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A
Absolute differential calculus — the original name for tensor calculus developed around 1890.
Absolute geometry — geometry based on Euclidean geometry. It is sometimes referred to as neutral geometry as its axiom system is neutral to the parallel postulate. It is an extension of ordered geometry.
Abstract algebra — the study of algebraic structures and their properties. Originally it was known as modern algebra.
Abstract differential geometry — a form of differential geometry without the notion of smoothness from calculus. Instead it is built using sheaf theory and sheaf cohomology.
Abstract harmonic analysis — a modern branch of harmonic analysis that extends upon the generalized Fourier transforms that can be defined on locally compact groups.
Abstract homotopy theory
Additive combinatorics
Additive number theory — a part of number theory that studies subsets of integers and their behaviour under addition.
Affine geometry — a branch of geometry that is centered on the study of geometric properties that remain unchanged by affine transformations. It can be described as a generalization of Euclidean geometry.
Affine geometry of curves — the study of curves in affine space.
Affine differential geometry — a type of differential geometry dedicated to differential invariants under volume-preserving affine transformations.
Algebra — a major part of pure mathematics centered around operations and relations. Beginning with elementary algebra, it introduces the concept of variables and how these can be manipulated towards problem solving; known as equation solving. Generalizations of operations and relations have shown different algebraic structures for different kinds of variables. This has lead to the more modern development; abstract algebra.
Algebraic analysis — a branch that deals with systems of linear partial differential equations by using methods from sheaf theory and complex analysis to study the properties and generalizations of functions. It combines algebraic geometry and algebraic topology and was started by Mikio Sato.
Algebraic combinatorics — an area that employs methods of abstract algebra to problems of combinatorics. It also refers to the application of methods from combinatorics to problems in abstract algebra.
Algebraic computation — see symbolic computation.
Algebraic geometry — a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties.
Algebraic graph theory — a branch of graph theory in which methods are taken from algebra and employed to problems about graphs. The methods are commonly taken from group theory and linear algebra.
Algebraic K-theory
Algebraic number theory — a part of algebraic geometry devoted to the study of the points of the algebraic varieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and is also said to study algebraic structures related to algebraic integers.
Algebraic statistics — the use of algebra to advance statistics, although the term is sometimes restricted to label the use of algebraic geometry and commutative algebra in statistics.
Algebraic topology — a branch that uses tools from abstract algebra to study topological spaces.
Anabelian geometry
Analysis — a rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimal calculus. (Then it was known as infinitesimal analysis.)
Analytic combinatorics — part of enumerative combinatorics in which methods of complex analysis are applied to generating functions
Analytic geometry — this can refer to the study of geometry using a coordinate system, which is most elementary. Alternatively it can refer to the geometry of analytic varieties. In this respect it is essentialy equivalent to real and complex algebraic geometry.
Analytic number theory — part of number theory using methods of analysis (as opposed to algebraic number theory)
Applied mathematics — concerns itself with mathematical methods that can applied to practical and theoretical problems. Typically the methods used are for science, engineering, finance, economics and logistics.
Approximation theory — part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials)
Arakelov theory
Arithmetic — commonly refers to the branch known as elementary arithmetic dedicated to the usage of addition, subtraction, multiplication and division. Arithmetic also includes higher arithmetic referring to advanced results from number theory.
Arithmetic algebraic geometry — see arithmetic geometry
Arithmetic combinatorics
Arithmetic dynamics
Arithmetic geometry
Arithmetic topology
Asymptotic analysis
Asymptotic combinatorics
Arithmetical algebraic geometry — an alternative name for arithmetic algebraic geometry
Asymptotic geometric analysis
Asymptotic theory
Axiomatic geometry — see synthetic geometry.
Axiomatic set theory
[edit] B
Bifurcation theory
Birational geometry — a part of algebraic geometry that deals with the geometry of an algebraic variety that is dependent only on its function field.
Bolyai-Lobachevskian geometry — see hyperbolic geometry.
Borel–Moore homology
[edit] C
C*-algebra theory
Cartesian geometry — see analytic geometry
Calculus — a branch focused limits, functions, derivatives, integrals and infinite series. It forms the basis of classical analysis, and historically was called the calculus of infinitesimals or infinitesimal calculus. Now it can refer to a system of calculation guided by symbolic manipulation.
Calculus of infinitesimals — also known as infinitesimal calculus. It is a branch of calculus built upon the concepts of infinitesimals.
