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307 relations: 't Hooft operator, Abel–Jacobi map, Abelian integral, Adelic algebraic group, Adhémar Jean Claude Barré de Saint-Venant, Affine connection, Affine differential geometry, Affine focal set, Aharonov–Bohm effect, Albanese variety, Algebraic K-theory, Almost complex manifold, Almost symplectic manifold, Ambient space, Andreotti–Norguet formula, Angular momentum, Anomaly (physics), Antisymmetric tensor, Arrangement of hyperplanes, Atiyah–Bott fixed-point theorem, Élie Cartan, Barrett–Crane model, Basic element, Beniamino Segre, BF model, Bianchi classification, Bundle gerbe, Calabi conjecture, Calculus on Manifolds (book), Calibrated geometry, Canonical bundle, Canonical form, Carathéodory–Jacobi–Lie theorem, Cartan connection, Cartan formalism (physics), Cartan formula, Cauchy–Riemann equations, Chain complex, Characteristic class, Chern–Simons form, Chow group, Circle bundle, Clifford algebra, Clifford bundle, Closed and exact differential forms, Coadjoint representation, Coherent sheaf, Cohomology, Commutative ring, Compact group, ..., Compactification (physics), Complex differential form, Connection (mathematics), Connection form, Conservative vector field, Contact (mathematics), Contact geometry, Cotangent space, Cotton tensor, Courant bracket, Covariant transformation, CR manifold, Crofton formula, Cross product, Cup product, Current (mathematics), Curvature form, Curvature of Riemannian manifolds, Curve-shortening flow, D-brane, D-module, Darboux's theorem, De Rham cohomology, Degree of a continuous mapping, Deligne cohomology, Density on a manifold, Derivation (differential algebra), Differentiable manifold, Differential, Differential (infinitesimal), Differential (mathematics), Differential calculus over commutative algebras, Differential forms on a Riemann surface, Differential geometry, Differential geometry of surfaces, Differential graded algebra, Differential ideal, Differential of a function, Differential of the first kind, Directional derivative, Discrete exterior calculus, Donald C. Spencer, Dual number, Dual photon, Duality (mathematics), Dyadics, Ehresmann connection, Eilenberg–MacLane space, Einstein field equations, Electromagnetic tensor, Elwin Bruno Christoffel, Enzo Martinelli, Equivariant differential form, Erich Kähler, Exact differential, Exterior algebra, Exterior derivative, False diffusion, Federigo Enriques, Finite element exterior calculus, Finsler manifold, Fisher information metric, Flat vector bundle, Floer homology, Form, Frame bundle, Frölicher–Nijenhuis bracket, Frobenius theorem (differential topology), Fundamental class, Fundamental theorem of calculus, Funk transform, G-structure on a manifold, Gaetano Fichera, Gauge anomaly, Gauge theory, Generalizations of the derivative, Generalized complex structure, Generalized function, Generalized Gauss–Bonnet theorem, Geometric algebra, Geometric calculus, Geometric genus, Geometric topology, Georges de Rham, Glossary of classical algebraic geometry, Gluon field strength tensor, Gradient, Gradient theorem, Grassmann number, Gravitation (book), Green's theorem, Green–Schwarz mechanism, Grunsky matrix, Gysin homomorphism, Harmonic differential, Harold Edwards (mathematician), Harry Bateman, Helmholtz decomposition, Hermann Grassmann, Hitchin functional, Hodge conjecture, Hodge star operator, Hodge theory, Hodge–de Rham spectral sequence, Homological integration, Inclusion map, Inexact differential, Injective sheaf, Inner product space, Instanton, Integrability conditions for differential systems, Integral, Integral geometry, Integration along fibers, Interior product, Intersection number, Invariant differential operator, Inverse problem for Lagrangian mechanics, Itô calculus, J. Arthur Seebach Jr., Jet bundle, Johann Friedrich Pfaff, K-theory (physics), Kalb–Ramond field, Kaluza–Klein theory, Kazimierz Żorawski, Kähler differential, Kähler manifold, Kelvin–Stokes theorem, Kodaira vanishing theorem, Komar superpotential, Lagrangian (field theory), Lagrangian system, Laplace operator, Laplace operators in differential geometry, Laplace–Beltrami operator, L² cohomology, Lebesgue integration, Lee Hwa Chung theorem, Leibniz integral rule, Leibniz's notation, Lie algebra cohomology, Lie derivative, Lie point symmetry, Lie theory, Liouville's theorem (Hamiltonian), List of differential geometry topics, List of German inventors and discoverers, List of lemmas, List of multivariable calculus topics, List of textbooks in electromagnetism, Logarithm, Logarithmic derivative, Logarithmic form, M2-brane, Manifold, Mathematical descriptions of