View source for Vector calculus

{{short description|Calculus of vector-valued functions}}
{{distinguish|Geometric calculus|Matrix calculus}}
{{More footnotes|date=February 2016}}
{{Calculus}}

'''Vector calculus''' or '''vector analysis''' is a branch of mathematics concerned with the [[derivative|differentiation]] and [[integral|integration]] of [[vector field]]s, primarily in three-dimensional [[Euclidean space]], <math>\mathbb{R}^3.</math><ref>{{Cite book |last1=Kreyszig |first1=Erwin |title=Advanced Engineering Mathematics |last2=Kreyszig |first2=Herbert |last3=Norminton |first3=E. J. |date=2011 |publisher=John Wiley |isbn=978-0-470-45836-5 |edition=10th |location=Hoboken, NJ}}</ref> The term ''vector calculus'' is sometimes used as a synonym for the broader subject of [[multivariable calculus]], which spans vector calculus as well as [[partial derivative|partial differentiation]] and [[multiple integral|multiple integration]]. Vector calculus plays an important role in [[differential geometry]] and in the study of [[partial differential equation]]s. It is used extensively in physics and engineering, especially in the description of [[electromagnetic field]]s, [[gravitational field]]s, and [[fluid flow]].

Vector calculus was developed from the theory of [[quaternion]]s by [[J. Willard Gibbs]] and [[Oliver Heaviside]] near the end of the 19th century, and most of the notation and terminology was established by Gibbs and [[Edwin Bidwell Wilson]] in their 1901 book, ''[[Vector Analysis]]'', though earlier mathematicians such as [[Isaac Newton]] pioneered the field.<ref name=":17">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA26 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=26, 82–83 |language=en |doi=10.1142/q0108}}</ref> In its standard form using the [[cross product]], vector calculus does not generalize to higher dimensions, but the alternative approach of [[geometric algebra]], which uses the [[Exterior algebra|exterior product]], does (see ''{{Section link|#Generalizations}}'' below for more).

== Basic objects ==

=== Scalar fields ===
{{Main|Scalar field}}
A [[scalar field]] associates a [[scalar (mathematics)|scalar]] value to every point in a space. The scalar is a mathematical number representing a [[scalar (physics)|physical quantity]]. Examples of scalar fields in applications include the [[temperature]] distribution throughout space, the [[pressure]] distribution in a fluid, and spin-zero quantum fields (known as [[Scalar boson|scalar bosons]]), such as the [[Higgs field]]. These fields are the subject of [[scalar field theory]].

=== Vector fields ===
{{Main|Vector field}}

A [[vector field]] is an assignment of a [[vector (geometry)|vector]] to each point in a [[Space (mathematics)|space]].<ref name="Galbis-2012-p12">{{cite book|author1=Galbis, Antonio |author2=Maestre, Manuel |title=Vector Analysis Versus Vector Calculus|publisher=Springer|year=2012|isbn=978-1-4614-2199-3|page=12|url=https://books.google.com/books?id=tdF8uTn2cnMC&pg=PA12}}</ref> A vector field in the plane, for instance, can be visualized as a collection of arrows with a given [[Magnitude_(mathematics)#Vector_spaces|magnitude]] and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some [[force]], such as the [[magnetic field|magnetic]] or [[gravity|gravitational]] force, as it changes from point to point. This can be used, for example, to calculate [[Work (physics)|work]] done over a line.

=== Vectors and pseudovectors ===
In more advanced treatments, one further distinguishes [[pseudovector]] fields and [[pseudoscalar]] fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, the [[Curl (mathematics)|curl]] of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in [[geometric algebra]], as described below.

== Vector algebra ==
{{main|Euclidean vector#Basic properties}}
The algebraic (non-differential) operations in vector calculus are referred to as ''vector algebra'', being defined for a vector space and then applied [[pointwise]] to a vector field. The basic algebraic operations consist of:

{| class="wikitable" style="text-align:center"
|+Notations in vector calculus
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
|-
![[Vector addition]]
|<math>\mathbf{v}_1 + \mathbf{v}_2</math>
|Addition of two vectors, yielding a vector.
|-
!scope="row"|[[Scalar multiplication]]
|<math>a \mathbf{v}</math>
|Multiplication of a scalar and a vector, yielding a vector.
|-
!scope="row"|[[Dot product]]
|<math>\mathbf{v}_1 \cdot \mathbf{v}_2</math>
|Multiplication of two vectors, yielding a scalar.
|-
!scope="row"|[[Cross product]]
|<math>\mathbf{v}_1 \times \mathbf{v}_2</math>
|Multiplication of two vectors in <math>\mathbb R^3</math>, yielding a (pseudo)vector.
|}

