation]] and [[Integral|integration]] of functions involving multiple variables (''multivariate''), rather than just one.<ref name="CourantJohn1999">{{cite book|author1=Richard Courant|author2=Fritz John|title=Introduction to Calculus and Analysis Volume II/2|date=14 December 1999|publisher=Springer Science & Business Media|isbn=978-3-540-66570-0}}</ref> 

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

==Introduction==
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 of a function|domain]] is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:
# There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D;
# There are multiple extended objects associated with the dimension; for example, for a 1D function, it must be represented as a curve on the 2D Cartesian plane, but a function with two variables is a surface in 3D, while curves can also live in 3D space.

The consequence of the first difference is the difference in the definition of the limit and differentiation. Directional limits and [[Directional derivative|derivative]]s 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 integral|line integral]]s, [[Surface integral|surface integral]]s and [[Volume integral|volume integral]]s. Due to the non-uniqueness of these integrals, an [[Antiderivative|antiderivative]] or indefinite integral cannot be properly defined.

== Limits ==
A study of [[Limit of a function|limits]] and [[Continuous function|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 <math>s(t): \mathbb{R} \to \mathbb{R}^n</math> in n-dimensional Euclidean space. Any function <math>f(\overrightarrow{x}): \mathbb{R}^n \to \mathbb{R}^m</math> can then be projected on the path as a 1D function <math>f(s(t))</math>. The limit of <math>f</math> to the point <math>s(t_0)</math> along the path <math>s(t)</math> can hence be defined as

{{NumBlk|:|<math>\lim_{\overrightarrow{x} \to s(t_0)} f(\overrightarrow{x}) = \lim_{t \to t_0} f(s(t))</math>|{{EquationRef|1}}}}

Note that the value of this limit can be dependent on the form of <math>s(t)</math>, i.e. the path chosen, not just the point which the limit approaches.<ref name="CourantJohn1999"/>{{rp|19–22}} For example, consider the function <math>f(x,y) = \frac{x^2y}{x^4+y^2}</math>. If the point <math>(0,0)</math> is approached through the line <math>y=kx</math>, or in parametric form:

[[File:((x^2)(y))⁄((x^4)+(y^2)).png|thumb|Plot of the function {{math|''f''(''x'', ''y'') {{=}} (''x''²y)/(''x''{{sup|4}} + ''y''{{sup|2}})}}]]

{{NumBlk|:|<math>x(t) = t, y(t) = kt</math>|{{EquationRef|2}}}}

Then the limit along the path will be:

{{NumBlk|:|<math>\lim_{t \to 0} f(x(t),y(t)) = \lim_{t \to 0} \frac{k^2 t^3}{k^4 t^4 + t^2} = 0</math>|{{EquationRef|3}}}}

On the other hand, if the path <math>y=\pm x^2</math> (or parametrically, <math>x(t)=t, y(t)=\pm t^2</math>) is chosen, then the limit becomes:

{{NumBlk|:|<math>\lim_{t \to 0} f(x(t),y(t)) = \lim_{t \to 0} \frac{\pm t^4}{t^4 + t^4} = \pm \frac{1}{2}</math>|{{EquationRef|4}}}}

Since taking different paths towards the same point yields different values, a general limit at the point <math>(0,0)</math> 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 <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> that the limit of <math>f</math> to some point <math>x_0 \in \mathbb{R}^n</math> is L, if and only if

{{NumBlk|:|<math>\lim_{t \to t_0} f(s(t)) = L</math>|{{EquationRef|5}}}}

for all continuous functions <math>s(t): \mathbb{R} \to \mathbb{R}^n</math> such that <math>s(t_0)=x_0</math>.

=== Continuity ===
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 <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> that <math>f</math> is continuous at the point <math>x_0</math>, if and only if

{{NumBlk|:|<math>\lim_{t \to t_0} f(s(t)) = f(x_0)</math>|{{EquationRef|5}}}}

for all continuous functions <math>s(t): \mathbb{R} \to \mathbb{R}^n</math> such that <math>s(t_0)=x_0</math>.

