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Vector calculus

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Calculus of vector-valued functions
Not to be confused withGeometric calculus orMatrix calculus.
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Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Vector calculus orvector analysis is a branch of mathematics concerned with thedifferentiation andintegration ofvector fields, primarily in three-dimensionalEuclidean space,R3.{\displaystyle \mathbb {R} ^{3}.}[1] The termvector calculus is sometimes used as a synonym for the broader subject ofmultivariable calculus, which spans vector calculus as well aspartial differentiation andmultiple integration. Vector calculus plays an important role indifferential geometry and in the study ofpartial differential equations. It is used extensively in physics and engineering, especially in the description ofelectromagnetic fields,gravitational fields, andfluid flow.

Vector calculus was developed from the theory ofquaternions byJ. Willard Gibbs andOliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs andEdwin Bidwell Wilson in their 1901 book,Vector Analysis, though earlier mathematicians such asIsaac Newton pioneered the field.[2] In its standard form using thecross product, vector calculus does not generalize to higher dimensions, but the alternative approach ofgeometric algebra, which uses theexterior product, does (see§ Generalizations below for more).

Basic objects

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Scalar fields

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Main article:Scalar field

Ascalar field associates ascalar value to every point in a space. The scalar is a mathematical number representing aphysical quantity. Examples of scalar fields in applications include thetemperature distribution throughout space, thepressure distribution in a fluid, and spin-zero quantum fields (known asscalar bosons), such as theHiggs field. These fields are the subject ofscalar field theory.

Vector fields

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Main article:Vector field

Avector field is an assignment of avector to each point in aspace.[3] A vector field in the plane, for instance, can be visualized as a collection of arrows with a givenmagnitude 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 someforce, such as themagnetic orgravitational force, as it changes from point to point. This can be used, for example, to calculatework done over a line.

Vectors and pseudovectors

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In more advanced treatments, one further distinguishespseudovector fields andpseudoscalar fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, thecurl 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 ingeometric algebra, as described below.

Vector algebra

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Main article:Euclidean vector § Basic properties

The algebraic (non-differential) operations in vector calculus are referred to asvector algebra, being defined for a vector space and then appliedpointwise to a vector field. The basic algebraic operations consist of:

Notations in vector calculus
OperationNotationDescription
Vector additionv1+v2{\displaystyle \mathbf {v} _{1}+\mathbf {v} _{2}}Addition of two vectors, yielding a vector.
Scalar multiplicationav{\displaystyle a\mathbf {v} }Multiplication of a scalar and a vector, yielding a vector.
Dot productv1⋅v2{\displaystyle \mathbf {v} _{1}\cdot \mathbf {v} _{2}}Multiplication of two vectors, yielding a scalar.
Cross productv1×v2{\displaystyle \mathbf {v} _{1}\times \mathbf {v} _{2}}Multiplication of two vectors inR3{\displaystyle \mathbb {R} ^{3}}, yielding a (pseudo)vector.

Also commonly used are the twotriple products:

Vector calculus triple products
OperationNotationDescription
Scalar triple productv1⋅(v2×v3){\displaystyle \mathbf {v} _{1}\cdot \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}The dot product of the cross product of two vectors.
Vector triple productv1×(v2×v3){\displaystyle \mathbf {v} _{1}\times \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}The cross product of the cross product of two vectors.

Operators and theorems

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Main article:Vector calculus identities

Differential operators

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Main articles:Gradient,Divergence,Curl (mathematics), andLaplacian

Vector calculus studies variousdifferential operators defined on scalar or vector fields, which are typically expressed in terms of thedel operator (∇{\displaystyle \nabla }), also known as "nabla". The three basicvector operators are:[4]

Differential operators in vector calculus
OperationNotationDescriptionNotational
analogy		Domain/Range
Gradientgrad⁡(f)=∇f{\displaystyle \operatorname {grad} (f)=\nabla f}Measures the rate and direction of change in a scalar field.Scalar multiplicationMaps scalar fields to vector fields.
Divergencediv⁡(F)=∇⋅F{\displaystyle \operatorname {div} (\mathbf {F} )=\nabla \cdot \mathbf {F} }Measures the scalar of a source or sink at a given point in a vector field.Dot productMaps vector fields to scalar fields.
Curlcurl⁡(F)=∇×F{\displaystyle \operatorname {curl} (\mathbf {F} )=\nabla \times \mathbf {F} }Measures the tendency to rotate about a point in a vector field inR3{\displaystyle \mathbb {R} ^{3}}.Cross productMaps vector fields to (pseudo)vector fields.
f denotes a scalar field andF denotes a vector field

Also commonly used are the two Laplace operators:

Laplace operators in vector calculus
OperationNotationDescriptionDomain/Range
LaplacianΔf=∇2f=∇⋅∇f{\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}Measures the difference between the value of the scalar field with its average on infinitesimal balls.Maps between scalar fields.
Vector Laplacian∇2F=∇(∇⋅F)−∇×(∇×F){\displaystyle \nabla ^{2}\mathbf {F} =\nabla (\nabla \cdot \mathbf {F} )-\nabla \times (\nabla \times \mathbf {F} )}Measures the difference between the value of the vector field with its average on infinitesimal balls.Maps between vector fields.
f denotes a scalar field andF denotes a vector field

A quantity called theJacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as achange of variables during integration.