Calculus of variations
Catastrophe theory — a branch of bifurcation theory from dynamical systems theory, and also a special case of the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries.
Category theory
Chaos theory — the study of the behaviour of dynamical systems that are highly sensitive to their initial conditions.
Class field theory — a branch of algebraic number theory that studies abelian extensions of number fields.
Classical differential geometry — also known as Euclidean differential geometry. see Euclidean differential geometry.
Classical algebraic topology
Classical analysis — usually refers to the more traditional topics of analysis such as real analysis and complex analysis. It includes any work that does not use techniques from functional analysis and is sometimes called hard analysis. However it may also refer to mathematical analysis done according to the principles of classical mathematics.
Classical differential calculus
Classical Diophantine geometry
Classical Euclidean geometry — see Euclidean geometry
Classical geometry — may refer to solid geometry or classical Euclidean geometry. See geometry
Classical invariant theory — the form of invariant theory that deals with describing polynomial functions that are invariant under transformations from a given linear group.
Classical mathematics — the standard approach to mathematics based on classical logic and ZFC set theory.
Classical projective geometry
Clifford analysis
Cobordism theory
Cohomology theory
Combinatorial analysis
Combinatorial commutative algebra — a discipline viewed as the intersection between commutative algebra and combinatorics. It frequently employs methods from one to address problems arising in the other. Polyhedral geometry also plays a significant role.
Combinatorial design theory
Combinatorial game theory
Combinatorial geometry
Combinatorial group theory
Combinatorial number theory
Combinatorial set theory — also known as Infinitary combinatorics. see infinitary combinatorics
Combinatorial theory
Combinatorial topology — an old name for algebraic topology, when topological invariants of spaces were regarded as derived from combinatorial decompositions.
Combinatorics
Commutative algebra — branch of abstract algebra studying commutative rings.
Complex algebra
Complex algebraic geometry — the main stream of algebraic geometry devoted to the study of the complex points of the algebraic varieties.
Complex analysis — part of analysis that deals with functions of a complex variable.
Complex analytic geometry
Complex differential geometry — a branch of differential geometry; the study of complex manifolds.
Complex geometry — the study of complex manifolds and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry.
Complexity theory
Computable analysis
Computable model theory
Computability theory
Computational algebraic geometry
Computational complexity theory
Computational geometry
Computational group theory
Computational mathematics
Computational number theory
Computational real algebraic geometry
Computational synthetic geometry
Computational topology
Computer algebra — see symbolic computation
Conformal geometry
Conformal geometric algebra
Constructive analysis — mathematical analysis done according to the principles of constructive mathematics. This differs from classical analysis.
Constructive function theory
Constructive mathematics — mathematics which tends to use intuitionistic logic. Essentially that is classical logic but without the assumption that the law of the excluded middle is an axiom.
Constructive quantum field theory
Contact geometry — a branch of differential geometry and topology, closely related to and considered the odd-dimensional counterpart of symplectic geometry.
Convex analysis
Convex geometry — part of geometry devoted to the study of convex sets.
Coordinate geometry — see analytic geometry
CR geometry — a branch of differential geometry, being the study of CR manifolds.
[edit] D
Descriptive set theory
Differential algebraic geometry — adapts methods and concepts from algebraic geometry to systems of algebraic differential equations.
Differential calculus — a subfield of calculus concerned with derivatives or the rates that quantities change. It is one of two traditional divisions of calculus, the other being integral calculus.
Differential Galois theory — studies the Galois groups of differential fields.
Differential geometry — uses techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on differentiable manifolds. It is closely related to differential topology.
Differential geometry of curves
Differential geometry of surfaces
Differential topology — a branch of topology that deals with differentiable functions on differentiable manifolds.
Diffiety theory
Diophantine geometry
Discrepancy theory
Discrete computational geometry
Discrete differential geometry
Discrete dynamics
Discrete exterior calculus
Discrete geometry
Discrete mathematics
Discrete Morse theory — a combinatorial adaption of Morse theory.
Distance geometry
Domain theory
Dynamical systems theory
[edit] E
Econometrics — the application of mathematical and statistical methods to economic data.
Elementary algebra — a fundamental form of algebra extending on elementary arithmetic to include the concept of variables.
Elementary arithmetic — the simplified portion of arithmetic considered necessary for primary education. It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includes the concept of fractions and negative numbers.
Elementary mathematics
Elementary group theory
Elimination theory
Elliptic geometry — a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is constructed in elliptic space.
Enumerative combinatorics — an area of combinatorics that deals with the number of ways that certain patterns can be formed.
