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162-470: Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress , elasticity , quantum mechanics , fluid mechanics , moment of inertia , ...), electrodynamics ( electromagnetic tensor , Maxwell tensor , permittivity , magnetic susceptibility , ...), and general relativity ( stress–energy tensor , curvature tensor , ...). In applications, it

324-406: A 1 e 1 , … , a k e k {\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}} is a basis of G , for some nonzero integers a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} . For details, see Free abelian group § Subgroups . In

486-506: A i , j v i = ∑ i = 1 n ( ∑ j = 1 n a i , j y j ) v i . {\displaystyle \mathbf {x} =\sum _{j=1}^{n}y_{j}\mathbf {w} _{j}=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}=\sum _{i=1}^{n}{\biggl (}\sum _{j=1}^{n}a_{i,j}y_{j}{\biggr )}\mathbf {v} _{i}.} The change-of-basis formula results then from

648-661: A k , b k . But many square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis . The geometric notions of an affine space , projective space , convex set , and cone have related notions of basis . An affine basis for an n -dimensional affine space

810-410: A , b ) + ( c , d ) = ( a + c , b + d ) {\displaystyle (a,b)+(c,d)=(a+c,b+d)} and scalar multiplication λ ( a , b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda }

972-438: A Hilbert basis (linear programming) . For a probability distribution in R with a probability density function , such as the equidistribution in an n -dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one , which is due to the fact that n linearly dependent vectors x 1 , ..., x n in R should satisfy

1134-499: A Platonist by Stephen Hawking , a view Penrose discusses in his book, The Road to Reality . Hawking referred to himself as an "unashamed reductionist" and took issue with Penrose's views. Mathematics provides a compact and exact language used to describe the order in nature. This was noted and advocated by Pythagoras , Plato , Galileo, and Newton. Some theorists, like Hilary Putnam and Penelope Maddy , hold that logical truths, and therefore mathematical reasoning, depend on

1296-405: A field . For example, scalars can come from a ring . But the theory is then less geometric and computations more technical and less algorithmic. Tensors are generalized within category theory by means of the concept of monoidal category , from the 1960s. An elementary example of a mapping describable as a tensor is the dot product , which maps two vectors to a scalar. A more complex example

1458-488: A frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with motion in the absence of gravitational fields and the general theory of relativity with motion and its connection with gravitation . Both quantum theory and the theory of relativity find applications in many areas of modern physics. While physics itself aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking,

1620-464: A linearly independent set L of n elements of V , one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L , having its other elements in S , and having the same number of elements as S . Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require

1782-406: A set B of vectors in a vector space V is called a basis ( pl. : bases ) if every element of V may be written in a unique way as a finite linear combination of elements of B . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B . The elements of a basis are called basis vectors . Equivalently, a set B

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1944-427: A (potentially multidimensional) array. Just as a vector in an n - dimensional space is represented by a one-dimensional array with n components with respect to a given basis , any tensor with respect to a basis is represented by a multidimensional array. For example, a linear operator is represented in a basis as a two-dimensional square n × n array. The numbers in the multidimensional array are known as

2106-448: A basis of R . More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let e i = ( 0 , … , 0 , 1 , 0 , … , 0 ) {\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)} be

2268-580: A basis of V . By definition of a basis, every v in V may be written, in a unique way, as v = λ 1 b 1 + ⋯ + λ n b n , {\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},} where the coefficients λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are scalars (that is, elements of F ), which are called

2430-595: A correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect: I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot. Tensors and tensor fields were also found to be useful in other fields such as continuum mechanics . Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors , and

2592-388: A free abelian group is a free abelian group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then there is a basis e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} of H and an integer 0 ≤ k ≤ n such that

2754-420: A hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for. Physics is a branch of fundamental science (also called basic science). Physics is also called " the fundamental science" because all branches of natural science including chemistry, astronomy, geology, and biology are constrained by laws of physics. Similarly, chemistry

2916-399: A lower index of an ( n , m ) -tensor produces an ( n − 1, m − 1) -tensor; this corresponds to moving diagonally up and to the left on the table. Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms

3078-502: A representation of GL( n ) on W (that is, a group homomorphism ρ : GL ( n ) → GL ( W ) {\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)} ). Then a tensor of type ρ {\displaystyle \rho } is an equivariant map T : F → W {\displaystyle T:F\to W} . Equivariance here means that When ρ {\displaystyle \rho }

3240-514: A ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above. This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic . Constructions of spaces of tensors based on

3402-465: A specific practical application as a goal, other than the deeper insight into the phenomema themselves. Applied physics is a general term for physics research and development that is intended for a particular use. An applied physics curriculum usually contains a few classes in an applied discipline, like geology or electrical engineering. It usually differs from engineering in that an applied physicist may not be designing something in particular, but rather