Also commonly used are the two [[triple product]]s:
{| class="wikitable" style="text-align:center"
|+Vector calculus triple products
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
|-
!scope="row"|[[Scalar triple product]]
|<math>\mathbf{v}_1\cdot\left( \mathbf{v}_2\times\mathbf{v}_3 \right)</math>
|The dot product of the cross product of two vectors.
|-
!scope="row"|[[Vector triple product]]
|<math>\mathbf{v}_1\times\left( \mathbf{v}_2\times\mathbf{v}_3 \right)</math>
|The cross product of the cross product of two vectors.
|}

== Operators and theorems ==
{{main|Vector calculus identities}}
=== Differential operators ===
{{main|Gradient|Divergence|Curl (mathematics)|Laplacian}}
Vector calculus studies various [[differential operator]]s defined on scalar or vector fields, which are typically expressed in terms of the [[del]] operator (<math>\nabla</math>), also known as "nabla". The three basic [[vector operator]]s are:<ref>{{Cite web|title=Differential Operators|url=http://192.168.1.121/math2/differential-operators/|access-date=2020-09-17|website=Math24|language=en-US}}{{Dead link|date=July 2025 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
{| class="wikitable" style="text-align:center"
|+Differential operators in vector calculus
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
!scope="col"|[[Notation_for_differentiation#Notation_in_vector_calculus|Notational<br/>analogy]]
!scope="col"|Domain/Range
|-
!scope="row"|[[Gradient]]
|<math>\operatorname{grad}(f)=\nabla f</math>
|Measures the rate and direction of change in a scalar field.
|[[Scalar multiplication]]
|Maps scalar fields to vector fields.
|-
!scope="row"|[[Divergence]]
|<math>\operatorname{div}(\mathbf{F})=\nabla\cdot\mathbf{F}</math>
|Measures the scalar of a source or sink at a given point in a vector field.
|[[Dot product]]
|Maps vector fields to scalar fields.
|-
!scope="row"|[[Curl (mathematics)|Curl]]
|<math>\operatorname{curl}(\mathbf{F})=\nabla\times\mathbf{F}</math>
|Measures the tendency to rotate about a point in a vector field in <math>\mathbb R^3</math>.
|[[Cross product]]
|Maps vector fields to (pseudo)vector fields.
|-
!scope="row" colspan=5|{{mvar|f}} denotes a scalar field and {{mvar|F}} denotes a vector field
|}

Also commonly used are the two Laplace operators:

{| class="wikitable" style="text-align:center"
|+Laplace operators in vector calculus
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
!scope="col"|Domain/Range
|-
!scope="row"|[[Laplace operator|Laplacian]]
|<math>\Delta f=\nabla^2 f=\nabla\cdot \nabla f</math>
|Measures the difference between the value of the scalar field with its average on infinitesimal balls.
|Maps between scalar fields.
|-
!scope="row"|[[Vector Laplacian]]
|<math>\nabla^2\mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla \times (\nabla \times \mathbf{F})</math>
|Measures the difference between the value of the vector field with its average on infinitesimal balls.
|Maps between vector fields.
|-
!scope="row" colspan=4|{{mvar|f}} denotes a scalar field and {{mvar|F}} denotes a vector field
|}

A quantity called the [[Jacobian matrix and determinant|Jacobian matrix]] is useful for studying functions when both the domain and range of the function are multivariable, such as a [[change of variables]] during integration.

=== Integral theorems ===
The three basic vector operators have corresponding theorems which generalize the [[fundamental theorem of calculus]] to higher dimensions:

{| class="wikitable" style="text-align:center"
|+Integral theorems of vector calculus
|-
!scope="col"| Theorem
!scope="col"| Statement
!scope="col"| Description
|-
!scope="row"| [[Gradient theorem]]
| <math> \int_{L \subset \mathbb R^n}\!\!\! \nabla\varphi\cdot d\mathbf{r} \ =\ \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)\ \  \text{ for }\ \ L = L[p\to q]   </math>
| The [[line integral]] of the gradient of a scalar field over a [[curve]] {{math|''L''}} is equal to the change in the scalar field between the endpoints {{math|''p''}} and {{math|''q''}} of the curve.
|-
!scope="row"| [[Divergence theorem]]
| <math> \underbrace{ \int \!\cdots\! \int_{V \subset \mathbb R^n} }_n (\nabla \cdot \mathbf{F}) \, dV 
\  = \ \underbrace{ \oint \!\cdots\! \oint_{\partial V} }_{n-1} \mathbf{F} \cdot d \mathbf{S} </math>
| The integral of the divergence of a vector field over an {{mvar|n}}-dimensional solid {{math|''V''}} is equal to the [[flux]] of the vector field through the {{math|(''n''−1)}}-dimensional closed boundary surface of the solid.
|-
!scope="row"| [[Kelvin–Stokes theorem|Curl (Kelvin–Stokes) theorem]]
| <math> \iint_{\Sigma\subset\mathbb R^3} (\nabla \times \mathbf{F}) \cdot d\mathbf{\Sigma} \ =\ \oint_{\partial \Sigma} \mathbf{F} \cdot d \mathbf{r} </math>
| The integral of the curl of a vector field over a [[Surface (topology)|surface]] {{math|Σ}} in <math>\mathbb R^3</math> is equal to the circulation of the vector field around the closed curve bounding the surface.
|-
!scope="row" colspan=5|<math>\varphi</math> denotes a scalar field and {{mvar|F}} denotes a vector field
|}

