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123-402: In mathematics and physics , a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold ) or of the physical space . Tensor fields are used in differential geometry , algebraic geometry , general relativity , in the analysis of stress and strain in material object, and in numerous applications in

246-428: A ′ = h 2 + k 2 + 2 h k sin ⁡ θ ′ , b ′ = h 2 + k 2 − 2 h k sin ⁡ θ ′ a = a ′ + b ′ 2 , b =

369-1262: A ′ − b ′ 2 s = h k sin ⁡ θ ′ ω = 2 arcsin ⁡ b ′ a ′ {\displaystyle {\begin{aligned}h&={\frac {1}{R}}{\sqrt {{{\left({\frac {\partial x}{\partial \varphi }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \varphi }}\right)}^{2}}}}\\[4pt]k&={\frac {1}{R\cos \varphi }}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\[4pt]\sin \theta '&={\frac {1}{R^{2}hk\cos \varphi }}\left({{\frac {\partial y}{\partial \varphi }}{\frac {\partial x}{\partial \lambda }}-{\frac {\partial x}{\partial \varphi }}{\frac {\partial y}{\partial \lambda }}}\right)\\[4pt]a'&={\sqrt {{h^{2}}+{k^{2}}+2hk\sin \theta '}},\quad b'={\sqrt {{h^{2}}+{k^{2}}-2hk\sin \theta '}}\\[4pt]a&={\frac {a'+b'}{2}},\quad b={\frac {a'-b'}{2}}\\[4pt]s&=hk\sin \theta '\\[4pt]\omega &=2\arcsin {\frac {b'}{a'}}\end{aligned}}} where φ {\displaystyle \varphi } and λ {\displaystyle \lambda } are

492-475: A module over the ring of smooth functions, C ( M ), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any commutative ring . As a motivating example, consider the space Ω 1 ( M ) = T 1 0 ( M ) {\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)} of smooth covector fields ( 1-forms ), also

615-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

738-539: A coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates. Following Schouten (1951) and McConnell (1957) , the concept of a tensor relies on a concept of a reference frame (or coordinate system ), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems. For example, coordinates belonging to

861-411: A different type (although this is not usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C) vector fields on M , X ( M ) := T 0 1 ( M ) {\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)} (see the section on notation above) as a single space —

984-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

1107-439: A fresh notion, the covariant derivative . This handles the formulation of variation of a tensor field along a vector field . The original absolute differential calculus notion, which was later called tensor calculus , led to the isolation of the geometric concept of connection . An extension of the tensor field idea incorporates an extra line bundle L on M . If W is the tensor product bundle of V with L , then W

1230-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

1353-470: A function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously. A frequent example application of this general rule is showing that the Levi-Civita connection , which is a mapping of smooth vector fields ( X , Y ) ↦ ∇ X Y {\displaystyle (X,Y)\mapsto \nabla _{X}Y} taking

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1476-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

1599-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

1722-512: A module M , we call A a tensor field on M . Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold , which is a topological space . Let M be a manifold , for instance the Euclidean plane R . Definition. A tensor field of type ( p , q )

1845-505: A module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ω and a vector field X , we define Because of the pointwise nature of everything involved, the action of ω ~ {\displaystyle {\tilde {\omega }}} on X is a C ( M )-linear map, that is, for any p in M and smooth function f . Thus we can regard covector fields not just as sections of

1968-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

2091-437: A more traditional explanation see the tensor density article. One feature of the bundle of densities (again assuming orientability) L is that L is well-defined for real number values of s ; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a half-density , the case where s = ⁠ 1 / 2 ⁠ . In general we can take sections of W ,

2214-557: A pair of vector fields to a vector field, does not define a tensor field on M . This is because it is only R {\displaystyle \mathbb {R} } -linear in Y (in place of full C ( M )-linearity, it satisfies the Leibniz rule, ∇ X ( f Y ) = ( X f ) Y + f ∇ X Y {\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y} )). Nevertheless, it must be stressed that even though it