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3564-426: A speed much less than the speed of light. These theories continue to be areas of active research today. Chaos theory , an aspect of classical mechanics, was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Newton (1642–1727). These central theories are important tools for research into more specialized topics, and any physicist, regardless of their specialization,

3726-399: A subfield of mechanics , is used in the building of bridges and other static structures. The understanding and use of acoustics results in sound control and better concert halls; similarly, the use of optics creates better optical devices. An understanding of physics makes for more realistic flight simulators , video games, and movies, and is often critical in forensic investigations. With

3888-407: A tensor as a multilinear map, it is conventional to identify the double dual V of the vector space V , i.e., the space of linear functionals on the dual vector space V , with the vector space V . There is always a natural linear map from V to its double dual, given by evaluating a linear form in V against a vector in V . This linear mapping is an isomorphism in finite dimensions, and it

4050-404: A tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field , often referred to simply as a tensor. In this context, a coordinate basis is often chosen for the tangent vector space . The transformation law may then be expressed in terms of partial derivatives of

4212-404: A tensor, for the sign change under transformations changing the orientation. Physics Physics is the scientific study of matter , its fundamental constituents , its motion and behavior through space and time , and the related entities of energy and force . Physics is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics

4374-597: A term derived from φύσις ( phúsis 'origin, nature, property'). Astronomy is one of the oldest natural sciences . Early civilizations dating before 3000 BCE, such as the Sumerians , ancient Egyptians , and the Indus Valley Civilisation , had a predictive knowledge and a basic awareness of the motions of the Sun, Moon, and stars. The stars and planets, believed to represent gods, were often worshipped. While

4536-580: A third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol ε i j k {\displaystyle \varepsilon _{ijk}} nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems. This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type ( n , m ) , where n

4698-399: A vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is,

4860-420: Is n + 1 {\displaystyle n+1} points in general linear position . A projective basis is n + 2 {\displaystyle n+2} points in general position, in a projective space of dimension n . A convex basis of a polytope is the set of the vertices of its convex hull . A cone basis consists of one point by edge of a polygonal cone. See also

5022-572: Is a linear isomorphism from the vector space F n {\displaystyle F^{n}} onto V . In other words, F n {\displaystyle F^{n}} is the coordinate space of V , and the n -tuple φ − 1 ( v ) {\displaystyle \varphi ^{-1}(\mathbf {v} )} is the coordinate vector of v . The inverse image by φ {\displaystyle \varphi } of b i {\displaystyle \mathbf {b} _{i}}

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5184-422: Is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups. A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as

5346-443: Is a basis if it satisfies the two following conditions: The scalars a i {\displaystyle a_{i}} are called the coordinates of the vector v with respect to the basis B , and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional . In this case, the finite subset can be taken as B itself to check for linear independence in

5508-416: Is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B . In other words, a basis is a linearly independent spanning set . A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of

5670-413: Is a basis of V . Since L max belongs to X , we already know that L max is a linearly independent subset of V . If there were some vector w of V that is not in the span of L max , then w would not be an element of L max either. Let L w = L max ∪ { w } . This set is an element of X , that is, it is a linearly independent subset of V (because w

5832-400: Is a manifestation of the so-called measure concentration phenomenon . The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n -dimensional cube [−1, 1] as a function of dimension, n . A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If

5994-734: Is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame ). Let, as usual, F n {\displaystyle F^{n}} be the set of the n -tuples of elements of F . This set is an F -vector space, with addition and scalar multiplication defined component-wise. The map φ : ( λ 1 , … , λ n ) ↦ λ 1 b 1 + ⋯ + λ n b n {\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}

6156-521: Is an element of X . Therefore, L Y is an upper bound for Y in ( X , ⊆) : it is an element of X , that contains every element of Y . As X is nonempty, and every totally ordered subset of ( X , ⊆) has an upper bound in X , Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L max of X satisfying the condition that whenever L max ⊆ L for some element L of X , then L = L max . It remains to prove that L max

6318-461: Is an ordered basis, and R = ( R j i ) {\displaystyle R=\left(R_{j}^{i}\right)} is an invertible n × n {\displaystyle n\times n} matrix, then the action is given by Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL( n ). Let W be a vector space and let ρ {\displaystyle \rho } be

6480-529: Is any real number. A simple basis of this vector space consists of the two vectors e 1 = (1, 0) and e 2 = (0, 1) . These vectors form a basis (called the standard basis ) because any vector v = ( a , b ) of R may be uniquely written as v = a e 1 + b e 2 . {\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.} Any other pair of linearly independent vectors of R , such as (1, 1) and (−1, 2) , forms also

6642-531: Is called a physicist . Physics is one of the oldest academic disciplines . Over much of the past two millennia, physics, chemistry , biology , and certain branches of mathematics were a part of natural philosophy , but during the Scientific Revolution in the 17th century, these natural sciences branched into separate research endeavors. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry , and