Enumerative geometry
Equivariant noncommutative algebraic geometry
Ergodic Ramsey theory — a branch where problems are motivated by additive combinatorics and solved using ergodic theory.
Ergodic theory — the study of dynamical systems with an invariant measure, and related problems.
Euclidean geometry
Euclidean differential geometry — also known as classical differential geometry. See differential geometry.
Euler calculus
Experimental mathematics
Extraordinary cohomology theory
Extremal combinatorics — a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.
Extremal graph theory
[edit] F
Field theory — branch of algebra studying fields.
Finite geometry
Finite model theory
Finsler geometry — a branch of differential geometry whose main object of study is the Finsler manifold (a generalisation of a Riemannian manifold).
First order arithmetic
Fourier analysis
Fractional calculus — a branch of analysis that studies the possibility of taking real or complex powers of the differentiation operator.
Fractional dynamics — investigates the behaviour of objects and systems that are described by differentiation and integration of fractional orders using methods of fractional calculus.
Fredholm theory — part of spectral theory studying integral equations.
Function theory — part of analysis devoted to properties of functions, especially functions of a complex variable (see complex analysis).
Functional analysis
Functional calculus
Fuzzy arithmetic
Fuzzy geometry
Fuzzy Galois theory
Fuzzy mathematics — a branch of mathematics based on fuzzy set theory and fuzzy logic.
Fuzzy measure theory
Fuzzy qualitative trigonometry
Fuzzy set theory — a form of set theory that studies fuzzy sets, that is sets that have degrees of membership.
Fuzzy topology
[edit] G
Galois cohomology — an application of homological algebra, it is the study of group cohomology of Galois modules.
Galois theory — named after variste Galois, it is a branch of abstract algebra providing a connection between field theory and group theory.
Galois geometry
Game theory
Gauge theory
General topology — also known as point-set topology, it is a branch of topology studying the properties of topological spaces and structures defined on them. It differs from other branches of topology as the topological spaces do not have to be similar to manifolds.
Generalized trigonometry — developments of trigonometric methods from the application to real numbers of Euclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, rational trigonometry, universal hyperbolic trigonometry, fuzzy qualitative trigonometry, operator trigonometry and lattice trigonometry.
Geometric algebra — an alternative approach to classical, computational and relativistic geometry. It shows a natural correspondence between geometric entities and elements of algebra.
Geometric analysis — a discipline that uses methods from differential geometry to study partial differential equations as well as the applications to geometry.
Geometric calculus
Geometric combinatorics
Geometric invariant theory
Geometric graph theory
Geometric group theory
Geometric measure theory
Geometric topology — a branch of topology studying manifolds and mappings between them; in particular the embedding of one manifold into another.
Geometry— a branch of mathematics concerned with shape and the properties of space. Classically it arose as what is now known as solid geometry; this was concerning practical knowledge of length, area and volume. It was then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry. The use of coordinates by Ren Descartes gave rise to Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including projective geometry, differential geometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of topology.
Geometry of numbers — initiated by Hermann Minkowski, it is a branch of number theory studying convex bodies and integer vectors.
Global analysis — the study of differential equations on manifolds and the relationship between differential equations and topology.
Global arithmetic dynamics
Graph theory — a branch of discrete mathematics devoted to the study of graphs. It has many applications in physical, biological and social systems.
Group representation theory
Group theory
Gyrotrigonometry — a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vector space in Euclidean geometry.)
[edit] H
Hard analysis — see classical analysis
Harmonic analysis — part of analysis concerned with the representations of functions in terms of waves. It generalizes the notions of Fourier series and Fourier transforms from the Fourier analysis.
High-dimensional topology
Higher arithmetic
Higher category theory
Hodge theory
Homological algebra — the study of homology in general algebraic settings.
Homology theory
Homotopy theory
Hyperbolic geometry — also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is a non-Euclidean geometry looking at hyperbolic space.
hyperbolic trigonometry — the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.
Hypercomplex analysis
Hyperfunction theory
[edit] I
Ideal theory — once the precursor name for commutative algebra, it is the theory of ideals in commutative rings.
Idempotent analysis
Incidence geometry
Inconsistent mathematics — see paraconsistent mathematics.
Infinitary combinatorics — an expansion of ideas in combinatorics to account for infinite sets.
Infinitesimal analysis — once a synonym for infinitesimal calculus
Infinitesimal calculus — see calculus of infinitesimals
Information geometry
Integral calculus
Integral geometry
Intersection theory
Intuitionistic type theory
Invariant theory — studies how group actions on algebraic varieties affect functions.