2337-484: A plane. Linear scale has not been preserved in this projection, as O A ′ ≆ O A {\displaystyle {OA'\ncong OA}} and O B ′ ≆ O B {\displaystyle OB'\ncong OB} . Because ∠ M ′ O A ′ ≆ ∠ M O A {\displaystyle {\angle M'OA'\ncong \angle MOA}} , we know that there

2460-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

2583-555: A product of k copies of X ( M ) {\displaystyle {\mathfrak {X}}(M)} and l copies of Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} into C ( M ), it turns out that it arises from a tensor field on M if and only if it is multilinear over C ( M ). Namely C ∞ ( M ) {\displaystyle C^{\infty }(M)} -module of tensor fields of type ( k , l ) {\displaystyle (k,l)} over M

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2706-412: A radius of 1, but when dealing with a Tissot indicatrix, one deals with ellipses of infinitesimal radius. Even though the radii of the original circle and its distortion ellipse will all be infinitesimal, by employing differential calculus the ratios between them can still be meaningfully calculated. For example, if the ratio between the radius of the input circle and a projected circle is equal to 1, then

2829-450: A rotation during projection. For a given point, it is common in the literature to represent the scale along the meridian as h {\displaystyle h} and the scale along the parallel as k {\displaystyle k} . Unless the projection is conformal, all angles except the one subtended by the semi-major axis and semi-minor axis of the ellipse may have changed as well. A particular angle will have changed

2952-484: A routine way – again independently of coordinates, as mentioned in the introduction. We therefore can give a definition of tensor field , namely as a section of some tensor bundle . (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of

3075-408: A scaling along the basis, Λ {\displaystyle \Lambda } , and a subsequent second rotation, U {\displaystyle U} . For understanding distortion, the first rotation is irrelevant, as it rotates the axes of the circle but has no bearing on the final orientation of the ellipse. The next operation, represented by the diagonal singular value matrix, scales

3198-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

3321-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

3444-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

3567-438: A system of functions v k {\displaystyle v^{k}} of the coordinates that, under such an affine transformation undergoes a transformation This is precisely the requirement needed to ensure that the quantity v k e k {\displaystyle v^{k}\mathbf {e} _{k}} is an invariant object that does not depend on the coordinate system chosen. More generally,

3690-408: A tensor of valence ( p , q ) has p downstairs indices and q upstairs indices, with the transformation law being The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable , analytic , etc.). A covector field is a function v k {\displaystyle v_{k}} of the coordinates that transforms by

3813-476: A tensor section is not only a linear map on the vector space of sections, but a C ( M )-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus,

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3936-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

4059-486: Is g x ( v , w ) {\displaystyle g_{x}(v,w)} . The field g {\displaystyle g} could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is Tissot's indicatrix . In general, we want to specify tensor fields in

4182-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

4305-408: Is flat " and "a field is always a ring ". Tissot%27s indicatrix A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians and parallels. These schematics are important in

4428-413: Is a vector space and the tensor bundle is a special kind of vector bundle . The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold M . For example, a vector space of one dimension depending on an angle could look like a Möbius strip or alternatively like a cylinder . Given a vector bundle V over M ,

4551-404: Is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the geometric nature of tensor quantities ; this kind of descent argument justifies abstractly the whole theory. The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it

4674-473: Is a bundle of vector spaces of just the same dimension as V . This allows one to define the concept of tensor density , a 'twisted' type of tensor field. A tensor density is the special case where L is the bundle of densities on a manifold , namely the determinant bundle of the cotangent bundle . (To be strictly accurate, one should also apply the absolute value to the transition functions – this makes little difference for an orientable manifold .) For

4797-686: Is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition: which is the transformation law for a scalar density of weight +2. Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

4920-416: Is a function of the equatorial radius, R {\displaystyle R} , and eccentricity, e {\displaystyle e} : The element of distance on the sphere, d s {\displaystyle ds} is defined by the first fundamental form : whose coefficients are defined as: Computing the necessary derivatives gives: where M {\displaystyle M}