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6804-413: Is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical. The problems in this field start with a " mathematical model of a physical situation " (system) and a "mathematical description of a physical law" that will be applied to that system. Every mathematical statement used for solving has

6966-408: Is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field . In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing

7128-410: Is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if

7290-419: Is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies not subject to an acceleration), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics ),

7452-400: Is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in particle accelerators . On this scale, ordinary, commonsensical notions of space, time, matter, and energy are no longer valid. The two chief theories of modern physics present a different picture of

7614-469: Is customary to refer to B o l d {\displaystyle B_{\mathrm {old} }} and B n e w {\displaystyle B_{\mathrm {new} }} as the old basis and the new basis , respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has expressions involving the old coordinates, and if one wants to obtain equivalent expressions in terms of

7776-421: Is denoted, as usual, by ⊆ . Let Y be a subset of X that is totally ordered by ⊆ , and let L Y be the union of all the elements of Y (which are themselves certain subsets of V ). Since ( Y , ⊆) is totally ordered, every finite subset of L Y is a subset of an element of Y , which is a linearly independent subset of V , and hence L Y is linearly independent. Thus L Y

7938-732: Is equal to 1, is a countable Hamel basis. In the study of Fourier series , one learns that the functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions f satisfying ∫ 0 2 π | f ( x ) | 2 d x < ∞ . {\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .} The functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are linearly independent, and every function f that

8100-525: Is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials ) is also a basis. (Such a set of polynomials is called a polynomial sequence .) But there are also many bases for F [ X ] that are not of this form. Many properties of finite bases result from the Steinitz exchange lemma , which states that, for any vector space V , given a finite spanning set S and

8262-433: Is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V , which is usually taken to be a particular vector space of some geometrical significance like

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8424-425: Is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics , electromagnetism , and special relativity. Classical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics , optics , thermodynamics, and electromagnetism. Classical mechanics

8586-429: Is generally concerned with matter and energy on the normal scale of observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on a very large or very small scale. For example, atomic and nuclear physics study matter on the smallest scale at which chemical elements can be identified. The physics of elementary particles is on an even smaller scale since it

8748-444: Is given by polynomial rings . If F is a field, the collection F [ X ] of all polynomials in one indeterminate X with coefficients in F is an F -vector space. One basis for this space is the monomial basis B , consisting of all monomials : B = { 1 , X , X 2 , … } . {\displaystyle B=\{1,X,X^{2},\ldots \}.} Any set of polynomials such that there

8910-777: Is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinite-dimensional normed vector space that is complete (i.e. X is a Banach space ), then any Hamel basis of X is necessarily uncountable . This is a consequence of the Baire category theorem . The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional ( non-complete ) normed spaces that have countable Hamel bases. Consider c 00 {\displaystyle c_{00}} ,

9072-414: Is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T . This motivates viewing multilinear maps as the intrinsic objects underlying tensors. In viewing

9234-442: Is not in the span of L max , and L max is independent). As L max ⊆ L w , and L max ≠ L w (because L w contains the vector w that is not contained in L max ), this contradicts the maximality of L max . Thus this shows that L max spans V . Hence L max is linearly independent and spans V . It is thus a basis of V , and this proves that every vector space has

9396-731: Is now meant by a tensor. Gibbs introduced dyadics and polyadic algebra , which are also tensors in the modern sense. The contemporary usage was introduced by Woldemar Voigt in 1898. Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus , and originally presented in 1892. It was made accessible to many mathematicians by the publication of Ricci-Curbastro and Tullio Levi-Civita 's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications). In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in

9558-593: Is often called the central science because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and reactions of matter (chemistry's focus on the molecular and atomic scale distinguishes it from physics ). Structures are formed because particles exert electrical forces on each other, properties include physical characteristics of given substances, and reactions are bound by laws of physics, like conservation of energy , mass , and charge . Fundamental physics seeks to better explain and understand phenomena in all spheres, without

9720-411: Is often then expedient to identify V with its double dual. For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property as explained here and here . A type ( p , q ) tensor is defined in this context as an element of

9882-439: Is often useful to express the coordinates of a vector x with respect to B o l d {\displaystyle B_{\mathrm {old} }} in terms of the coordinates with respect to B n e w . {\displaystyle B_{\mathrm {new} }.} This can be done by the change-of-basis formula , that is described below. The subscripts "old" and "new" have been chosen because it

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10044-500: Is possible only in discrete steps proportional to their frequency. This, along with the photoelectric effect and a complete theory predicting discrete energy levels of electron orbitals , led to the theory of quantum mechanics improving on classical physics at very small scales. Quantum mechanics would come to be pioneered by Werner Heisenberg , Erwin Schrödinger and Paul Dirac . From this early work, and work in related fields,