As with limits, being continuous along ''one'' path <math>s(t)</math> does not imply multivariate continuity.

Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example.<ref name="CourantJohn1999" />{{rp|17–19}} For example, for a real-valued function <math>f: \mathbb{R}^2 \to \mathbb{R}</math> with two real-valued parameters, <math>f(x,y)</math>, continuity 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>.

Consider
:<math>
f(x,y)=
\begin{cases}
\frac{y}{x}-y & \text{if}\quad 0 \leq y < x \leq 1 \\
\frac{x}{y}-x & \text{if}\quad 0 \leq x < y \leq 1 \\
1-x & \text{if}\quad 0 < 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>(0,1)\times (0,1)</math>. Furthermore, the functions defined for constant <math>x</math> and <math>y</math> and <math>0 \le a \le 1</math> by
:<math>g_a(x) = f(x,a)\quad</math> and <math>\quad h_a(y) = f(a,y)\quad</math>
are continuous. Specifically,
:<math>g_0(x) = f(x,0) = h_0(0,y) = f(0,y) = 0</math> for all {{mvar|x}} and {{mvar|y}}. Therefore, <math>f(0,0)=0</math> and moreover, along the coordinate axes, <math>\lim_{x \to 0} f(x,0) = 0</math> and <math>\lim_{y \to 0} f(0,y) = 0</math>. Therefore the function is continuous along both individual arguments.

However, consider the parametric path <math>x(t) = t, y(t) = t</math>. The parametric function becomes

{{NumBlk|:|<math>
f(x(t),y(t))=
\begin{cases}
1-t & \text{if}\quad t > 0 \\
0 & \text{everywhere else}.
\end{cases}
</math>|{{EquationRef|6}}}}

Therefore,

{{NumBlk|:|<math>\lim_{t \to 0^+} f(x(t),y(t)) = 1 \neq f(0,0) = 0</math>|{{EquationRef|7}}}}

It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates.

===Theorems regarding multivariate limits and continuity ===
* All properties of linearity and superposition from single-variable calculus carry over to multivariate calculus.
* '''Composition''': If <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> and <math>g: \mathbb{R}^m \to \mathbb{R}^p</math> are both multivariate continuous functions at the points <math>x_0 \in \mathbb{R}^n</math> and <math>f(x_0) \in \mathbb{R}^m</math> respectively, then <math>g \circ f: \mathbb{R}^n \to \mathbb{R}^p</math> is also a multivariate continuous function at the point <math>x_0</math>.
* '''Multiplication''': If <math>f: \mathbb{R}^n \to \mathbb{R}</math> and <math>g: \mathbb{R}^n \to \mathbb{R}</math> are both continuous functions at the point <math>x_0 \in \mathbb{R}^n</math>, then <math>fg: \mathbb{R}^n \to \mathbb{R}</math> is continuous at <math>x_0</math>, and <math>\frac{f}{g}: \mathbb{R}^n \to \mathbb{R}</math> is also continuous at <math>x_0</math> provided that <math>g(x_0) \neq 0</math>.
* If <math>f: \mathbb{R}^n \to \mathbb{R}</math> is a continuous function at point <math>x_0 \in \mathbb{R}^n</math>, then <math>|f|</math> is also continuous at the same point.
* If <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> is Lipschitz continuous (with the appropiate normed spaces as needed) in the neighbourhood of the point <math>x_0 \in \mathbb{R}^n</math>, then <math>f</math> is multivariate continuous at <math>x_0</math>.

{{Collapse top|Proof|expand=true}}
From the Lipschitz continuity condition for <math>f</math> we have

{{NumBlk|:|<math>|f(s(t))-f(s(t_0))| \leq K|s(t)-s(t_0)|</math>|{{EquationRef|8}}}}

where <math>K</math> is the Lipschitz constant. Note also that, as <math>s(t)</math> is continuous at <math>t_0</math>, for every <math>\delta > 0</math> there exists a <math>\epsilon > 0</math> such that <math>|s(t)-s(t_0)| < \delta</math> <math>\forall |t-t_0| < \epsilon</math>.