Integral theorems

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The three basic vector operators have corresponding theorems which generalize thefundamental theorem of calculus to higher dimensions:

Integral theorems of vector calculus
TheoremStatementDescription
Gradient theorem∫L⊂Rn∇φ⋅dr = φ(q)−φ(p)   for   L=L[p→q]{\displaystyle \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]}Theline integral of the gradient of a scalar field over acurveL is equal to the change in the scalar field between the endpointsp andq of the curve.
Divergence theorem∫⋯∫V⊂Rn⏟n(∇⋅F)dV = ∮⋯∮∂V⏟n−1F⋅dS{\displaystyle \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} }The integral of the divergence of a vector field over ann-dimensional solidV is equal to theflux of the vector field through the(n−1)-dimensional closed boundary surface of the solid.
Curl (Kelvin–Stokes) theorem∬Σ⊂R3(∇×F)⋅dΣ = ∮∂ΣF⋅dr{\displaystyle \iint _{\Sigma \subset \mathbb {R} ^{3}}(\nabla \times \mathbf {F} )\cdot d\mathbf {\Sigma } \ =\ \oint _{\partial \Sigma }\mathbf {F} \cdot d\mathbf {r} }The integral of the curl of a vector field over asurfaceΣ inR3{\displaystyle \mathbb {R} ^{3}} is equal to the circulation of the vector field around the closed curve bounding the surface.
φ{\displaystyle \varphi } denotes a scalar field andF denotes a vector field

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

Green's theorem of vector calculus
TheoremStatementDescription
Green's theorem∬A⊂R2(∂M∂x−∂L∂y)dA = ∮∂A(Ldx+Mdy){\displaystyle \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)}The integral of the divergence (or curl) of a vector field over some regionA inR2{\displaystyle \mathbb {R} ^{2}} equals the flux (or circulation) of the vector field over the closed curve bounding the region.
For divergence,F = (M, −L). For curl,F = (L,M, 0).L andM are functions of(x,y).

Applications

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Linear approximations

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Main article:Linear approximation

Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable functionf(x,y) with real values, one can approximatef(x,y) for(x,y) close to(a,b) by the formula

f(x,y) ≈ f(a,b)+∂f∂x(a,b)(x−a)+∂f∂y(a,b)(y−b).{\displaystyle 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).}

The right-hand side is the equation of the plane tangent to the graph ofz =f(x,y) at(a,b).

Optimization

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Main article:Mathematical optimization

For a continuously differentiablefunction of several real variables, a pointP (that is, a set of values for the input variables, which is viewed as a point inRn) iscritical if all of thepartial derivatives of the function are zero atP, or, equivalently, if itsgradient is zero. The critical values are the values of the function at the critical points.

If the function issmooth, or, at least twice continuously differentiable, a critical point may be either alocal maximum, alocal minimum or asaddle point. The different cases may be distinguished by considering theeigenvalues of theHessian matrix of second derivatives.

ByFermat's theorem, all localmaxima 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

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Vector calculus can also be generalized to other3-manifolds andhigher-dimensional spaces.

Different 3-manifolds

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Vector calculus is initially defined forEuclidean 3-space,R3,{\displaystyle \mathbb {R} ^{3},} which has additional structure beyond simply being a 3-dimensional real vector space, namely: anorm (giving a notion of length) defined via aninner product (thedot product), which in turn gives a notion of angle, and anorientation, which gives a notion of left-handed and right-handed. These structures give rise to avolume form, and also thecross 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 thecoordinate 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 symmetricnondegenerate 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 (thespecial orthogonal groupSO(3)).

More generally, vector calculus can be defined on any 3-dimensional orientedRiemannian manifold, or more generallypseudo-Riemannian manifold. This structure simply means that thetangent 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 nondegeneratemetric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.

Other dimensions

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Most of the analytic results are easily understood, in a more general form, using the machinery ofdifferential 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 (yieldingharmonic 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 beingk-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 to0,1,n − 1 orn 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[5] (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or7 dimensions can a cross product be defined (generalizations in other dimensionalities either requiren−1{\displaystyle n-1} vectors to yield 1 vector, or are alternativeLie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated atCurl § Generalizations; in brief, the curl of a vector field is abivector field, which may be interpreted as thespecial 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(n2)=12n(n−1){\displaystyle \textstyle {{\binom {n}{2}}={\frac {1}{2}}n(n-1)}} dimensions of rotations inn dimensions).

There are two important alternative generalizations of vector calculus. The first,geometric algebra, usesk-vector fields instead of vector fields (in 3 or fewer dimensions, everyk-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 theexterior product, which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yieldsClifford algebras 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 usesdifferential forms (k-covector fields) instead of vector fields ork-vector fields, and is widely used in mathematics, particularly indifferential geometry,geometric topology, andharmonic analysis, in particular yieldingHodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to theexterior 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 ofStokes' 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 identifiesk-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 identifiesk-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

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References

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Citations

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  1. ^Kreyszig, Erwin; Kreyszig, Herbert; Norminton, E. J. (2011).Advanced Engineering Mathematics (10th ed.). Hoboken, NJ: John Wiley.ISBN 978-0-470-45836-5.
  2. ^Rowlands, Peter (2017).Newton and the Great World System.World Scientific Publishing. pp. 26,82–83.doi:10.1142/q0108.ISBN 978-1-78634-372-7.
  3. ^Galbis, Antonio; Maestre, Manuel (2012).Vector Analysis Versus Vector Calculus. Springer. p. 12.ISBN 978-1-4614-2199-3.
  4. ^"Differential Operators".Math24. Retrieved2020-09-17.[permanent dead link]
  5. ^Lizhong Peng & Lei Yang (1999) "The curl in seven dimensional space and its applications",Approximation Theory and Its Applications 15(3): 66 to 80doi:10.1007/BF02837124

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