10206-743: Is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that lim n → ∞ ∫ 0 2 π | a 0 + ∑ k = 1 n ( a k cos ⁡ ( k x ) + b k sin ⁡ ( k x ) ) − f ( x ) | 2 d x = 0 {\displaystyle \lim _{n\to \infty }\int _{0}^{2\pi }{\biggl |}a_{0}+\sum _{k=1}^{n}\left(a_{k}\cos \left(kx\right)+b_{k}\sin \left(kx\right)\right)-f(x){\biggr |}^{2}dx=0} for suitable (real or complex) coefficients

10368-513: Is that not every module has a basis. A module that has a basis is called a free module . Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions . A module over the integers is exactly the same thing as an abelian group . Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of

10530-474: Is the Cauchy stress tensor T , which takes a directional unit vector v as input and maps it to the stress vector T , which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product , where two vectors are mapped to

10692-530: Is the Kronecker delta , which functions similarly to the identity matrix , and has the effect of renaming indices ( j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will ( commutativity ), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of

10854-404: Is the n -tuple e i {\displaystyle \mathbf {e} _{i}} all of whose components are 0, except the i th that is 1. The e i {\displaystyle \mathbf {e} _{i}} form an ordered basis of F n {\displaystyle F^{n}} , which is called its standard basis or canonical basis . The ordered basis B

11016-418: Is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a symmetric monoidal category that encodes their most important properties, rather than the specific models of those categories. In many applications, especially in differential geometry and physics, it is natural to consider

11178-481: Is the image by φ {\displaystyle \varphi } of the canonical basis of F n {\displaystyle F^{n}} . It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of F n {\displaystyle F^{n}} , and that every linear isomorphism from F n {\displaystyle F^{n}} onto V may be defined as

11340-451: Is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2) -tensor; an inner product is an example of a (0, 2) -tensor, but not all (0, 2) -tensors are inner products. In the (0, M ) -entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of

11502-533: Is the smallest infinite cardinal, the cardinal of the integers. The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces , Banach spaces , or Fréchet spaces . The preference of other types of bases for infinite-dimensional spaces

11664-404: Is thus given as, Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type ( p , q ) . The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of

11826-431: Is using physics or conducting physics research with the aim of developing new technologies or solving a problem. The approach is similar to that of applied mathematics . Applied physicists use physics in scientific research. For instance, people working on accelerator physics might seek to build better particle detectors for research in theoretical physics. Physics is used heavily in engineering. For example, statics,

11988-681: Is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from Hom 2 ⁡ ( U ∗ × V ∗ ; F ) {\displaystyle \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)} and Hom ⁡ ( U ∗ ⊗ V ∗ ; F ) {\displaystyle \operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)} . Tensor products can be defined in great generality – for example, involving arbitrary modules over

12150-516: Is ε-orthogonal to y if | ⟨ x , y ⟩ | / ( ‖ x ‖ ‖ y ‖ ) < ε {\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon } (that is, cosine of the angle between x and y is less than ε ). In high dimensions, two independent random vectors are with high probability almost orthogonal, and

12312-454: The Archaic period (650 BCE – 480 BCE), when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause. They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment; for example, atomism was found to be correct approximately 2000 years after it

12474-527: The Industrial Revolution as energy needs increased. The laws comprising classical physics remain widely used for objects on everyday scales travelling at non-relativistic speeds, since they provide a close approximation in such situations, and theories such as quantum mechanics and the theory of relativity simplify to their classical equivalents at such scales. Inaccuracies in classical mechanics for very small objects and very high velocities led to

12636-470: The Islamic Golden Age developed it further, especially placing emphasis on observation and a priori reasoning, developing early forms of the scientific method . The most notable innovations under Islamic scholarship were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl , Al-Kindi , Ibn al-Haytham , Al-Farisi and Avicenna . The most notable work

12798-485: The Riemann curvature tensor . The exterior algebra of Hermann Grassmann , from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms , as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics, and Hassler Whitney popularized

12960-508: The Scientific Revolution . Galileo cited Philoponus substantially in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan , a teacher in the faculty of arts at the University of Paris , developed the concept of impetus. It was a step toward the modern ideas of inertia and momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during

13122-601: The Standard Model of particle physics was derived. Following the discovery of a particle with properties consistent with the Higgs boson at CERN in 2012, all fundamental particles predicted by the standard model, and no others, appear to exist; however, physics beyond the Standard Model , with theories such as supersymmetry , is an active area of research. Areas of mathematics in general are important to this field, such as

13284-431: The axiom of choice or a weaker form of it, such as the ultrafilter lemma . If V is a vector space over a field F , then: If V is a vector space of dimension n , then: Let V be a vector space of finite dimension n over a field F , and B = { b 1 , … , b n } {\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}} be