Hence, for every <math>\alpha > 0</math>, choose <math>\delta = \frac{\alpha}{K}</math>; there exists an <math>\epsilon > 0</math> such that for all <math>t</math> satisfying <math>|t-t_0| < \epsilon</math>, <math>|s(t)-s(t_0)| < \delta</math>, and <math>|f(s(t)) - f(s(t_0))| \leq K|s(t)-s(t_0)| < K\delta = \alpha</math>. Hence <math>\lim_{t \to t_0} f(s(t))</math> converges to <math>f(s(t_0))</math> regardless of the precise form of <math>s(t)</math>.

{{Collapse bottom}}

== Differentiation ==
{{main article|Partial derivative|Directional derivative}}

=== Directional derivative ===
The derivative of a single-variable function is defined as

{{NumBlk|:|<math>\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}</math>|{{EquationRef|9}}}}

Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function <math>f: \mathbb{R}^n \to \mathbb{R}</math> along some path <math>s(t): \mathbb{R} \to \mathbb{R}^n</math>:

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0+h))-f(s(t_0))}{|s(t_0+h)-s(t_0)|}</math>|{{EquationRef|10}}}}

Unlike limits, for which the value depends on the exact form of the path <math>s(t)</math>, it can be shown that the derivative along the path depends only on the tangent vector of the path at <math>s(t_0)</math>, i.e. <math>s'(t_0)</math>, provided that <math>f</math> is Lipschitz continuous at <math>s(t_0)</math>, and that the limit exits for at least one such path.
<!-- I am not sure in the slightest I got these conditions right. I will look them up at some point, but in the meantime, if you have a better way to put it, please do. This comment will be removed after the reconstruction is finished.-->

{{collapse top|Proof|expand=true}}
For <math>s(t)</math> continuous up to the first derivative (this statement is well defined as <math>s</math> is a function of one variable), we can write the Taylor expansion of <math>s</math> around <math>t_0</math> using [[Taylor's theorem]] to construct the remainder:

{{NumBlk|:|<math>s(t) = s(t_0) + s'(\tau) (t-t_0) </math>|{{EquationRef|11}}}}

where <math>\tau \in [t_0,t]</math>.

Substituting this into {{EquationNote|10}},

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0)+s'(\tau)h)-f(s(t_0))}{|s'(\tau)h|}</math>|{{EquationRef|12}}}}

where <math>\tau(h) \in [t_0,t_0+h]</math>.

Lipschitz continuity gives us <math>|f(x)-f(y)| \leq K|x-y|</math> for some finite <math>K</math>, <math>\forall x,y\in \mathbb{R}^n</math>. It follows that <math>|f(x+O(h))-f(x)| \sim O(h)</math>.

Note also that given the continuity of <math>s'(t)</math>, <math>s'(\tau) = s'(t_0)+O(h)</math> as <math> h \to 0</math>.

Substituting these two conditions into {{EquationNote|12}},

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0)+s'(t_0)h)-f(s(t_0))+O(h^2)}{|s'(t_0)h|+O(h^2)}</math>|{{EquationRef|13}}}}

whose limit depends only on <math>s'(t_0)</math> as the dominant term.

{{collapse bottom}}

It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function <math>f:\mathbb{R}^n \to \mathbb{R}</math> along the unit vector <math>\hat{\bold{u}}</math> at some point <math>x_0 \in \mathbb{R}^n</math> is

{{NumBlk|:|<math>\nabla_{\hat{\bold{u}}} f(x_0) = \lim_{t \to 0} \frac{f(x_0+\hat{\bold{u}} t) - f(x_0)}{t}</math>|{{EquationRef|14}}}}
<!-- Do limits need normed spaces too, or is it just derivatives? -->

or, when expressed in terms of ordinary differentiation,

{{NumBlk|:|<math>\nabla_{\hat{\bold{u}}} f(x_0) = \left . \frac{df(x_0+\hat{\bold{u}}t)}{dt} \right |_{t=0}</math>|{{EquationRef|15}}}}

which is a well defined expression because <math>f(x_0+\hat{\bold{u}}t)</math> is a scalar function with one variable in <math>t</math>.