the electromagnetic field, Mathieu transformation, Maurer–Cartan form, Maxwell's equations, Mayer–Vietoris sequence, Metric tensor, Michael Hutchings (mathematician), Minkowski space, Mixmaster universe, Module (mathematics), Modulo (jargon), Multilinear algebra, Multilinear form, Multisymplectic integrator, Multivariable calculus, Multivector, Novikov–Shubin invariant, Omega, One-form, Orientability, Orientation sheaf, P-adic Hodge theory, P-form electrodynamics, Paracompact space, Picard theorem, Poincaré–Hopf theorem, Poisson bracket, Polyhedral space, Polyvector field, Positive form, Predual, Presymplectic form, Products in algebraic topology, Projective space, Prym differential, Pullback, Pullback (cohomology), Pullback (differential geometry), Quantum differential calculus, Quantum geometry, Ramond–Ramond field, Rational homotopy theory, Real analysis, Real coordinate space, Recurrent tensor, Representation theory, Representation theory of diffeomorphism groups, Representation theory of the Lorentz group, Resolution (algebra), Ricci calculus, Riemannian connection on a surface, Saint-Venant's compatibility condition, Scalar field, Schouten–Nijenhuis bracket, Secondary calculus and cohomological physics, Sesquilinear form, Sheaf (mathematics), Sheaf cohomology, Signature operator, Sphere, Stochastic differential equation, Stokes' theorem, Supersymmetric theory of stochastic dynamics, Supersymmetry, Surface integral, Symmetry of second derivatives, Symplectic geometry, Symplectic integrator, Symplectic manifold, Tensor, Tensor contraction, Tensor density, Tensor field, Tensor product, Tensor product bundle, Tensor product of modules, Timeline of manifolds, Topological recursion, Torsion tensor, Total derivative, Translation surface, Two-vector, Valya algebra, Variational bicomplex, Varifold, Vasiliev equations, Vector calculus, Vector field, Vector space, Vector-valued differential form, Vertical and horizontal bundles, Vladimir Miklyukov, Volume, Volume element, Volume form, Weil cohomology theory, Weitzenböck identity, Wirtinger inequality (2-forms), Work (thermodynamics), 2 × 2 real matrices, 3-manifold. Expand index (257 more) »
't Hooft operator
In theoretical physics, a 't Hooft operator, introduced by Gerard 't Hooft in the 1978 paper "On the phase transition towards permanent quark confinement", is a dual version of the Wilson loop in which the electromagnetic potential A is replaced by its electromagnetic dual Amag, where the exterior derivative of A is equal to the Hodge dual of the exterior derivative of Amag.
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Abel–Jacobi map
In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety.
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Abelian integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form where R(x,w) is an arbitrary rational function of the two variables x and w, which are related by the equation where F(x,w) is an irreducible polynomial in w, whose coefficients \varphi_j(x), j.
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Adelic algebraic group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A.
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Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant (23 August 1797, Villiers-en-Bière, Seine-et-Marne – 6 January 1886, Saint-Ouen, Loir-et-Cher) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as the Saint-Venant equations that are a fundamental set of equations used in modern hydraulic engineering.
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Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.
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Affine differential geometry
Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under volume-preserving affine transformations.
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Affine focal set
In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines.
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Aharonov–Bohm effect
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (V, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.
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Albanese variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
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Algebraic K-theory
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.
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Almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space.
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Almost symplectic manifold
In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M that is everywhere non-singular.