13446-725: The column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is X = A Y . {\displaystyle X=AY.} The formula can be proven by considering the decomposition of the vector x on the two bases: one has x = ∑ i = 1 n x i v i , {\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},} and x = ∑ j = 1 n y j w j = ∑ j = 1 n y j ∑ i = 1 n

13608-434: The complex numbers ), with F replacing ⁠ R {\displaystyle \mathbb {R} } ⁠ as the codomain of the multilinear maps. By applying a multilinear map T of type ( p , q ) to a basis { e j } for V and a canonical cobasis { ε } for V , a ( p + q ) -dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T

13770-427: The components of the tensor. They are denoted by indices giving their position in the array, as subscripts and superscripts , following the symbolic name of the tensor. For example, the components of an order 2 tensor T could be denoted T ij  , where i and j are indices running from 1 to n , or also by T   j . Whether an index is displayed as a superscript or subscript depends on

13932-513: The coordinates of v over B . However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, 3 b 1 + 2 b 2 {\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}} and 2 b 1 + 3 b 2 {\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}} have

14094-408: The dimension or the number of ways of an array, which is why a tensor is sometimes referred to as an m -dimensional array or an m -way array. The total number of indices is also called the order , degree or rank of a tensor, although the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the basis of

14256-579: The empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world, which may explain the peculiar relation between these fields. Physics uses mathematics to organise and formulate experimental results. From those results, precise or estimated solutions are obtained, or quantitative results, from which new predictions can be made and experimentally confirmed or negated. The results from physics experiments are numerical data, with their units of measure and estimates of

14418-585: The exact sciences are descended from late Babylonian astronomy . Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey ; later Greek astronomers provided names, which are still used today, for most constellations visible from the Northern Hemisphere . Natural philosophy has its origins in Greece during

14580-452: The n -tuple with all components equal to 0, except the i th, which is 1. Then e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} is a basis of F n , {\displaystyle F^{n},} which is called the standard basis of F n . {\displaystyle F^{n}.} A different flavor of example

14742-420: The same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If

14904-539: The standard consensus that the laws of physics are universal and do not change with time, physics can be used to study things that would ordinarily be mired in uncertainty . For example, in the study of the origin of the Earth, a physicist can reasonably model Earth's mass, temperature, and rate of rotation, as a function of time allowing the extrapolation forward or backward in time and so predict future or prior events. It also allows for simulations in engineering that speed up

15066-424: The tangent space to a manifold. In this approach, a type ( p , q ) tensor T is defined as a multilinear map , where V is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers , ⁠ R {\displaystyle \mathbb {R} } ⁠ . More generally, V can be taken over any field F (e.g.

15228-460: The tensor product . From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem ). Correspondingly there are types of tensors at work in many branches of abstract algebra , particularly in homological algebra and representation theory . Multilinear algebra can be developed in greater generality than for scalars coming from

15390-435: The 16th and 17th centuries, and Isaac Newton 's discovery and unification of the laws of motion and universal gravitation (that would come to bear his name). Newton also developed calculus , the mathematical study of continuous change, which provided new mathematical methods for solving physical problems. The discovery of laws in thermodynamics , chemistry , and electromagnetics resulted from research efforts during

15552-434: The above definition. It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation , or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering

15714-435: The angle between the vectors was within π/2 ± 0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of

15876-524: The array (or its generalization in other definitions), p + q in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type ( p , q ) is also called a ( p , q ) -tensor for short. This discussion motivates the following formal definition: Definition. A tensor of type ( p , q ) is an assignment of a multidimensional array to each basis f = ( e 1 , ..., e n ) of an n -dimensional vector space such that, if we apply

16038-396: The basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis , which is therefore not simply an unstructured set , but a sequence , an indexed family , or similar; see § Ordered bases and coordinates below. The set R of the ordered pairs of real numbers is a vector space under the operations of component-wise addition (

16200-433: The boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in these and other academic disciplines such as mathematics and philosophy. Advances in physics often enable new technologies . For example, advances in the understanding of electromagnetism , solid-state physics , and nuclear physics led directly to

16362-411: The case of the real numbers R viewed as a vector space over the field Q of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , where ℵ 0 {\displaystyle \aleph _{0}} ( aleph-nought )

16524-435: The chain) is recorded. For each n , 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented. Let V be any vector space over some field F . Let X be the set of all linearly independent subsets of V . The set X is nonempty since the empty set is an independent subset of V , and it is partially ordered by inclusion, which

16686-604: The change of basis then the multidimensional array obeys the transformation law The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. An equivalent definition of a tensor uses the representations of the general linear group . There is an action of the general linear group on the set of all ordered bases of an n -dimensional vector space. If f = ( f 1 , … , f n ) {\displaystyle \mathbf {f} =(\mathbf {f} _{1},\dots ,\mathbf {f} _{n})}

