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thogonal 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 spectral analysis harmonic analysis p adic analysis p adic numbers algebraic analysis tropical analysis measure theory microlocal analysis representation theory functions continuous function special functions limit series infinity mathematics portal v t e industrial and applied mathematics computational algorithms design analysis automata theory automated theorem proving coding theory computational geometry constraint satisfaction constraint programming computational logic cryptography computational group theory information theory statistics mathematical software arbitrary precision arithmetic finite element analysis tensor software interactive geometry software optimization software statistical software numerical analysis software numerical libraries proof assistants solvers discrete computer algebra computational number theory combinatorics graph theory discrete geometry analysis approximation theory clifford analysis clifford algebra differential equations ordinary differential equations partial differential equations stochastic differential equations differential geometry differential forms gauge theory geometric analysis dynamical systems chaos theory control theory functional analysis operator algebra operator theory harmonic analysis fourier analysis multilinear algebra exterior geometric tensor vector multivariable calculus exterior geometric tensor vector numerical analysis numerical linear algebra numerical methods for ordinary differential equations numerical methods for partial differential equations validated numerics variational calculus probability theory distributions random variables stochastic processes analysis path integral stochastic variational calculus mathematical physics analytical mechanics lagrangian hamiltonian field theory classical conformal effective gauge quantum statistical topological perturbation theory in quantum mechanics potential theory string theory bosonic topological supersymmetry supersymmetric quantum mechanics supersymmetric theory of stochastic dynamics algebraic structures algebra of physical space particle physics and representation theory feynman integral poisson algebra quantum group renormalization group spacetime algebra superalgebra supersymmetry algebra decision sciences game theory operations research optimization social choice theory statistics mathematical economics mathematical finance other applications biology chemistry psychology sociology the unreasonable effectiveness of mathematics in the natural sciences related mathematics organizations society for industrial and applied mathematics japan society for industrial and applied mathematics société de mathématiques appliquées et industrielles international council for industrial and applied mathematics european community on computational methods in applied sciences category mathematics portal outline topics list v t e calculus precalculus binomial theorem concave function continuous function factorial finite difference free variables and bound variables graph of a function linear function radian rolle s theorem secant slope tangent limits indeterminate form limit of a function one sided limit limit of a sequence order of approximation ε δ definition of limit differential calculus derivative second derivative partial derivative differential differential operator mean value theorem notation leibniz s notation newton s notation rules of differentiation linearity power sum chain l hôpital s product general leibniz s rule quotient other techniques implicit differentiation inverse function rule logarithmic derivative related rates stationary points first derivative test second derivative test extreme value theorem maximum and minimum further applications newton s method taylor s theorem differential equation ordinary differential equation partial differential equation stochastic differential equation integral calculus antiderivative arc length riemann integral basic properties constant of integration fundamental theorem of calculus differentiating under the integral sign integration by parts integration by substitution trigonometric euler tangent half angle substitution partial fractions in integration quadratic integral trapezoidal rule volumes washer method shell method integral equation integro differential equation vector calculus derivatives curl directional derivative divergence gradient laplacian basic theorems line integrals green s stokes gauss multivariable calculus divergence theorem geometric hessian matrix jacobian matrix and determinant lagrange multiplier line integral matrix multiple integral partial derivative surface integral volume integral advanced topics differential forms exterior derivative generalized stokes theorem tensor calculus sequences and series arithmetico geometric sequence types of series alternating binomial fourier geometric harmonic infinite power maclaurin taylor telescoping tests of convergence abel s alternating series cauchy condensation direct comparison dirichlet s integral limit comparison ratio root term special functions and numbers bernoulli numbers e mathematical constant exponential function natural logarithm stirling s approximation history of calculus adequality brook taylor colin maclaurin generality of algebra gottfried wilhelm leibniz infinitesimal infinitesimal calculus isaac newton fluxion law of continuity leonhard euler method of fluxions the method of mechanical theorems lists integrals rational functions irrational algebraic functions exponential functions logarithmic functions hyperbolic functions inverse trigonometric functions inverse secant secant cubed list of limits list of derivatives miscellaneous topics complex calculus contour integral differential geometry manifold curvature of curves of surfaces tensor euler maclaurin formula gabriel s horn integration bee proof that 22 7 exceeds π regiomontanus angle maximization problem steinmetz solid authority control databases international gnd national united states japan czech republic 2 retrieved from https en wikipedia org w index php title vector_calculus oldid 1375179789 categories vector calculus mathematical 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