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Ambient space
An ambient space or ambient configuration space is the space surrounding an object.
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Andreotti–Norguet formula
The Andreotti–Norguet formula, first introduced by, is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function.
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Angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.
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Anomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory.
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Antisymmetric tensor
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.
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Arrangement of hyperplanes
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space.
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Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Barrett–Crane model
The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action.
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Basic element
In algebra, a basic element x with respect to an element y is an element of a cochain complex (C^*, d) (e.g., complex of differential forms on a manifold) that is closed: dx.
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Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
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BF model
The BF model is a topological field, which when quantized, becomes a topological quantum field theory.
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Bianchi classification
In mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of Lie algebras.
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Bundle gerbe
In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology.
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Calabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by and proved by.
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Calculus on Manifolds (book)
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief (146 pp.) monograph on the theory of vector-valued functions of several real variables (f: Rn→Rm) and differentiable manifolds in Euclidean space.
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Calibrated geometry
In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential ''p''-form φ (for some 0 ≤ p ≤ n) which is a calibration in the sense that.
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Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n.
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Canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression.
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Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.
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Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.
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Cartan formalism (physics)
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds.
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Cartan formula
In mathematics, Cartan formula can mean.
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Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.
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Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.
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Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
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Chern–Simons form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes.
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Chow group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space.
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Circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle \scriptstyle \mathbf^1.
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Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.
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Clifford bundle
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure.
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Closed and exact differential forms
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα.
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Coadjoint representation
In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation.
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Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space.
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Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions.
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Complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.
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Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
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Conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.
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Contact (mathematics)
In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives.
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Contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'.
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Cotangent space
In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below).
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Cotton tensor
In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor.
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Courant bracket
In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of ''p''-forms.
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Covariant transformation
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis.
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CR manifold
In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
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Crofton formula
In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.
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Cross product
In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.
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Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.
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Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Formally currents behave like Schwartz distributions on a space of differential forms.
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Curvature form
In differential geometry, the curvature form describes the curvature of a connection on a principal bundle.
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Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
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Curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature.
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D-brane
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named.
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D-module
In mathematics, a D-module is a module over a ring D of differential operators.
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Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem.
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De Rham cohomology
In mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.
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Degree of a continuous mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping.
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Deligne cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold.
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Density on a manifold
In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner.
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Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Differential
Differential may refer to.
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Differential (infinitesimal)
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity.
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Differential (mathematics)
In mathematics, differential refers to infinitesimal differences or to the derivatives of functions.
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Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms.
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Differential forms on a Riemann surface
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifying a Riemannian metric.
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
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Differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.
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Differential ideal
In the theory of differential forms, a differential ideal I is an algebraic ideal in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exterior differentiation d. In other words, for any form α in I, the exterior derivative dα is also in I. In the theory of differential algebra, a differential ideal I in a differential ring R is an ideal which is mapped to itself by each differential operator.
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Differential of a function
In calculus, the differential represents the principal part of the change in a function y.
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Differential of the first kind
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms.
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Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
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Discrete exterior calculus
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes.
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Donald C. Spencer
Donald Clayton Spencer (April 25, 1912 – December 23, 2001) was an American mathematician, known for work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of partial differential equations.
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Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2.
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Dual photon
In theoretical physics, the dual photon is a hypothetical elementary particle that is a dual of the photon under electric-magnetic duality which is predicted by some theoretical models and some results of M-theory in eleven dimensions.
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Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.
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Dyadics
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
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Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle.
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Eilenberg–MacLane space
In mathematics, and algebraic topology in particular, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name.
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Einstein field equations
The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
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Electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.
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Elwin Bruno Christoffel
Elwin Bruno Christoffel (November 10, 1829 – March 15, 1900) was a German mathematician and physicist.
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Enzo Martinelli
Enzo Martinelli (11 November 1911 – 27 August 1999 writes that his death year is 1998, unlike to, and, but it is probably a typographical error.) was an Italian mathematician, working in the theory of functions of several complex variables: he is best known for his work on the theory of integral representations for holomorphic functions of several variables, notably for discovering the Bochner–Martinelli formula in 1938, and for his work in the theory of multi-dimensional residues.
