Multivariable calculus

2025-09-07T21:41:51Z

2001:4C4E:11C9:1800:9C8C:EB35:E5BA:2ED9: /* Directional derivative */

{{Short description|Calculus of functions of several variables}}
{{One source|date=October 2015}}
{{Calculus}}

'''Multivariable calculus''' (also known as '''multivariate calculus''') is the extension of [[calculus]] in one [[Variable (mathematics)|variable]] to [[Function of several real variables|functions of several variables]]: the [[Differential calculus|differentiation]] and [[integral|integration]] of functions involving multiple variables (''[[multivariate (mathematics)|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 [[calculus on Euclidean space]]. The special case of calculus in three dimensional space is often called ''[[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, a 1D function is represented as a curve on the 2D [[Cartesian plane]], but a [[Scalar (mathematics)|scalar]]-valued function of 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 limits and continuity. Directional [[Limit of a function|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]]s, [[surface integral]]s and [[volume integral]]s. Due to the non-uniqueness of these integrals, an [[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 t^3}{t^4 + k^2 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>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 appropriate 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 exists for at least one such path.


{{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 generalize 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}}}}


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 ===
{{Main article|Partial derivative}}
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.<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]], 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.

[[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 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.<ref name="CourantJohn1999"/>{{rp|367ff}}

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

===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:<ref name="CourantJohn1999"/>{{rp|543ff}}
* [[Gradient theorem]]
* [[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 Stokes theorem|generalized 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]]s, and [[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]]s of surfaces, [[surface integral]]s, [[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]]s, [[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]] including [[gradient]], [[divergence]], and [[Curl (mathematics)|curl]].
|}
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]].

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 science|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 [[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]].

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

==References==
{{reflist}}

==External links==
{{Commons category|Multivariate calculus}}
* [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| ]]

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Multivariable calculus