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\mathbf {e} _{m}\right|\\[8pt]&=\left|\sum _{p}\sum _{k}\sum _{m}{\mathcal {E}}_{kmp}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{p}\right|\end{aligned}}} where E {\displaystyle {\mathcal {E}}} is the permutation sy... |
β β q i ( h 1 h 2 h 3 h i 2 v i ) {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{h_{1}h_{2}h_{3}}}~\sum _{i}{\frac {\partial }{\partial q^{i}}}\left({\cfrac {h_{1}h_{2}h_{3}}{h_{i}^{2}}}~v_{i}\right)} We can get an expression for the Laplacian in a similar manner by noting that g l i β Ο β q l = {... |
Then the covariant and contravariant basis vectors are b 1 = e r = b 1 b 2 = r e ΞΈ = r 2 b 2 b 3 = e z = b 3 {\displaystyle {\begin{aligned}\mathbf {b} _{1}&=\mathbf {e} _{r}=\mathbf {b} ^{1}\\\mathbf {b} _{2}&=r~\mathbf {e} _{\theta }=r^{2}~\mathbf {b} ^{2}\\\mathbf {b} _{3}&=\mathbf {e} _{z}=\mathbf {b} ^{3}\end{alig... |
e r + 1 r ( β v ΞΈ β ΞΈ + v r ) e ΞΈ β e ΞΈ + β v ΞΈ β z e ΞΈ β e z + β v z β r e z β e r + 1 r β v z β ΞΈ e z β e ΞΈ + β v z β z e z β e z {\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_... |
a normalized contravariant basis. In cylindrical polar coordinates these components are: S ^ 11 = S 11 =: S r r , S ^ 12 = S 12 r =: S r ΞΈ , S ^ 13 = S 13 =: S r z S ^ 21 = S 21 r =: S ΞΈ r , S ^ 22 = S 22 r 2 =: S ΞΈ ΞΈ , S ^ 23 = S 23 r =: S ΞΈ z S ^ 31 = S 31 =: S z r , S ^ 32 = S 32 r =: S z ΞΈ , S ^ 33 = S 33 =: S z z ... |
z β r e ΞΈ β e z β e r + 1 r [ β S ΞΈ z β ΞΈ + S r z ] e ΞΈ β e z β e ΞΈ + β S ΞΈ z β z e ΞΈ β e z β e z + β S z r β r e z β e r β e r + 1 r [ β S z r β ΞΈ β S z ΞΈ ] e z β e r β e ΞΈ + β S z r β z e z β e r β e z + β S z ΞΈ β r e z β e ΞΈ β e r + 1 r [ β S z ΞΈ β ΞΈ + S z r ] e z β e ΞΈ β e ΞΈ + β S z ΞΈ β z e z β e ΞΈ β e z + β S z z ... |
_{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e... |
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one. Multivariable calculus may be thought of as an el... |
point ( 0 , 0 ) {\displaystyle (0,0)} is approached through the line y = k x {\displaystyle y=kx} , or in parametric form: Then the limit along the path will be: On the other hand, if the path y = Β± x 2 {\displaystyle y=\pm x^{2}} (or parametrically, x ( t ) = t , y ( t ) = Β± t 2 {\displaystyle x(t)=t,\,y(t)=\pm t^{2}}... |
0\leq x<y\leq 1\\1-x&{\text{if}}\quad 0<x=y\\0&{\text{everywhere else}}.\end{cases}}} It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle ( 0 , 1 ) Γ ( 0 , 1 ) {\displaystyle (0,1)\times (0,1)} . Furthermore, the functions defined for constant x {\displaystyle x} ... |
0} . If f : R n β R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a continuous function at point x 0 β R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} , then | f | {\displaystyle |f|} is also continuous at the same point. If f : R n β R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} is Lipschitz continu... |
thought of as the directional derivative of the function along a coordinate axis. Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator ( β {\displaystyle \nabla } ) is used to define the concepts of gradient, divergence, a... |
outputs to produce, are modeled with multivariate calculus. Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus. == See also == List of multivariable calculus topics Multivariate statistics == References == == External links == UC Berkeley video lec... |
In mathematics, the tensor product V β W {\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated a bilinear map V Γ W β V β W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) , v β V , w β W {\di... |