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Equivariant differential form
In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map from the Lie algebra \mathfrak.
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Erich Kähler
Erich Kähler (16 January 1906, Leipzig – 31 May 2000, Wedel) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.
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Exact differential
In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
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False diffusion
False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations.
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Federigo Enriques
Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.
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Finite element exterior calculus
Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods in the calculus of differential forms.
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Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space, allowing to define the length of any smooth curve as Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.
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Fisher information metric
In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.
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Flat vector bundle
In mathematics, a vector bundle is said to be flat if it is endowed with an linear connection with vanishing curvature, ie. a flat connection.
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Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.
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Form
Form is the shape, visual appearance, or configuration of an object.
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Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.
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Frölicher–Nijenhuis bracket
In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold.
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Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations.
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Fundamental class
In mathematics, the fundamental class is a homology class associated to an oriented manifold M of dimension n, which corresponds to the generator of the homology group H_n(M;\mathbf)\cong\mathbf.
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Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.
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Funk transform
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere.
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G-structure on a manifold
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
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Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables.
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Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e. of a gauge theory.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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Generalizations of the derivative
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.
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Generalized complex structure
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure.
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Generalized function
In mathematics, generalized functions, or distributions, are objects extending the notion of functions.
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Generalized Gauss–Bonnet theorem
In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss and Pierre Ossian Bonnet) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature.
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Geometric algebra
The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars F and the vector space V. Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form.
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Geometric calculus
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration.
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Geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
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Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
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Georges de Rham
Georges de Rham (10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
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Glossary of classical algebraic geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck.
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Gluon field strength tensor
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
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Gradient
In mathematics, the gradient is a multi-variable generalization of the derivative.
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Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
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Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers.
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Gravitation (book)
Gravitation is a physics book on Einstein's theory of gravity, written by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler and originally published by W. H. Freeman and Company in 1973.
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Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.
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Green–Schwarz mechanism
The Green–Schwarz mechanism (sometimes called the Green–Schwarz anomaly cancellation mechanism) is the main discovery that started the first superstring revolution in superstring theory.
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Grunsky matrix
In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by in complex analysis and geometric function theory.
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Gysin homomorphism
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle.
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Harmonic differential
In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω*, are both closed.
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Harold Edwards (mathematician)
Harold Mortimer Edwards, Jr. (born August 6, 1936) is an American mathematician working in number theory, algebra, and the history and philosophy of mathematics.
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Harry Bateman
Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician.
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Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation.
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Hermann Grassmann
Hermann Günther Grassmann (Graßmann; April 15, 1809 – September 26, 1877) was a German polymath, known in his day as a linguist and now also as a mathematician.
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Hitchin functional
The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin.
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Hodge conjecture
In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of it.
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Hodge star operator
In mathematics, the Hodge isomorphism or Hodge star operator is an important linear map introduced in general by W. V. D. Hodge.
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Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, uses partial differential equations to study the cohomology groups of a smooth manifold M. The key tool is the Laplacian operator associated to a Riemannian metric on M. The theory was developed by Hodge in the 1930s as an extension of de Rham cohomology.
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Hodge–de Rham spectral sequence
In mathematics, the Hodge–de Rham spectral sequence (named after W. V. D. Hodge and Georges de Rham), also known as the Frölicher spectral sequence (after Alfred Frölicher) computes the cohomology of a complex manifold.
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Homological integration
In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds.
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Inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.
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Inexact differential
An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential.
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Injective sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext).
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Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
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Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics.
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Integrability conditions for differential systems
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space.
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Integration along fibers
In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where m is the dimension of the fiber, via "integration".
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Interior product
In mathematics, the interior product (aka interior derivative/, interior multiplication, inner multiplication, inner derivative, or inner derivation) is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold.
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Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency.
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Invariant differential operator
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type.
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Inverse problem for Lagrangian mechanics
In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.
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Itô calculus
Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process).
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J. Arthur Seebach Jr.
J.
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Jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle.
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Johann Friedrich Pfaff
Johann Friedrich Pfaff (sometimes spelled Friederich; 22 December 1765 – 21 April 1825) was a German mathematician.
