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to vector fields divergence div 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 product maps vector fields to scalar fields curl curl f f displaystyle operatorname curl mathbf f nabla times mathbf f measures the tendency to rotate about a point in a vector field in r 3 displaystyle mathbb r 3 cross product maps vector fields to pseudo vector fields f denotes a scalar field and f denotes a vector field also commonly used are the two laplace operators laplace operators in vector calculus operation notation description domain range laplacian δ f 2 f 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 2 f 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 and f denotes a vector field a quantity called the 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 edit the three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions integral theorems of vector calculus theorem statement description gradient theorem l r n φ d r 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 the line integral of the gradient of a scalar field over a curve l is equal to the change in the scalar field between the endpoints p and q of the curve divergence theorem v r n n f d v v n 1 f d s 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 an n dimensional solid v is equal to the flux of the vector field through the n 1 dimensional closed boundary surface of the solid curl kelvin stokes theorem σ r 3 f d σ σ f d r 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 a surface σ in r 3 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 and f denotes a vector field in two dimensions the divergence and curl theorems reduce to the green s theorem green s theorem of vector calculus theorem statement description green s theorem a r 2 m x l y d a a l d x m d y 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 region a in r 2 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 and m are functions of x y applications edit linear approximations edit main article linear approximation linear approximations are used to replace complicated functions with linear functions that are almost the same given a differentiable function f x y with real values one can approximate f 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 of z f x y at a b optimization edit main article mathematical optimization for a continuously differentiable function of several real variables a point p that is a set of values for the input variables which is viewed as a point in r n is critical if all of the partial derivatives of the function are zero at 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 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 eigenvalues of the hessian matrix of second derivatives by 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 edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed august 2019 learn how and when to remove this message vector calculus can also be generalized to other 3 manifolds and higher dimensional spaces different 3 manifolds edit vector calculus is initially defined for euclidean 3 space r 3 displaystyle mathbb r 3 which has additional structure beyond simply being a 3 dimensional real vector space namely a norm giving a notion of length defined via an inner product the dot product which in turn gives a notion of angle and an 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 5 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 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 edit 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 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 0 1 n 1 or 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 6 and trivially in dimension 0 or 1 is the curl of a vector field a vector field and only in 3 or 7 dimensions can a cross product be defined generalizations in other dimensionalities either require n 1 displaystyle n 1 vectors to yield 1 vector or are alternative lie algebras which are more general antisymmetric bilinear products the generalization of grad and div and how curl may be generalized is elaborated at 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 n 2 1 2 n n 1 displaystyle textstyle binom n 2 frac 1 2 n n 1 dimensions of rotations in n dimensions there are two important alternative generalizations of vector calculus the first geometric algebra uses k vector fields instead of vector fields in 3 or fewer dimensions every 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 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 uses differential forms k covector fields instead of vector fields or 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 7 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 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 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 tensor fields edit the scalar and vector fields above are specific cases of tensor fields in differential geometry a vector field is defined as a map from a manifold m textstyle m to its tangent bundle defined as the disjoint union of all tangent spaces one tangent space for each point of m textstyle m v m t m t m p m t p m textstyle v m to tm tm equiv bigsqcup _ p in m t_ p m similarly a covector field is a map from a manifold to its cotangent bundle defined analogously to the above expression but replacing t p m displaystyle t_ p m with t p m displaystyle t_ p m a p q displaystyle p q tensor can be formed by taking a tensor product of a p 0 displaystyle p 0 tensor and a 0 q displaystyle 0 q tensor which themselves can be formed by taking p displaystyle p and q displaystyle q fold tensor products of vectors and covectors respectively thus a p q displaystyle p q tensor field is a map from a manifold to bundles of p displaystyle p and q displaystyle q fold tensor products of the tangent spaces and cotangent spaces to which vectors and covectors belong respectively t m x m j 1 p t p m i 1 q t p m displaystyle t m to bigsqcup _ x in m bigotimes _ j 1 p t_ p m bigotimes _ i 1 q t_ p m with this generalization we see that the scalar and vector fields defined above are just 0 0 displaystyle 0 0 and 1 0 displaystyle 1 0 tensor fields see also edit mathematics portal conservative vector field directional derivative geometric calculus helmholtz decomposition laplacian vector field solenoidal vector field tensor vector algebra relations vector calculus identities references edit citations edit kreyszig erwin kreyszig herbert norminton e j 2011 advanced engineering mathematics 10th ed hoboken nj john wiley isbn 978 0 470 45836 5 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 galbis antonio maestre manuel 2012 vector analysis versus vector calculus springer p 12 isbn 978 1 4614 2199 3 differential operators math24 retrieved 2020 09 17 permanent dead link salma jama fantice lin 2025 05 31 exploring 3 manifolds pdf 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 bachman david 2012 stokes theorem in bachman david ed a geometric approach to differential forms boston birkhäuser pp 83 100 doi 10 1007 978 0 8176 8304 7_6 isbn 978 0 8176 8304 7 retrieved 2026 05 03 sources edit sandro caparrini 2002 the discovery of the vector representation of moments and angular velocity archive for history of exact sciences 56 151 81 crowe michael j 1967 a history of vector analysis the evolution of the idea of a vectorial system reprint ed dover publications isbn 978 0 486 67910 5 marsden j e 1976 vector calculus w h freeman company isbn 978 0 7167 0462 1 schey h m 2005 div grad curl and all that an informal text on vector calculus w w norton company isbn 978 0 393 92516 6 barry spain 1965 vector analysis 2nd edition link from internet archive chen to tai 1995 a historical study of vector analysis technical report rl 915 radiation laboratory university of michigan external links edit the feynman lectures on physics vol ii ch 2 differential calculus of vector fields vector analysis encyclopedia of mathematics ems press 2001 1994 vector algebra encyclopedia of mathematics ems press 2001 1994 a survey of the improper use of in vector analysis 1994 tai chen to 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 v t e major topics in mathematical analysis calculus integration differentiation differential equations ordinary partial stochastic fundamental theorem of calculus calculus of variations vector calculus tensor calculus matrix calculus lists of integrals table of derivatives real analysis complex analysis hypercomplex analysis quaternionic analysis functional analysis fourier analysis least squares spe...