universal property. When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the universal property, that is that tensor products exist. === From bases === Let V and W be two vector spaces over a field... |
W x v y w B ( v , w ) {\displaystyle B(x,y)=\sum _{v\in B_{V}}\sum _{w\in B_{W}}x_{v}y_{w}\,B(v,w)} Hence, we see that the value of B {\displaystyle B} for any ( x , y ) β V Γ W {\displaystyle (x,y)\in V\times W} is uniquely and totally determined by the values that it takes on β B V Γ B W {\displaystyle B_{V}\times B_... |
with this definition, the map β : ( x , y ) β¦ x β y {\displaystyle {\otimes }:(x,y)\mapsto x\otimes y} is a bilinear map from V Γ W {\displaystyle V\times W} to V β W {\displaystyle V\otimes W} satisfying the universal property that any construction of the tensor product satisfies (see below). If arranged into a rectan... |
w 1 , w 2 β W {\displaystyle w,w_{1},w_{2}\in W} and β s β F {\displaystyle s\in F} β . Then, the tensor product is defined as the quotient space: V β W = L / R , {\displaystyle V\otimes W=L/R,} and the image of ( v , w ) {\displaystyle (v,w)} in this quotient is denoted β v β w {\displaystyle v\otimes w} β . It is strai... |
may set Z = C m n {\displaystyle Z=\mathbb {C} ^{mn}} and define the bilinear map as T : C m Γ C n β C m n ( x , y ) = ( ( x 1 , β¦ , x m ) , ( y 1 , β¦ , y n ) ) β¦ ( x i y j ) j = 1 , β¦ , n i = 1 , β¦ , m {\displaystyle {\begin{aligned}T:\mathbb {C} ^{m}\times \mathbb {C} ^{n}&\to \mathbb {C} ^{mn}\\(x,y)=((x_{1},\ldots ... |
=== The tensor product is associative in the sense that, given three vector spaces β U , V , W {\displaystyle U,V,W} β , there is a canonical isomorphism: ( U β V ) β W β
U β ( V β W ) , {\displaystyle (U\otimes V)\otimes W\cong U\otimes (V\otimes W),} that maps ( u β v ) β w {\displaystyle (u\otimes v)\otimes w} to β u... |
f β g ) ( u β w ) = f ( u ) β g ( w ) . {\displaystyle (f\otimes g)(u\otimes w)=f(u)\otimes g(w).} One has: f β g = ( f β Z ) β ( U β g ) = ( V β g ) β ( f β W ) . {\displaystyle f\otimes g=(f\otimes Z)\circ (U\otimes g)=(V\otimes g)\circ (f\otimes W).} In terms of category theory, this means that the tensor product is... |
, 2 a 1 , 2 b 2 , 1 a 1 , 2 b 2 , 2 a 2 , 1 b 1 , 1 a 2 , 1 b 1 , 2 a 2 , 2 b 1 , 1 a 2 , 2 b 1 , 2 a 2 , 1 b 2 , 1 a 2 , 1 b 2 , 2 a 2 , 2 b 2 , 1 a 2 , 2 b 2 , 2 ] . {\displaystyle {\begin{aligned}{\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}}\otimes {\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\... |
respectively (i.e. F β T m 0 {\displaystyle F\in T_{m}^{0}} and β G β T n 0 {\displaystyle G\in T_{n}^{0}} β ), then the components of their tensor product are given by: ( F β G ) i 1 i 2 β― i m + n = F i 1 i 2 β― i m G i m + 1 i m + 2 i m + 3 β― i m + n . {\displaystyle (F\otimes G)_{i_{1}i_{2}\cdots i_{m+n}}=F_{i_{1}i_{2... |
n d ( V ) {\displaystyle \mathrm {End} (V)} by means of the diagonal action: for simplicity let us assume β r = s = 1 {\displaystyle r=s=1} β , then, for each β u β E n d ( V ) {\displaystyle u\in \mathrm {End} (V)} β , u ( a β b ) = u ( a ) β b β a β u β ( b ) , {\displaystyle u(a\otimes b)=u(a)\otimes b-a\otimes u^{*}(... |
) , {\displaystyle \dim(U\otimes V)=\dim(U)\dim(V),} which automatically gives the important fact that { u i β v j } {\displaystyle \{u_{i}\otimes v_{j}\}} forms a basis of U β V {\displaystyle U\otimes V} where { u i } , { v j } {\displaystyle \{u_{i}\},\{v_{j}\}} are bases of U and V. Furthermore, given three vector ... |
R B {\displaystyle A\otimes _{R}B} satisfies β Ο = f β Ο {\displaystyle \psi =f\circ \varphi } β , and this property determines Ο {\displaystyle \varphi } within group isomorphism. See the main article for details. === Tensor product of modules over a non-commutative ring === Let A be a right R-module and B be a left R-... |
