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87-461: In geometry , curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates , coined by

174-412: A Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity

261-435: A basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},}

348-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

435-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

522-426: A ring R , which are zero except for a finite number of indices , if we interpret 1 as 1 R , the unit in R . The existence of other 'standard' bases has become a topic of interest in algebraic geometry , beginning with work of Hodge from 1943 on Grassmannians . It is now a part of representation theory called standard monomial theory . The idea of standard basis in the universal enveloping algebra of

609-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

696-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

783-613: A coordinate system by two groups of basis vectors: Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates. Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: { b 1 , b 2 , b 3 } is the contravariant basis, and { b , b , b } is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other. Note

870-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

957-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

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1044-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

1131-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

1218-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

1305-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

1392-474: A problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in

1479-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

1566-798: A rectangular box using Cartesian coordinates, it is easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences , cartography , quantum mechanics , relativity , and engineering . For now, consider 3-D space . A point P in 3-D space (or its position vector r ) can be defined using Cartesian coordinates ( x , y , z ) [equivalently written ( x , x , x )], by r = x e x + y e y + z e z {\displaystyle \mathbf {r} =x\mathbf {e} _{x}+y\mathbf {e} _{y}+z\mathbf {e} _{z}} , where e x , e y , e z are

1653-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

1740-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

1827-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

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1914-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

2001-484: Is b 1 (notated h 1 above, with b reserved for unit vectors) and it is built on the q axis which is a tangent to that coordinate line at the point P . The axis q and thus the vector b 1 form an angle α {\displaystyle \alpha } with the Cartesian x axis and the Cartesian basis vector e 1 . It can be seen from triangle PAB that where | e 1 |, | b 1 | are

2088-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

2175-471: Is invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner). Consider the one-dimensional curve shown in Fig. 3. At point P , taken as an origin , x is one of the Cartesian coordinates, and q is one of the curvilinear coordinates. The local (non-unit) basis vector

2262-563: Is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

2349-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

2436-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

2523-507: Is any set and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta , equal to zero whenever i ≠ j and equal to 1 if i = j . This family is the canonical basis of the R -module ( free module ) R ( I ) {\displaystyle R^{(I)}} of all families f = ( f i ) {\displaystyle f=(f_{i})} from I into

2610-1070: Is commonly called monomial basis . For matrices M m × n {\displaystyle {\mathcal {M}}_{m\times n}} , the standard basis consists of the m × n -matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices e 11 = ( 1 0 0 0 ) , e 12 = ( 0 1 0 0 ) , e 21 = ( 0 0 1 0 ) , e 22 = ( 0 0 0 1 ) . {\displaystyle \mathbf {e} _{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\quad \mathbf {e} _{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {e} _{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad \mathbf {e} _{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.} By definition,

2697-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

Curvilinear coordinates - Misplaced Pages Continue

2784-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

2871-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

2958-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

3045-488: Is the metric tensor (see below). A vector can be specified with covariant coordinates (lowered indices, written v k ) or contravariant coordinates (raised indices, written v ). From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors. A key feature of the representation of vectors and tensors in terms of indexed components and basis vectors

3132-511: The Lamé coefficients (after Gabriel Lamé ) by and the curvilinear orthonormal basis vectors by These basis vectors may well depend upon the position of P ; it is therefore necessary that they are not assumed to be constant over a region. (They technically form a basis for the tangent bundle of R 3 {\displaystyle \mathbb {R} ^{3}} at P , and so are local to P .) In general, curvilinear coordinates allow

3219-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

3306-414: The coordinate curves . The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates. In the Cartesian system, the standard basis vectors can be derived from

3393-443: The standard basis vectors . It can also be defined by its curvilinear coordinates ( q , q , q ) if this triplet of numbers defines a single point in an unambiguous way. The relation between the coordinates is then given by the invertible transformation functions: The surfaces q = constant, q = constant, q = constant are called the coordinate surfaces ; and the space curves formed by their intersection in pairs are called

3480-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

3567-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

Curvilinear coordinates - Misplaced Pages Continue

3654-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

3741-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

3828-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

3915-401: The gradient , divergence , curl , and Laplacian ) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system. A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under

4002-619: The scalars v x {\displaystyle v_{x}} ,  v y {\displaystyle v_{y}} ,  v z {\displaystyle v_{z}} being the scalar components of the vector v . In the n - dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the standard basis consists of n distinct vectors { e i : 1 ≤ i ≤ n } , {\displaystyle \{\mathbf {e} _{i}:1\leq i\leq n\},} where e i denotes

4089-437: The y direction, and the vector e z points in the z direction. There are several common notations for standard-basis vectors, including { e x ,  e y ,  e z }, { e 1 ,  e 2 ,  e 3 }, { i ,  j ,  k }, and { x ,  y ,  z }. These vectors are sometimes written with a hat to emphasize their status as unit vectors ( standard unit vectors ). These vectors are

4176-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

4263-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

4350-459: The French mathematician Lamé , derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space ( R ) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane ; for example z = 0 defines the x - y plane. In the same space,

4437-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

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4524-812: The axes of the Cartesian coordinate system , so the basis with these vectors does not meet the definition of standard basis. There is a standard basis also for the ring of polynomials in n indeterminates over a field , namely the monomials . All of the preceding are special cases of the indexed family ( e i ) i ∈ I = ( ( δ i j ) j ∈ I ) i ∈ I {\displaystyle {(e_{i})}_{i\in I}=((\delta _{ij})_{j\in I})_{i\in I}} where I {\displaystyle I}