5043-419: Is a function of the equatorial radius, R {\displaystyle R} , and the ellipsoid eccentricity, e {\displaystyle e} : Substituting these values into the first fundamental form gives the formula for elemental distance on the ellipsoid: This result relates the measure of distance on the ellipsoid surface as a function of the spherical coordinate system. Recall that

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5166-410: Is a section where V is a vector bundle on M , V is its dual and ⊗ is the tensor product of vector bundles. Equivalently, it is a collection of elements T x ∈ V x ⊗ ( V x ) for all points x ∈ M , arranging into a smooth map T : M → V ⊗ ( V ). Elements T x are called tensors . Often we take V = TM to be the tangent bundle of M . Intuitively, a vector field

5289-599: Is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the w th power, is called a tensor density with weight w . Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n -forms (where n is the dimension of the space), as opposed to taking their values in just R . Higher "weights" then just correspond to taking additional tensor products with this space in

5412-420: Is an angular distortion. Because Area ⁡ ( A ′ B ′ C ′ D ′ ) ≠ Area ⁡ ( A B C D ) {\displaystyle \operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)} , we know there is an areal distortion. [REDACTED] The original circle in the above example had

5535-566: Is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface. Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field g {\displaystyle g} , such that given any two vectors v , w {\displaystyle v,w} at point x {\displaystyle x} , their inner product

5658-403: Is canonically isomorphic to C ∞ ( M ) {\displaystyle C^{\infty }(M)} -module of C ∞ ( M ) {\displaystyle C^{\infty }(M)} - multilinear forms This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to

5781-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

5904-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

6027-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

6150-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

6273-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

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6396-414: Is not a tensor field, it still qualifies as a geometric object with a component-free interpretation. The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity . In electromagnetism , the electric and magnetic fields are combined into an electromagnetic tensor field . It

6519-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

6642-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

6765-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

6888-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

7011-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

7134-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

7257-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

7380-576: Is through the differential geometry of surfaces. This approach lends itself well to modern numerical methods, as the parameters of Tissot's indicatrix can be computed using singular value decomposition (SVD) and central difference approximation . Let a 3D point, X ^ {\displaystyle {\hat {X}}} , on an ellipsoid be parameterized as: where ( λ , ϕ ) {\displaystyle (\lambda ,\phi )} are longitude and latitude, respectively, and N {\displaystyle N}

7503-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

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7626-431: Is worth noting that differential forms , used in defining integration on manifolds, are a type of tensor field. In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus . Even to formulate such equations requires

7749-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

7872-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

7995-408: The Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field v k {\displaystyle v^{k}} transforms by the inverse Jacobian. A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber

8118-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

8241-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

8364-455: The chain rule in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields. Abstractly, we can identify the chain rule as a 1- cocycle . It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying functorial properties of tensor constructions to

8487-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

8610-406: The n -dimensional real coordinate space R n {\displaystyle \mathbb {R} ^{n}} may be subjected to arbitrary affine transformations : (with n -dimensional indices, summation implied ). A covariant vector, or covector, is a system of functions v k {\displaystyle v_{k}} that transforms under this affine transformation by

8733-401: The physical sciences . As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over

8856-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

8979-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

9102-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

9225-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

9348-476: The Tissot indicatrix and the metric tensor of the map projection coordinate conversion. Tissot's theory was developed in the context of cartographic analysis . Generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid . Tissot's indicatrices illustrate linear, angular, and areal distortions of maps: In conformal maps, where each point preserves angles projected from

9471-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

9594-441: The chain rule itself; this is why they also are intrinsic (read, 'natural') concepts. What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, post hoc approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities

9717-422: The circle along its axes, deforming it to an ellipse. Thus, the singular values represent the scale factors along axes of the ellipse. The first singular value provides the semi-major axis, a {\displaystyle a} , and the second provides the semi-minor axis, b {\displaystyle b} , which are the directional scaling factors of distortion. Scale distortion can be computed as