16848-421: The components ( T v ) i {\displaystyle (Tv)^{i}} are given by ( T v ) i = T j i v j {\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}} . These components transform contravariantly, since The transformation law for an order p + q tensor with p contravariant indices and q covariant indices

17010-434: The concepts of space, time, and matter from that presented by classical physics. Classical mechanics approximates nature as continuous, while quantum theory is concerned with the discrete nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The theory of relativity is concerned with the description of phenomena that take place in

17172-409: The constant speed predicted by Maxwell's equations of electromagnetism. This discrepancy was corrected by Einstein's theory of special relativity , which replaced classical mechanics for fast-moving bodies and allowed for a constant speed of light. Black-body radiation provided another problem for classical physics, which was corrected when Planck proposed that the excitation of material oscillators

17334-515: The context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel ) or algebraic basis can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces , Schauder bases , and Markushevich bases on normed linear spaces . In

17496-443: The coordinate functions, defining a coordinate transformation, The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry , and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what

17658-400: The coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . This formula may be concisely written in matrix notation. Let A be

17820-424: The definition of a vector space by a ring , one gets the definition of a module . For modules, linear independence and spanning sets are defined exactly as for vector spaces, although " generating set " is more commonly used than that of "spanning set". Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces

17982-415: The density object it is falling through (e.g. density of air). He also stated that, when it comes to violent motion (motion of an object when a force is applied to it by a second object) that the speed that object moves, will only be as fast or strong as the measure of force applied to it. The problem of motion and its causes was studied carefully, leading to the philosophical notion of a " prime mover " as

18144-466: The development of a new technology. There is also considerable interdisciplinarity , so many other important fields are influenced by physics (e.g., the fields of econophysics and sociophysics ). Physicists use the scientific method to test the validity of a physical theory . By using a methodical approach to compare the implications of a theory with the conclusions drawn from its related experiments and observations, physicists are better able to test

18306-429: The development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory and Albert Einstein 's theory of relativity. Both of these theories came about due to inaccuracies in classical mechanics in certain situations. Classical mechanics predicted that the speed of light depends on the motion of the observer, which could not be resolved with

18468-407: The development of new experiments (and often related equipment). Physicists who work at the interplay of theory and experiment are called phenomenologists , who study complex phenomena observed in experiment and work to relate them to a fundamental theory . Theoretical physics has historically taken inspiration from philosophy; electromagnetism was unified this way. Beyond the known universe,

18630-548: The development of technologies that have transformed modern society, such as television, computers, domestic appliances , and nuclear weapons ; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus . The word physics comes from the Latin physica ('study of nature'), which itself is a borrowing of the Greek φυσική ( phusikḗ 'natural science'),

18792-422: The difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries later, during

18954-550: The earlier work of Bernhard Riemann , Elwin Bruno Christoffel , and others – as part of the absolute differential calculus . The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor . Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. A tensor may be represented as

19116-416: The equation det[ x 1 ⋯ x n ] = 0 (zero determinant of the matrix with columns x i ), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases. It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product , x

19278-682: The errors in the measurements. Technologies based on mathematics, like computation have made computational physics an active area of research. Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while mathematical statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data. The distinction

19440-470: The explanations for the observed positions of the stars were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy, as the stars were found to traverse great circles across the sky, which could not explain the positions of the planets . According to Asger Aaboe , the origins of Western astronomy can be found in Mesopotamia , and all Western efforts in

19602-863: The field of theoretical physics also deals with hypothetical issues, such as parallel universes , a multiverse , and higher dimensions . Theorists invoke these ideas in hopes of solving particular problems with existing theories; they then explore the consequences of these ideas and work toward making testable predictions. Experimental physics expands, and is expanded by, engineering and technology. Experimental physicists who are involved in basic research design and perform experiments with equipment such as particle accelerators and lasers , whereas those involved in applied research often work in industry, developing technologies such as magnetic resonance imaging (MRI) and transistors . Feynman has noted that experimentalists may seek areas that have not been explored well by theorists. Basis vectors In mathematics ,

19764-420: The individual matrix entries, this transformation law has the form T ^ j ′ i ′ = ( R − 1 ) i i ′ T j i R j ′ j {\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}} so

19926-826: The isomorphism that maps the canonical basis of F n {\displaystyle F^{n}} onto a given ordered basis of V . In other words, it is equivalent to define an ordered basis of V , or a linear isomorphism from F n {\displaystyle F^{n}} onto V . Let V be a vector space of dimension n over a field F . Given two (ordered) bases B old = ( v 1 , … , v n ) {\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})} and B new = ( w 1 , … , w n ) {\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})} of V , it