Inversive geometry
Inversive ring geometry
It calculus
Iwasawa theory
[edit] J
[edit] K
K-theory — originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II string theory. (In particular twisted K-theory.)
K-homology
Khler geometry — a branch of differential geometry, more specifically a union of Riemannian geometry, complex differential geometry and symplectic geometry. It is the study of Khler manifolds. (named after Erich Khler)
KK-theory
Klein geometry
Knot theory — part of topology dealing with knots
[edit] L
L-theory
Lattice theory
Large deviations theory — part of probability theory studying events of small probability (tail events).
Lattice trigonometry
Lie algebra theory
Lie group theory
Lie sphere geometry
Lie theory
Line geometry
Linear algebra – a branch of algebra studying linear spaces and linear maps. It has applications in fields such as abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized in operator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used are in multilinear algebra.
Linear functional analysis
Local algebra — sometimes applied to the theory of local rings.
Local arithmetic dynamics — also known as p-adic dynamics or nonarchimedean dynamics.
Local class field theory
Low-dimensional topology
[edit] M
Malliavin calculus
Mathematical logic
Mathematical physics — a part of mathematics that develops mathematical methods motivated by problems in physics.
Mathematical sciences — refers to academic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarial science.
Matrix algebra
Matrix calculus
Matrix theory
Matroid theory
Measure theory
Metric geometry
Microlocal analysis
Model theory
Modern algebra — see abstract algebra
Modern algebraic geometry — the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.
Modern invariant theory — the form of invariant theory that analyses the decomposition of representations into irreducibles.
Modular representation theory
Module theory
Molecular geometry
Morse theory — a part of differential topology, it analyzes the topological space of a manifold by studying differentiable functions on that manifold.
Motivic cohomology
Multilinear algebra — an extension of linear algebra building upon concepts of p-vectors and multivectors with Grassman algebra.
Multivariable calculus
Multiplicative number theory — a subfield of analytic number theory that deals with prime numbers, factorization and divisors.
Multiple-scale analysis
Multiplicative calculus
[edit] N
Neutral geometry — see absolute geometry
Nevanlinna theory — part of complex analysis studying the value distribution of meromorphic functions
Non-classical analysis
Non-Euclidean geometry
Non-standard analysis
Non-standard calculus
Nonarchimedean dynamics — also known as p-adic analysis or local arithmetic dynamics
Noncommutative algebraic geometry — a direction in noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects.
Noncommutative geometry
Noncommutative harmonic analysis — see representation theory
Noncommutative topology
Number theory — a branch of pure mathematics devoted primarily to the study of the integers. Originally it was known as arithmetic or higher arithmetic.
Numerical analysis
Numerical geometry
Numerical linear algebra
[edit] O
Operad theory — a type of abstract algebra concerned with prototypical algebras.
Operator geometry
Operator K-theory
Operator theory — part of functional analysis studying operators.
Operator trigonometry
Orbifold theory
Order theory — a branch that investigates the intuitive notion of order using binary relations.
Ordered geometry — a form of geometry omitting the notion of measurement but featuring the concept of intermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclidean geometry, absolute geometry and hyperbolic geometry.
Oriented elliptic geometry
Oriented spherical geometry
[edit] P
p-adic analysis — a branch of number theory that deals with the analysis of functions of p-adic numbers.
p-adic dynamics — an application of p-adic analysis looking at p-adic differential equations.
Parabolic geometry
Paraconsistent mathematics — sometimes called inconsistent mathematics, it is an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic.
Partition theory
Perturbation theory
Picard–Vessiot theory
Plane geometry
Point-set topology — see general topology
Pointless topology
Polyhedral combinatorics — a branch within combinatorics and discrete geometry that studies the problems of describing convex polytopes.
Polyhedral geometry
Possibility theory
Potential theory
Precalculus
Predicative mathematics
Probability theory
Probabilistic combinatorics
Probabilistic graph theory
Probabilistic number theory
Projective geometry — a form of geometry that studies geometric properties that are invariant under a projective transformation.
Projective differential geometry
Proof theory
Pseudo-Riemannian geometry — generalizes Riemannian geometry to the study of pseudo-Riemannian manifolds.
Pure mathematics — the part of mathematics that studies entirely abstract concepts.
[edit] Q
Quaternionic analysis
[edit] R
Ramsey theory — studies the conditions in which order must appear. It is named after Frank P. Ramsey.