9840-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

9963-458: The corresponding field concept is called a section of the bundle: for m varying over M , a choice of vector where V m is the vector space "at" m . Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces ) the whole apparatus explained at component-free treatment of tensors carries over in

10086-794: The cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there). In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on M as multilinear maps on vectors and covectors, we can regard general ( k , l ) tensor fields on M as C ( M )-multilinear maps defined on k copies of X ( M ) {\displaystyle {\mathfrak {X}}(M)} and l copies of Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} into C ( M ). Now, given any arbitrary mapping T from

10209-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

10332-411: The definition of the transform T {\displaystyle {\mathcal {T}}} represented by the indicatrix: This transform T {\displaystyle {\mathcal {T}}} encapsulates the mapping from the ellipsoid surface to the plane. Expressed in this form, SVD can be used to parcel out the important components of the local transformation. In order to extract

10455-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

10578-513: The desired distortion information, at any given location in the spherical coordinate system, the values of K {\displaystyle K} can be computed directly. The Jacobian, J {\displaystyle J} , can be computed analytically from the mapping function itself, but it is often simpler to numerically approximate the values at any location on the map using central differences . Once these values are computed, SVD can be applied to each transformation matrix to extract

10701-480: The ellipse degenerates into a circle, with the radius being equal to the scale factor. For equal-area such as the sinusoidal projection , the semi-major axis of the ellipse is the reciprocal of the semi-minor axis, such that every ellipse has equal area even as their eccentricities vary. For arbitrary projections, the shape and the area of the ellipses at each point are largely independent from one another. Another way to understand and derive Tissot's indicatrix

10824-517: The equation above, yielding: For the purposes of this computation, it is useful to express this relationship as a matrix operation: Now, in order to relate the distances on the ellipsoid surface to those on the plane, we need to relate the coordinate systems. From the chain rule, we can write: where J is the Jacobian matrix : Plugging in the matrix expression for d λ {\displaystyle d\lambda } and d ϕ {\displaystyle d\phi } yields

10947-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

11070-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

11193-452: The geometric model, the Tissot's indicatrices are all circles of size varying by location, possibly also with varying orientation (given the four circle quadrants split by meridians and parallels ). In equal-area projections , where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across

11316-420: The indicatrix is drawn with as a circle with an area of 1. The size that the indicatrix gets drawn on the map is arbitrary: they are all scaled by the same factor so that their sizes are proportional to one another. Like M {\displaystyle M} in the diagram, the axes from O {\displaystyle O} along the parallel and along the meridian may undergo a change of length and

11439-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

11562-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

11685-417: The latitude and longitude coordinates of a point, R {\displaystyle R} is the radius of the globe, and x {\displaystyle x} and y {\displaystyle y} are the point's resulting coordinates after projection. In the result for any given point, a {\displaystyle a} and b {\displaystyle b} are

11808-417: The local distortion information. Remember that, because distortion is local, every location on the map will have its own transformation. Recall the definition of SVD: It is the decomposition of the transformation, T {\displaystyle {\mathcal {T}}} , into a rotation in the source domain (i.e. the ellipsoid surface), V T {\displaystyle V^{T}} ,

11931-594: The manifold a tensor in the space where V is the tangent space at that point and V is the cotangent space . See also tangent bundle and cotangent bundle . Given two tensor bundles E → M and F → M , a linear map A : Γ( E ) → Γ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of E ∗ ⊗ F {\displaystyle \scriptstyle E^{*}\otimes F} if and only if it satisfies A ( fs ) = fA ( s ), for each section s in Γ( E ) and each smooth function f on M . Thus

12054-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

12177-442: The map. In the diagram below, the circle A B C D {\displaystyle ABCD} has unit area as defined on the surface of a sphere. The circle A ′ B ′ C ′ D ′ {\displaystyle {A'B'C'D'}} is the Tissot's indicatrix that results from some projection of A B C D {\displaystyle ABCD} onto