20088-445: The knowledge of previous scholars, he began to explain how light enters the eye. He asserted that the light ray is focused, but the actual explanation of how light projected to the back of the eye had to wait until 1604. His Treatise on Light explained the camera obscura , hundreds of years before the modern development of photography. The seven-volume Book of Optics ( Kitab al-Manathir ) influenced thinking across disciplines from

20250-400: The latter include such branches as hydrostatics , hydrodynamics and pneumatics . Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics , the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics , the physics of animal calls and hearing, and electroacoustics ,

20412-490: The laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match predictions provided by classical mechanics. Einstein contributed the framework of special relativity, which replaced notions of absolute time and space with spacetime and allowed an accurate description of systems whose components have speeds approaching

20574-412: The manipulation of audible sound waves using electronics. Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation , which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light. Heat is a form of energy, the internal energy possessed by

20736-419: The matrix R , where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, w i , of a covector (or row vector), w , transform with the matrix R itself, This is called a covariant transformation law, because the covector components transform by

20898-495: The matrix of the a i , j {\displaystyle a_{i,j}} , and X = [ x 1 ⋮ x n ] and Y = [ y 1 ⋮ y n ] {\displaystyle X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}} be

21060-409: The modern sense. In the 20th century, the subject came to be known as tensor analysis , and achieved broader acceptance with the introduction of Albert Einstein 's theory of general relativity , around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann . Levi-Civita then initiated

21222-429: The natural place of another, the less abundant element will automatically go towards its own natural place. For example, if there is a fire on the ground, the flames go up into the air in an attempt to go back into its natural place where it belongs. His laws of motion included: that heavier objects will fall faster, the speed being proportional to the weight and the speed of the object that is falling depends inversely on

21384-713: The new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates. Typically, the new basis vectors are given by their coordinates over the old basis, that is, w j = ∑ i = 1 n a i , j v i . {\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.} If ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} and ( y 1 , … , y n ) {\displaystyle (y_{1},\ldots ,y_{n})} are

21546-621: The number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in n -dimensional ball. Choose N independent random vectors from a ball (they are independent and identically distributed ). Let θ be a small positive number. Then for N random vectors are all pairwise ε-orthogonal with probability 1 − θ . This N growth exponentially with dimension n and N ≫ n {\displaystyle N\gg n} for sufficiently big n . This property of random bases

21708-451: The old basis vectors e j {\displaystyle \mathbf {e} _{j}} as, Here R i are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention , which will be used throughout this article. The components v of a column vector v transform with the inverse of

21870-572: The particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy. Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field , and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest. Classical physics

22032-398: The principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference . A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C ) is a linearly independent subset of V that spans V . This means that a subset B of V

22194-425: The same set of coefficients {2, 3} , and are different. It is therefore often convenient to work with an ordered basis ; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an origin ,

22356-590: The sixth century, Isidore of Miletus created an important compilation of Archimedes ' works that are copied in the Archimedes Palimpsest . In sixth-century Europe John Philoponus , a Byzantine scholar, questioned Aristotle 's teaching of physics and noted its flaws. He introduced the theory of impetus . Aristotle's physics was not scrutinized until Philoponus appeared; unlike Aristotle, who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics Philoponus wrote: But this

22518-436: The space of the sequences x = ( x n ) {\displaystyle x=(x_{n})} of real numbers that have only finitely many non-zero elements, with the norm ‖ x ‖ = sup n | x n | {\textstyle \|x\|=\sup _{n}|x_{n}|} . Its standard basis , consisting of the sequences having only one non-zero element, which

22680-405: The space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor. Raising an index on an ( n , m ) -tensor produces an ( n + 1, m − 1) -tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with

22842-412: The speed of light. Planck, Schrödinger, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales. Later, quantum field theory unified quantum mechanics and special relativity. General relativity allowed for a dynamical, curved spacetime, with which highly massive systems and the large-scale structure of

23004-412: The study of probabilities and groups . Physics deals with a wide variety of systems, although certain theories are used by all physicists. Each of these theories was experimentally tested numerous times and found to be an adequate approximation of nature. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at

23166-501: The tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1). Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above: where δ j k {\displaystyle \delta _{j}^{k}}

23328-405: The tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves . For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product ). In some applications, it

23490-456: The tensor product of vector spaces, A basis v i of V and basis w j of W naturally induce a basis v i ⊗ w j of the tensor product V ⊗ W . The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis { e i } for V and its dual basis { ε } , i.e. Using the properties of the tensor product, it can be shown that these components satisfy

23652-425: The theory of four elements . Aristotle believed that each of the four classical elements (air, fire, water, earth) had its own natural place. Because of their differing densities, each element will revert to its own specific place in the atmosphere. So, because of their weights, fire would be at the top, air underneath fire, then water, then lastly earth. He also stated that when a small amount of one element enters