Rational trigonometry — an reformulation of trigonometry in terms of spread and quadrance instead of angle and length.
Recreational mathematics
Recursion theory
Representation theory — a subfield of abstract algebra; it studies algebraic structures by representing their elements as linear transformations of vector spaces. It also studies modules over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.
Representation theory of groups
Real algebra — part of algebra relevant to real algebraic geometry.
Real algebraic geometry — the part of algebraic geometry that studies real points of the algebraic varieties.
Real analysis
Real analytic geometry
Real K-theory
Ribbon theory — a branch of topology studying ribbons.
Riemannian geometry — a branch of differential geometry that is more specifically, the study of Riemannian manifolds. It is named after Bernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus.
[edit] S
Scheme theory — the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaf theory to study algebraic varieties and is considered the central part of modern algebraic geometry.
Secondary calculus
Semialgebraic geometry — a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets.
Set-theoretic topology
Set theory
Sheaf theory
Sheaf cohomology
Sieve theory
Single operator theory — deals with the properties and classifications of single operators.
Singularity theory — a branch, notably of geometry; that studies the failure of manifold structure.
Smooth infinitesimal analysis — a rigorous reformation of infinitesimal calculus employing methods of category theory. As a theory, it is a subset of synthetic differential geometry.
Solid geometry
Spatial geometry
Spectral geometry — a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators.
Spectral graph theory — the study of properties of a graph using methods from matrix theory.
Spectral theory — part of operator theory extending the concepts of eigenvalues and eigenvectors from linear algebra and matrix theory.
Spectral theory of ordinary differential equations — part of spectral theory concerned with the spectrum and eigenfunction expansion associated with linear ordinary differential equations.
Spectrum continuation analysis — generalizes the concept of a Fourier series to non-periodic functions.
Spherical geometry — a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.
Spherical trigonometry — a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles.
Statistics — although the term may refer to the more general study of statistics, the term is used in mathematics to refer to the mathematical study of statistics and related fields. This includes probability theory.
Stochastic calculus
Stocahstic calculus of variations
Stochastic geometry — the study of random patterns of points
Stratified Morse theory
Super category theory
Super linear algebra
Surgery theory — a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.)
Symplectic geometry — a branch of differential geometry and topology whose main object of study is the symplectic manifold.
Systolic geometry — a branch of differential geometry studying systolic invariants of manifolds and polyhedra.
Systolic hyperbolic geometry — the study of systoles in hyperbolic geometry.
Symbolic computation — also known as algebraic computation and computer algebra. It refers to the techniques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulating them by the numerical quantities represented by them.
Symbolic dynamics
Synthetic differential geometry — a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic
Synthetic geometry — also known as axiomatic geometry, it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods.
[edit] T
Tensor analysis — the study of tensors, which play a role in differential geometry, mathematical physics, algebraic topology, multilinear algebra, homological algebra and representation theory.
Tensor calculus — an older term for tensor analysis.
Tensor theory — an alternative name for tensor analysis.
Theoretical physics — a branch primariliy of the science physics that uses mathematical models and abstraction of physics to rationalize and predict phenomena.
Time-scale calculus
Topology
Topological combinatorics — the application of methods from algebraic topology to solve problems in combinatorics.
Topological graph theory
Topological K-theory
Topos theory
Toric geometry
Transcendence theory — a branch of number theory that revolves around the transcendental numbers.
Transfinite order theory
Transformation geometry
Trigonometry — the study of triangles and the relationships between the length of their sides, and the angles between them. It is essential to many parts of applied mathematics.
Tropical analysis — see idempotent analysis
Tropical geometry
Twisted K-theory — a variation on K-theory, spanning abstract algebra, algebraic topology and operator theory.
Type theory
[edit] U
Umbral calculus
Universal algebra — a field studying algebraic structures themselves.
Universal hyperbolic trigonometry — an approach to hyperbolic trigonometry based on rational geometry.
[edit] V
Valuation theory
Variational analysis
Vector algebra — a part of linear algebra concerned with the operations of vector addition and scalar multiplication, although it may also refer to vector operations of vector calculus, including the dot and cross product. In this case it can be contrasted with geometric algebra which generalizes into higher dimensions.
Vector analysis — also known as vector calculus, see vector calculus.
Vector calculus — a branch of multivariable calculus concerned with differentiation and integration of vector fields. Primarily it is concerned with 3 dimensional Euclidean space.
[edit] W