It is not possible to define a unique scalar derivative without a direction; it is clear for example that <math>\nabla_{\hat{\bold{u}}}f(x_0) = - \nabla_{-\hat{\bold{u}}}f(x_0)</math>. It is also possible for directional derivatives to exist for some directions but not for others.

=== Partial derivative ===

The partial derivative generalizes the notion of the derivative to higher dimensions.  A partial derivative of a multivariable function is a [[Derivative|derivative]] with respect to one variable with all other variables held constant.<ref name="CourantJohn1999"/>{{rp|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|vector calculus]], the [[Del|del]] operator (<math>\nabla</math>) is used to define the concepts of [[Gradient|gradient]], [[Divergence|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.

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.<ref name="CourantJohn1999"/>{{rp|654ff}}

== Multiple integration ==
{{main article|Multiple integral}}
The multiple integral expands the concept of the [[Integral|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.<ref name="CourantJohn1999"/>{{rp|367ff}}

The [[Surface integral|surface integral]] and the [[Line integral|line integral]] are used to integrate over curved [[Manifold|manifold]]s such as [[Surface (mathematics)|surface]]s and [[Curve|curve]]s.

===Fundamental theorem of calculus in multiple dimensions===
In single-variable calculus, the [[Fundamental theorem of calculus|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:<ref name="CourantJohn1999"/>{{rp|543ff}}
* [[Gradient theorem]]
* [[Physics:Stokes' theorem#Special cases|Stokes' theorem]]
* [[Divergence theorem]]
* [[Green's theorem]].

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 [[Generalized Stokes theorem|Stokes' theorem]], which applies to the integration of differential forms over [[Differentiable manifold|manifolds]].<ref>{{Cite book|url=https://archive.org/details/SpivakM.CalculusOnManifolds_201703|title=Calculus on Manifolds|last=Spivak|first=Michael|publisher=W. A. Benjamin, Inc.|year=1965|isbn=9780805390216|location=New York}}</ref>

==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|line integral]]s, and [[Curvature|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|Area]]s of surfaces, [[Surface integral|surface integral]]s, [[Physics:Flux|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|directional derivative]]s, [[Level set|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|vector calculus]] including [[Gradient|gradient]], [[Divergence|divergence]], and [[Curl (mathematics)|curl]].
|}
Multivariable calculus can be applied to analyze [[Deterministic system|deterministic system]]s that have multiple [[Chemistry:Degrees of freedom (physics and chemistry)|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|system dynamics]].

Multivariate calculus is used in the [[Optimal control|optimal control]] of continuous time dynamic systems. It is used in [[Regression analysis|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:Social science|social science]] and [[Engineering:Engineering|engineering]] to model and study high-dimensional systems that exhibit deterministic behavior.  In [[Finance:Economics|economics]], for example, [[Philosophy:Consumer choice|consumer choice]] over a variety of goods, and [[Finance: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|stochastic calculus]].

== See also ==
* [[List of multivariable calculus topics]]
* [[Multivariate statistics]]

==References==
{{reflist}}

==External links==
* [https://www.youtube.com/watch?v=cw6pHhjhKmk UC Berkeley video lectures on Multivariable Calculus, Fall 2009, Professor Edward Frenkel]
* [https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38 MIT video lectures on Multivariable Calculus, Fall 2007]
* [http://www.math.gatech.edu/~cain/notes/calculus.html ''Multivariable Calculus'']: A free online textbook by George Cain and James Herod
* [http://math.etsu.edu/Multicalc/ ''Multivariable Calculus Online'']: A free online textbook by Jeff Knisley
* [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
* [http://www.stat.rice.edu/~dobelman/notes_papers/math/calculus.MV.pdf ''Multivariable Calculus''], Online text by Dr. Jerry Shurman

{{Industrial and applied mathematics}}

[[Category:Multivariable calculus| ]]

{{Sourceattribution|Multivariable calculus}}