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K-theory (physics)
In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-branes.
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Kalb–Ramond field
In theoretical physics in general and string theory in particular, the Kalb–Ramond field (named after Michael Kalb and Pierre Ramond), also known as the Kalb–Ramond B-field or Kalb–Ramond NS–NS B-field, is a quantum field that transforms as a two-form, i.e., an antisymmetric tensor field with two indices.
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Kaluza–Klein theory
In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time and considered an important precursor to string theory.
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Kazimierz Żorawski
Kazimierz Żorawski (June 22, 1866 – January 23, 1953) was a Polish mathematician.
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Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.
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Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.
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Kelvin–Stokes theorem
The Kelvin–Stokes theoremThis proof is based on the Lecture Notes given by Prof.
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Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero.
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Komar superpotential
In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert-Einstein Lagrangian \mathcal_\mathrm.
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Lagrangian (field theory)
Lagrangian field theory is a formalism in classical field theory.
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Lagrangian system
In mathematics, a Lagrangian system is a pair, consisting of a smooth fiber bundle and a Lagrangian density, which yields the Euler–Lagrange differential operator acting on sections of.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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Laplace operators in differential geometry
In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian.
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Laplace–Beltrami operator
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds.
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L² cohomology
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric.
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Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.
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Lee Hwa Chung theorem
The Lee Hwa Chung theorem is a theorem in symplectic topology.
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Leibniz integral rule
In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where -\infty, the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of f(x, t) with x is considered in taking the derivative.
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Leibniz's notation
dydx d2ydx2 In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and, respectively, just as and represent finite increments of and, respectively.
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Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras.
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Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow defined by another vector field.
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Lie point symmetry
Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs).
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Lie theory
In mathematics, the researcher Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.
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Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.
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List of differential geometry topics
This is a list of differential geometry topics.
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List of German inventors and discoverers
---- This is a list of German inventors and discoverers.
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List of lemmas
This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs).
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List of multivariable calculus topics
This is a list of multivariable calculus topics.
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List of textbooks in electromagnetism
Following is a list of notable textbooks in electromagnetism.
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Logarithm
In mathematics, the logarithm is the inverse function to exponentiation.
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Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f' is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f', scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule.
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Logarithmic form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind.
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M2-brane
In theoretical physics, an M2-brane, is a spatially extended mathematical object that appears in string theory and in related theories (e.g. M-theory, F-theory).
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Mathematical descriptions of the electromagnetic field
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature.
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Mathieu transformation
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form The transformation is named after the French mathematician Émile Léonard Mathieu.
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Maurer–Cartan form
In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of.
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Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
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Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.
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Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
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Michael Hutchings (mathematician)
Michael Lounsbery Hutchings is an American mathematician, a professor of mathematics at the University of California, Berkeley.
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Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.
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Mixmaster universe
The Mixmaster universe (named after Sunbeam Mixmaster, a brand of Sunbeam Products electric kitchen mixer) is a solution to Einstein field equations of general relativity studied by Charles Misner in an effort to better understand the dynamics of the early universe.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Modulo (jargon)
The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means "a small measure." It was introduced into mathematics by Carl Friedrich Gauss in 1801.
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Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra.
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Multilinear form
In abstract algebra and multilinear algebra, a multilinear form on V is a map of the type f: V^k \to K,where V is a vector space over the field K (or more generally, a module over a commutative ring), that is separately K-linear in each of its k arguments.
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Multisymplectic integrator
In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic.
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Multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.
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Multivector
A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between elements of the space is called an algebra; examples are matrix algebra and vector algebra.
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Novikov–Shubin invariant
In mathematics, a Novikov–Shubin invariant.
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Omega
Omega (capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and last letter of the Greek alphabet.
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One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space.
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Orientability
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.
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Orientation sheaf
In algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is (in the integer coefficients or some other coefficients).
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P-adic Hodge theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as '''Q'''''p'').
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P-form electrodynamics
In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
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Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
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Picard theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function.
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Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology.
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Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system.
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Polyhedral space
Polyhedral space is a certain metric space.