12300-623: The maximum and minimum scale factors, analogous to the semimajor and semiminor axes in the diagram; s {\displaystyle s} represents the amount of inflation or deflation in area, and ω {\displaystyle \omega } represents the maximum angular distortion. For conformal projections such as the Mercator projection , h = k {\displaystyle h=k} and θ = π 2 {\displaystyle \theta ={\pi \over 2}} , such that at each point

12423-1477: The most, and the value of that maximum change is known as the angular deformation, denoted as θ {\displaystyle \theta } . In general, which angle that is and how it is oriented do not figure prominently into distortion analysis; it is the magnitude of the change that is significant. The values of h {\displaystyle h} , k {\displaystyle k} , and θ {\displaystyle \theta } can be computed as follows: h = 1 R ( ∂ x ∂ φ ) 2 + ( ∂ y ∂ φ ) 2 k = 1 R cos ⁡ φ ( ∂ x ∂ λ ) 2 + ( ∂ y ∂ λ ) 2 sin ⁡ θ ′ = 1 R 2 h k cos ⁡ φ ( ∂ y ∂ φ ∂ x ∂ λ − ∂ x ∂ φ ∂ y ∂ λ )

12546-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

12669-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

12792-414: The origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it does make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion. As an advanced explanation of the tensor concept, one can interpret

12915-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

13038-639: The planar map. This can be expressed by the relation: where d s ( λ , 0 ) {\displaystyle ds(\lambda ,0)} and d s ( 0 , ϕ ) {\displaystyle ds(0,\phi )} represent the computation of d s {\displaystyle ds} along the longitudinal and latitudinal axes, respectively. Computation of d s ( λ , 0 ) {\displaystyle ds(\lambda ,0)} and d s ( 0 , ϕ ) {\displaystyle ds(0,\phi )} can be performed directly from

13161-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

13284-456: The purpose of Tissot's indicatrix is to relate how distances on the sphere change when mapped to a planar surface. Specifically, the desired relation is the transform T {\displaystyle {\mathcal {T}}} that relates differential distance along the bases of the spherical coordinate system to differential distance along the bases of the Cartesian coordinate system on

13407-582: The range. A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the Einstein–Hilbert action in general relativity. The most common example of a scalar 1-density is the volume element , which in the presence of a metric tensor g is the square root of its determinant in coordinates, denoted det g {\displaystyle {\sqrt {\det g}}} . The metric tensor

13530-400: The rule The list of Cartesian coordinate basis vectors e k {\displaystyle \mathbf {e} _{k}} transforms as a covector, since under the affine transformation e k ↦ A k i e i {\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}} . A contravariant vector is

13653-402: The set of infinitely-differentiable tensor fields on M . Thus, are the sections of the ( m , n ) tensor bundle on M that are infinitely-differentiable. A tensor field is an element of this set. There is another more abstract (but often useful) way of characterizing tensor fields on a manifold M , which makes tensor fields into honest tensors (i.e. single multilinear mappings), though of

13776-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

13899-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

14022-415: The study of map projections, both to illustrate distortion and to provide the basis for the calculations that represent the magnitude of distortion precisely at each point. Because the infinitesimal circles represented by the ellipses on the map all have the same area on the underlying curved geometric model, the distortion imposed by the map projection is evident. There is a one-to-one correspondence between

14145-456: The tangent bundle TM = T ( M ) might sometimes be written as to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M . This should not be confused with the very similar looking notation in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M . Curly (script) letters are sometimes used to denote

14268-417: The tensor product of V with L , and consider tensor density fields with weight s . Half-densities are applied in areas such as defining integral operators on manifolds, and geometric quantization . When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M , we get back to a situation where a tensor field is synonymous with a tensor 'sitting at

14391-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

14514-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

14637-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

14760-406: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

14883-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

15006-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

15129-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

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