In two dimensions, the divergence and curl theorems reduce to the Green's theorem:

{| class="wikitable" style="text-align:center"
|+Green's theorem of vector calculus
|-
! scope="col"| Theorem
! scope="col"| Statement
! scope="col"| Description
|-
!scope="row"| [[Green's theorem]]
| <math> \iint_{A\,\subset\mathbb R^2} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA \ =\ \oint_{\partial A} \left ( L\, dx + M\, dy \right ) </math>|| The integral of the divergence (or curl) of a vector field over some region {{math|''A''}} in <math>\mathbb R^2</math> equals the flux (or circulation) of the vector field over the closed curve bounding the region.
|-
!scope="row" colspan=5|For divergence, {{math|1=''F'' = (''M'', −''L'')}}. For curl, {{math|1=''F'' = (''L'', ''M'', 0)}}. {{mvar|L}} and {{mvar|M}} are functions of {{math|(''x'', ''y'')}}.
|}

== Applications ==

=== Linear approximations ===
{{main|Linear approximation}}
Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function {{math|''f''(''x'', ''y'')}} with real values, one can approximate {{math|''f''(''x'', ''y'')}} for {{math|(''x'', ''y'')}} close to {{math|(''a'', ''b'')}} by the formula
: <math>f(x,y)\ \approx\ f(a,b)+\tfrac{\partial f}{\partial x} (a,b)\,(x-a)+\tfrac{\partial f}{\partial y}(a,b)\,(y-b).</math>

The right-hand side is the equation of the plane tangent to the graph of {{math|1=''z'' = ''f''(''x'', ''y'')}} at {{nowrap|{{math|(''a'', ''b'')}}.}}

=== Optimization ===
{{main|Mathematical optimization}}
For a continuously differentiable [[function of several real variables]], a point {{math|''P''}} (that is, a set of values for the input variables, which is viewed as a point in {{math|'''R'''<sup>''n''</sup>}}) is '''critical''' if all of the [[partial derivative]]s of the function are zero at {{math|''P''}}, or, equivalently, if its [[gradient]] is zero. The critical values are the values of the function at the critical points.

If the function is [[smooth function|smooth]], or, at least twice continuously differentiable, a critical point may be either a [[local maximum]], a [[local minimum]] or a [[saddle point]]. The different cases may be distinguished by considering the [[eigenvalue]]s of the [[Hessian matrix]] of second derivatives.

By [[Fermat's theorem (stationary points)|Fermat's theorem]], all local [[maxima and minima]] of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

== Generalizations ==
{{Unreferenced section|date=August 2019}}
Vector calculus can also be generalized to other [[3-manifolds]] and [[higher dimension|higher-dimensional]] spaces.

=== Different 3-manifolds ===
Vector calculus is initially defined for [[Euclidean space|Euclidean 3-space]], <math>\mathbb{R}^3,</math> which has additional structure beyond simply being a 3-dimensional real vector space, namely: a [[norm (mathematics)|norm]] (giving a notion of length) defined via an [[inner product]] (the [[dot product]]), which in turn gives a notion of angle, and an [[orientability|orientation]], which gives a notion of left-handed and right-handed. These structures give rise to a [[volume form]], and also the [[cross product]], which is used pervasively in vector calculus.

The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the [[coordinate system]] to be taken into account.

Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric [[nondegenerate form]]) and an orientation; this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the [[special orthogonal group]] {{math|SO(3)}}).

More generally, vector calculus can be defined on any 3-dimensional oriented [[Riemannian manifold]], or more generally [[pseudo-Riemannian manifold]]. This structure simply means that the [[tangent space]] at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate [[metric tensor]] and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.

=== Other dimensions ===
Most of the analytic results are easily understood, in a more general form, using the machinery of [[differential geometry]], of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding [[harmonic analysis]]), while curl and cross product do not generalize as directly.