right S-module, where β ( a β b ) s := a β ( b s ) {\displaystyle (a\otimes b)s:=a\otimes (bs)} β . If A is a (S,R)-bimodule and B is a (R,T)-bimodule, then A β R B {\displaystyle A\otimes _{R}B} is a (S,T)-bimodule, where the left and right actions are defined in the same way as the previous two examples. If R is a com... |
Z/nZ, which is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived tensor product. == Tensor product of algebras == Let R be a commutative ring. The tensor product of R-modules applies, in particular, if A and B are R-alg... |
format n Γ n Γ β― Γ n {\displaystyle n\times n\times \cdots \times n} with entries ( a i 1 i 2 β― i d ) {\displaystyle (a_{i_{1}i_{2}\cdots i_{d}})} lying in an algebraically closed field K {\displaystyle K} of characteristic zero. Such a tensor A β ( K n ) β d {\displaystyle A\in (K^{n})^{\otimes d}} defines polynomial ... |
Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the LittlewoodβRichardson rule. === Tensor product of quadratic forms === === Tensor product of multilinear forms === Given two multilinear forms f ( x 1 , β¦ , x k ) {\displaystyle ... |
V\wedge V:=V\otimes V{\big /}{\bigl \{}v_{1}\otimes v_{2}+v_{2}\otimes v_{1}\mid (v_{1},v_{2})\in V^{2}{\bigr \}}.} The image of v 1 β v 2 {\displaystyle v_{1}\otimes v_{2}} in the exterior product is usually denoted v 1 β§ v 2 {\displaystyle v_{1}\wedge v_{2}} and satisfies, by construction, β v 1 β§ v 2 = β v 2 β§ v 1 {... |
from the original on 7 May 2021. Grillet, Pierre A. (2007). Abstract Algebra. Springer Science+Business Media, LLC. ISBN 978-0387715674. Halmos, Paul (1974). Finite dimensional vector spaces. Springer. ISBN 0-387-90093-4. Hungerford, Thomas W. (2003). Algebra. Springer. ISBN 0387905189. Lang, Serge (2002), Algebra, Gra... |
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initia... |
is generally nonlinear, but it can (by use of a Taylor series) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is isotropic, the linearized stressβstrain relationship is called Hooke's law, which is often presumed to apply up to the elastic lim... |
and the elastic modulus is the pascal (Pa). This unit is defined as force per unit area, generally a measurement of pressure, which in mechanics corresponds to stress. The pascal and therefore elasticity have the dimension Lβ1β
Mβ
Tβ2. For most commonly used engineering materials, the elastic modulus is on the scale of g... |
material" models (for which stress can be derived from a scalar "elastic potential" function). === Hypoelastic materials === A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria: The Cauchy stress Ο {\displaystyle {\boldsymbol {\sigm... |
the hyperelastic model may be written alternatively as Ο = 2 J F β W β C F T where J := det F . {\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.} == Applications ... |
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W {\displaystyle W} of the ... |
The representation is given by in the standard basis by u β Γ v β = [ u 1 u 2 ] [ 0 1 β 1 0 ] [ v 1 v 2 ] = u 1 v 2 β u 2 v 1 {\displaystyle {\vec {u}}\times {\vec {v}}={\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}{\begin{bmatrix}0&1\\-1&0\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}=u_{1}v_{2}-u_{2}v_{1}} If we... |
weights that is the negation of that presented here. In contrast to the meaning used in this article, in general relativity "pseudotensor" sometimes means an object that does not transform like a tensor or relative tensor of any weight. === Tensor and pseudotensor densities === For example, a mixed rank-two (authentic)... |
Ξ΄ β x Β― Ο΅ β x Ξ² T Β― Ο΅ Ξ΄ . {\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\... |