4611-458: The chain rule, dq 1 can be expressed as: If the displacement d r is such that dq 2 = dq 3 = 0, i.e. the position vector r moves by an infinitesimal amount along the coordinate axis q 2 =const and q 3 =const, then: Dividing by dq 1 , and taking the limit dq 1 → 0: or equivalently: Now if the displacement d r is such that dq 1 = dq 3 =0, i.e. the position vector r moves by an infinitesimal amount along

4698-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

4785-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

4872-533: The coordinate axis q 1 =const and q 3 =const, then: Dividing by dq 2 , and taking the limit dq 2 → 0: or equivalently: And so forth for the other dot products. Alternative Proof: and the Einstein summation convention is implied. A vector v can be specified in terms of either basis, i.e., Using the Einstein summation convention, the basis vectors relate to the components by and where g

4959-497: The coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere , which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars , vectors , or tensors . Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as

5046-407: The derivative of the location of point P with respect to the local coordinate Applying the same derivatives to the curvilinear system locally at point P defines the natural basis vectors: Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis . All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are

5133-904: The directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts. It follows that the component (projection) of b 1 on the x axis is If q = q ( x 1 , x 2 , x 3 ) and x i = x i ( q , q , q ) are smooth (continuously differentiable) functions the transformation ratios can be written as ∂ q i ∂ x j {\displaystyle {\cfrac {\partial q^{i}}{\partial x_{j}}}} and ∂ x i ∂ q j {\displaystyle {\cfrac {\partial x_{i}}{\partial q^{j}}}} . That is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to

5220-452: The distance PE is infinitesimally small. Then PE measured on the q axis almost coincides with PE measured on the q line. At the same time, the ratio PD/PE ( PD being the projection of PE on the x axis) becomes almost exactly equal to cos ⁡ α {\displaystyle \cos \alpha } . Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and d q . Then Thus,

5307-505: The dot product as: Consider an infinitesimal displacement d r = d x ⋅ e x + d y ⋅ e y + d z ⋅ e z {\displaystyle d\mathbf {r} =dx\cdot \mathbf {e} _{x}+dy\cdot \mathbf {e} _{y}+dz\cdot \mathbf {e} _{z}} . Let dq 1 , dq 2 and dq 3 denote the corresponding infinitesimal changes in curvilinear coordinates q 1 , q 2 and q 3 respectively. By

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5394-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

5481-593: The following important equality: b i ⋅ b j = δ j i {\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}} wherein δ j i {\displaystyle \delta _{j}^{i}} denotes the generalized Kronecker delta . In the Cartesian coordinate system ( e x , e y , e z ) {\displaystyle (\mathbf {e} _{x},\mathbf {e} _{y},\mathbf {e} _{z})} , we can write

5568-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

5655-400: The general case appears later on this page. In orthogonal curvilinear coordinates, since the total differential change in r is so scale factors are h i = | ∂ r ∂ q i | {\displaystyle h_{i}=\left|{\frac {\partial \mathbf {r} }{\partial q^{i}}}\right|} In non-orthogonal coordinates

5742-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

5829-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

5916-408: The influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R . Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in

6003-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

6090-714: The length of d r = d q 1 h 1 + d q 2 h 2 + d q 3 h 3 {\displaystyle d\mathbf {r} =dq^{1}\mathbf {h} _{1}+dq^{2}\mathbf {h} _{2}+dq^{3}\mathbf {h} _{3}} is the positive square root of d r ⋅ d r = d q i d q j h i ⋅ h j {\displaystyle d\mathbf {r} \cdot d\mathbf {r} =dq^{i}dq^{j}\mathbf {h} _{i}\cdot \mathbf {h} _{j}} (with Einstein summation convention ). The six independent scalar products g ij = h i . h j of

6177-518: The magnitudes of the two basis vectors, i.e., the scalar intercepts PB and PA . PA is also the projection of b 1 on the x axis. However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reasons: The angles that the q line and that axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P . Let point E be located very close to P , so close that

6264-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

6351-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

6438-536: The natural basis vectors h i not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering , particularly fluid mechanics and continuum mechanics , require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of

6525-407: The natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g ij are the components of the metric tensor , which has only three non zero components in orthogonal coordinates: g 11 = h 1 h 1 , g 22 = h 2 h 2 , g 33 = h 3 h 3 . Spatial gradients, distances, time derivatives and scale factors are interrelated within

6612-407: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Standard basis Here the vector e x points in the x direction, the vector e y points in

6699-759: The oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem ,

6786-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

6873-427: The other system. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of

6960-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

7047-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

7134-400: The same at all points are global bases , and can be associated only with linear or affine coordinate systems . For this article e is reserved for the standard basis (Cartesian) and h or b is for the curvilinear basis. These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define

7221-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

7308-762: The standard basis is a sequence of orthogonal unit vectors . In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e. , v 1 = ( 3 2 , 1 2 ) {\displaystyle v_{1}=\left({{\sqrt {3}} \over 2},{1 \over 2}\right)\,} v 2 = ( 1 2 , − 3 2 ) {\displaystyle v_{2}=\left({1 \over 2},{-{\sqrt {3}} \over 2}\right)\,} are also orthogonal unit vectors, but they are not aligned with

7395-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

7482-417: The vector with a 1 in the i th coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces , whose definition involves coefficients , such as polynomials and matrices . In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and

7569-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

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