From a general point of view, the various fields in (3-dimensional) vector calculus are uniformly seen as being {{math|''k''}}-vector fields: scalar fields are 0-vector fields, vector fields are 1-vector fields, pseudovector fields are 2-vector fields, and pseudoscalar fields are 3-vector fields. In higher dimensions there are additional types of fields (scalar, vector, pseudovector or pseudoscalar corresponding to {{math|0}}, {{math|1}}, {{math|''n'' &minus; 1}} or {{math|''n''}} dimensions, which is exhaustive in dimension 3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.

In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7<ref>Lizhong Peng & Lei Yang (1999) "The curl in seven dimensional space and its applications", ''Approximation Theory and Its Applications'' 15(3): 66 to 80 {{doi|10.1007/BF02837124}}</ref> (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or [[seven-dimensional cross product|7]] dimensions can a cross product be defined (generalizations in other dimensionalities either require <math>n-1</math> vectors to yield 1 vector, or are alternative [[Lie algebra]]s, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at ''[[Curl (mathematics)#Generalizations|Curl § Generalizations]]''; in brief, the curl of a vector field is a [[bivector]] field, which may be interpreted as the [[special orthogonal Lie algebra]] of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ – there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally <math>\textstyle{\binom{n}{2}=\frac{1}{2}n(n-1)}</math> dimensions of rotations in {{math|''n''}} dimensions).

There are two important alternative generalizations of vector calculus. The first, [[geometric algebra]], uses [[multivector|{{math|''k''}}-vector]] fields instead of vector fields (in 3 or fewer dimensions, every {{math|''k''}}-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the [[exterior product]], which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yields [[Clifford algebra]]s as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.

The second generalization uses [[differential form]]s ({{math|''k''}}-covector fields) instead of vector fields or {{math|''k''}}-vector fields, and is widely used in mathematics, particularly in [[differential geometry]], [[geometric topology]], and [[harmonic analysis]], in particular yielding [[Hodge theory]] on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to the [[exterior derivative]] of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of [[Stokes' theorem]].

From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear.
From the point of view of geometric algebra, vector calculus implicitly identifies {{math|''k''}}-vector fields with vector fields or scalar functions: 0-vectors and 3-vectors with scalars, 1-vectors and 2-vectors with vectors. From the point of view of differential forms, vector calculus implicitly identifies {{math|''k''}}-forms with scalar fields or vector fields: 0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturally takes as input a vector field or 1-form, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.

== See also ==
{{Portal|Mathematics}}
{{div col|colwidth=20em}}
* [[Vector calculus identities]]
* [[Vector algebra relations]]
* [[Directional derivative]]
* [[Conservative vector field]]
* [[Solenoidal vector field]]
* [[Laplacian vector field]]
* [[Helmholtz decomposition]]
* [[Tensor]]
* [[Geometric calculus]]
{{div col end}}

== References ==
=== Citations ===
{{reflist}}

=== Sources ===
{{refbegin}}
* Sandro Caparrini (2002) "[https://link.springer.com/article/10.1007%2Fs004070200001?LI=true The discovery of the vector representation of moments and angular velocity]", Archive for History of Exact Sciences 56:151–81.
* {{cite book | first = Michael J. |last = Crowe  | authorlink=Michael J. Crowe|title = A History of Vector Analysis : The Evolution of the Idea of a Vectorial System | publisher = Dover Publications |edition = reprint | year= 1967 | isbn = 978-0-486-67910-5 | title-link = A History of Vector Analysis }}
* {{cite book | first = J. E. |last = Marsden | title = Vector Calculus | publisher = W. H. Freeman & Company | year = 1976 | isbn = 978-0-7167-0462-1}}
* {{cite book | first = H. M. |last = Schey | title = Div Grad Curl and all that: An informal text on vector calculus | publisher = W. W. Norton & Company | year= 2005 | isbn = 978-0-393-92516-6}}
* Barry Spain (1965) [https://archive.org/details/VectorAnalysis Vector Analysis], 2nd edition, link from [[Internet Archive]].
* Chen-To Tai (1995). ''[http://deepblue.lib.umich.edu/handle/2027.42/7868 A historical study of vector analysis]''. Technical Report RL 915, Radiation Laboratory, University of Michigan.
{{refend}}

== External links ==
* [https://feynmanlectures.caltech.edu/II_02.html The Feynman Lectures on Physics Vol. II Ch. 2: Differential Calculus of Vector Fields]
* {{springer|title=Vector analysis|id=p/v096360}}
* {{springer|title=Vector algebra|id=p/v096350}}
* [https://deepblue.lib.umich.edu/handle/2027.42/7869 A survey of the improper use of ∇ in vector analysis] (1994) Tai, Chen-To
* [https://books.google.com/books?id=R5IKAAAAYAAJ Vector Analysis:] A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of [[Willard Gibbs]]) by [[Edwin Bidwell Wilson]], published 1902.

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