is a non-singular matrix and a rank-two tensor density of weight W {\displaystyle W} with covariant indices then its matrix inverse will be a rank-two tensor density of weight β W {\displaystyle -W} with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and co... |
( g ΞΌ Ξ½ ) {\displaystyle {g}=\det \left({g}_{\mu \nu }\right)} is the determinant of the metric tensor g ΞΌ Ξ½ . {\displaystyle {g}_{\mu \nu }.} === Use of metric tensor to manipulate tensor densities === Consequently, an even tensor density, T Ξ½ β¦ ΞΌ β¦ , {\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots },} of weigh... |
{\begin{aligned}g_{\kappa \lambda ;\alpha }&=0\\\left({\sqrt {-g}}\;^{W}\right)_{;\alpha }&=\left({\sqrt {-g}}\;^{W}\right)_{,\alpha }-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^{W}={\frac {W}{2}}g^{\kappa \lambda }g_{\kappa \lambda ,\alpha }{\sqrt {-g}}\;^{W}-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^... |
β x Ξ± ] ) β 1 . {\displaystyle \epsilon _{\alpha _{1}\cdots \epsilon _{\alpha _{N}}}={\bar {\epsilon }}_{\beta _{1}\cdots \epsilon _{\beta _{N}}}{\frac {\partial {\bar {x}}^{\beta _{1}}}{\partial x^{\alpha _{1}}}}\cdots {\frac {\partial {\bar {x}}^{\beta _{N}}}{\partial x^{\alpha _{N}}}}\left(\det \left[{\frac {\partia... |
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which ar... |
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 the dimension or the number of ways of an array, which is why a tenso... |
(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 β 1 T R {\displaystyle {\hat {T}}=R^{-1}TR} . For the individual matrix en... |
(Tv)^{i}=T_{j}^{i}v^{j}} . These components transform contravariantly, since ( T v ^ ) i β² = T ^ j β² i β² v ^ j β² = [ ( R β 1 ) i i β² T j i R j β² j ] [ ( R β 1 ) k j β² v k ] = ( R β 1 ) i i β² ( T v ) i . {\displaystyle \left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i... |
f ] {\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]} R j 1 β² j 1 β― R j q β² j q . {\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.} 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 ... |
_{q{\text{ copies}}}\rightarrow \mathbf {R} ,} where Vβ 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, β R {\displaystyle \mathbb {R} } β . More generally, V can be taken over any field F (e.g. the complex numbers), wit... |
obtained from a basis {ei} for V and its dual basis {Ξ΅j}, i.e. T = T j 1 β¦ j q i 1 β¦ i p e i 1 β β― β e i p β Ξ΅ j 1 β β― β Ξ΅ j q . {\displaystyle T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\b... |
specific models of those categories. === Tensor fields === In many applications, especially in differential geometry and physics, it is natural to consider 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 objec... |
mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915β17, 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... |
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 o... |
is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stres... |
a tensor of different type. === Tensor product === The tensor product takes two tensors, S and T, and produces a new tensor, S β T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e., ( S β T ) ( v 1 , β¦ , v n , v ... |
as a linear combination T = v 1 β w 1 β Ξ± 1 + v 2 β w 2 β Ξ± 2 + β― + v N β w N β Ξ± N . {\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.} The contraction of T on the first and last slots is then the vector Ξ± 1 ( v 1 ) w 1 + Ξ± 2 ( v... |
mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second-order tensor is needed. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surfac... |