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Polyvector field
In mathematics, a multivector field, polyvector field of degree k, or k-vector field, on a manifold M, is a section of the kth exterior power of the tangent bundle, \Lambda^k TM.
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Positive form
In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).
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Predual
In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the space of bounded operators is the space of trace class operators.
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Presymplectic form
In Geometric Mechanics a presymplectic form is a closed differential 2-form of constant rank on a manifold.
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Products in algebraic topology
In algebraic topology, several types of products are defined on homological and cohomological theories.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Prym differential
In mathematics, a Prym differential of a Riemann surface is a differential form on the universal covering space that transforms according to some complex character of the fundamental group.
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Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product.
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Pullback (cohomology)
In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map.
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Pullback (differential geometry)
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*.
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Quantum differential calculus
In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A over a field k means the specification of a space of differential forms over the algebra.
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Quantum geometry
In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to Planck length.
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Ramond–Ramond field
In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory.
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Rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored.
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Real analysis
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.
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Real coordinate space
In mathematics, real coordinate space of dimensions, written R (also written with blackboard bold) is a coordinate space that allows several (''n'') real variables to be treated as a single variable.
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Recurrent tensor
In mathematics, a recurrent tensor, with respect to a connection \nabla on a manifold M, is a tensor T for which there is a one-form ω on M such that.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Representation theory of diffeomorphism groups
In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M.
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Representation theory of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity.
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Resolution (algebra)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category.
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Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.
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Riemannian connection on a surface
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.
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Saint-Venant's compatibility condition
In the mathematical theory of elasticity the strain \varepsilon is related to a displacement field \ u by where 1\le i,j \le 3.
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Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.
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Schouten–Nijenhuis bracket
In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields.
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Secondary calculus and cohomological physics
In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation.
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Sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Sheaf cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.
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Signature operator
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four.
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
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Stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.
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Stokes' theorem
In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
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Supersymmetric theory of stochastic dynamics
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise.
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Supersymmetry
In particle physics, supersymmetry (SUSY) is a theory that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin.
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Surface integral
In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.
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Symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a function of n variables.
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Symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
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Symplectic integrator
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems.
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Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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Tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual.
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Tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept.
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Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Tensor product bundle
In differential geometry, the tensor product of vector bundles E, F is a vector bundle, denoted by E ⊗ F, whose fiber over a point x is the tensor product of vector spaces Ex ⊗ Fx.
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Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps.
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Timeline of manifolds
This is a timeline of manifolds, one of the major geometric concepts of mathematics.
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Topological recursion
In mathematics, Topological Recursion is a recursive definition of invariants of spectral curves.
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Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.
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Total derivative
In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.
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Translation surface
In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations.
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Two-vector
A two-vector is a tensor of type (2,0) and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).
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Valya algebra
In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms: 1.
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Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations.
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Varifold
In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in differential geometry.
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Vasiliev equations
Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space.
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Vector calculus
Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Vector-valued differential form
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms.
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Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.
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Vladimir Miklyukov
Vladimir Michaelovich Miklyukov (Миклюков, Владимир Михайлович, also spelled Miklioukov or Mikljukov) (8 January 1944 – October 2013) was a Russian educator in mathematics, and head of the Superslow Process workgroup based at Volgograd State University.
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Volume
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
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Volume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.
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Volume form
In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).
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Weil cohomology theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups.
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Weitzenböck identity
In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol.
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Wirtinger inequality (2-forms)
In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold M, the exterior kth power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable) (2k)-vector ζ of unit volume, is bounded above by k!.
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Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred by the system to its surroundings, that is fully accounted for solely by macroscopic forces exerted on the system by factors external to it, that is to say, factors in its surroundings.
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2 × 2 real matrices
In mathematics, the associative algebra of real matrices is denoted by M(2, R).
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3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
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0-form, 2-form, 3-form, Differential 1-form, Differential forms, Differential k-form, Exterior differential form, Exterior form, Integration of a differential form, Integration on manifolds, K-form, P-form, Two-form, Zero-Form, Zero-form.
References
[1] https://en.wikipedia.org/wiki/Differential_form