23814-553: The theory of visual perception to the nature of perspective in medieval art, in both the East and the West, for more than 600 years. This included later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to Johannes Kepler . The translation of The Book of Optics had an impact on Europe. From it, later European scholars were able to build devices that replicated those Ibn al-Haytham had built and understand

23976-426: The transformation law for a type ( p , q ) tensor. Moreover, the universal property of the tensor product gives a one-to-one correspondence between tensors defined in this way and tensors defined as multilinear maps. This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual: The last line

24138-402: The transformation matrix and its inverse cancel, so that expressions like v i e i {\displaystyle {v}^{i}\,\mathbf {e} _{i}} can immediately be seen to be geometrically identical in all coordinate systems. Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts

24300-612: The transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript). As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array T {\displaystyle T} that transforms under a change of basis matrix R = ( R i j ) {\displaystyle R=\left(R_{i}^{j}\right)} by T ^ = R − 1 T R {\displaystyle {\hat {T}}=R^{-1}TR} . For

24462-402: The transformation properties of the tensor, described below. Thus while T ij and T   j can both be expressed as n -by- n matrices, and are numerically related via index juggling , the difference in their transformation laws indicates it would be improper to add them together. The total number of indices ( m ) required to identify each component uniquely is equal to

24624-552: The ultimate source of all motion in the world (Book 8 of his treatise Physics ). The Western Roman Empire fell to invaders and internal decay in the fifth century, resulting in a decline in intellectual pursuits in western Europe. By contrast, the Eastern Roman Empire (usually known as the Byzantine Empire ) resisted the attacks from invaders and continued to advance various fields of learning, including physics. In

24786-408: The uniqueness of the decomposition of a vector over a basis, here B old {\displaystyle B_{\text{old}}} ; that is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . If one replaces the field occurring in

24948-423: The universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed. Physics, as with the rest of science, relies on the philosophy of science and its " scientific method " to advance knowledge of the physical world. The scientific method employs a priori and a posteriori reasoning as well as

25110-573: The use of Bayesian inference to measure the validity of a given theory. Study of the philosophical issues surrounding physics, the philosophy of physics , involves issues such as the nature of space and time , determinism , and metaphysical outlooks such as empiricism , naturalism , and realism . Many physicists have written about the philosophical implications of their work, for instance Laplace , who championed causal determinism , and Erwin Schrödinger , who wrote on quantum mechanics. The mathematical physicist Roger Penrose has been called

25272-988: The validity of a theory in a logical, unbiased, and repeatable way. To that end, experiments are performed and observations are made in order to determine the validity or invalidity of a theory. A scientific law is a concise verbal or mathematical statement of a relation that expresses a fundamental principle of some theory, such as Newton's law of universal gravitation. Theorists seek to develop mathematical models that both agree with existing experiments and successfully predict future experimental results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although theory and experiment are developed separately, they strongly affect and depend upon each other. Progress in physics frequently comes about when experimental results defy explanation by existing theories, prompting intense focus on applicable modelling, and when new theories generate experimentally testable predictions , which inspire

25434-451: The values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing ε i j k {\displaystyle \varepsilon _{ijk}} not being

25596-525: The vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis . The components of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors ), where the new basis vectors e ^ i {\displaystyle \mathbf {\hat {e}} _{i}} are expressed in terms of

25758-573: The way vision works. Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics . Major developments in this period include the replacement of the geocentric model of the Solar System with the heliocentric Copernican model , the laws governing the motion of planetary bodies (determined by Kepler between 1609 and 1619), Galileo's pioneering work on telescopes and observational astronomy in

25920-410: Was The Book of Optics (also known as Kitāb al-Manāẓir), written by Ibn al-Haytham, in which he presented the alternative to the ancient Greek idea about vision. In his Treatise on Light as well as in his Kitāb al-Manāẓir , he presented a study of the phenomenon of the camera obscura (his thousand-year-old version of the pinhole camera ) and delved further into the way the eye itself works. Using

26082-482: Was influential for about two millennia. His approach mixed some limited observation with logical deductive arguments, but did not rely on experimental verification of deduced statements. Aristotle's foundational work in Physics, though very imperfect, formed a framework against which later thinkers further developed the field. His approach is entirely superseded today. He explained ideas such as motion (and gravity ) with

26244-536: Was proposed by Leucippus and his pupil Democritus . During the classical period in Greece (6th, 5th and 4th centuries BCE) and in Hellenistic times , natural philosophy developed along many lines of inquiry. Aristotle ( Greek : Ἀριστοτέλης , Aristotélēs ) (384–322 BCE), a student of Plato , wrote on many subjects, including a substantial treatise on " Physics " – in the 4th century BC. Aristotelian physics

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