=== The properties of tensors, especially tensor decomposition, have enabled their use in machine learning to embed higher dimensional data in artificial neural networks. This notion of tensor differs significantly from that in other areas of mathematics and physics, in the sense that a tensor is usually regarded as a ... |
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density Ο must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula ... |
sense that it is a function of the coordinate system transforming functorially under coordinate changes. Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles. === Spinors === When changing from one orthonormal basis (called a frame) to another by a ro... |
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. ... |
is invertible in the ground field K, then one can rewrite the fundamental identity above in the form u v + v u = 2 β¨ u , v β© 1 for all u , v β V , {\displaystyle uv+vu=2\langle u,v\rangle 1\ {\text{ for all }}u,v\in V,} where β¨ u , v β© = 1 2 ( Q ( u + v ) β Q ( u ) β Q ( v ) ) {\displaystyle \langle u,v\rangle ={\frac ... |
V β A such that j ( v ) 2 = Q ( v ) 1 A for all v β V {\displaystyle j(v)^{2}=Q(v)1_{A}{\text{ for all }}v\in V} (where 1A denotes the multiplicative identity of A), there is a unique algebra homomorphism f : B β A such that the following diagram commutes (i.e. such that f β i = j): The quadratic form Q may be replaced... |
property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras. == Basis and dimension == Since V comes equipped with a quadratic form Q, in characteristic not equal to 2 there exist bases for V that are ort... |
+ β― + v p 2 β v p + 1 2 β β― β v p + q 2 , {\displaystyle Q(v)=v_{1}^{2}+\dots +v_{p}^{2}-v_{p+1}^{2}-\dots -v_{p+q}^{2},} where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q.... |
bilinear form (or scalar product) v β
w = v 1 w 1 + v 2 w 2 + v 3 w 3 . {\displaystyle v\cdot w=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.} Now introduce the Clifford product of vectors v and w given by v w + w v = 2 ( v β
w ) . {\displaystyle vw+wv=2(v\cdot w).} Denote a set of orthogonal unit vectors of R3 as {e1, e2, e3}, th... |
+ v 2 w 2 + v 3 w 3 . {\displaystyle d(v,w)=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.} This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane. The Clifford product of vectors v and w is given by v w + w v = β 2 d ( v , w ) . {\displaystyle vw+wv=-2\,d(v,w).} Note the negative sign is introdu... |
non-zero vector x such that Q(x) = a, then Cl(V, Q) is algebra-isomorphic to a K-algebra generated by an element x that satisfies x2 = a, the quadratic algebra K[X] / (X2 β a). In particular, if a = 0 (that is, Q is the zero quadratic form) then Cl(V, Q) is algebra-isomorphic to the dual numbers algebra over K. If a is... |
Ο β S k sgn β‘ ( Ο ) v Ο ( 1 ) β― v Ο ( k ) {\displaystyle f_{k}(v_{1},\ldots ,v_{k})={\frac {1}{k!}}\sum _{\sigma \in \mathrm {S} _{k}}\operatorname {sgn}(\sigma )\,v_{\sigma (1)}\cdots v_{\sigma (k)}} where the sum is taken over the symmetric group on k elements, Sk. Since fk is alternating, it induces a unique linear ... |
{Cl} ^{[i]}(V,Q)\operatorname {Cl} ^{[j]}(V,Q)=\operatorname {Cl} ^{[i+j]}(V,Q)} where the bracketed superscripts are read modulo 2. This gives Cl(V, Q) the structure of a Z2-graded algebra. The subspace Cl[0](V, Q) forms a subalgebra of Cl(V, Q), called the even subalgebra. The subspace Cl[1](V, Q) is called the odd p... |
Ξ± and the transpose. We call this operation Clifford conjugation denoted x Β― {\displaystyle {\bar {x}}} x Β― = Ξ± ( x t ) = Ξ± ( x ) t . {\displaystyle {\bar {x}}=\alpha (x^{\mathrm {t} })=\alpha (x)^{\mathrm {t} }.} Of the two antiautomorphisms, the transpose is the more fundamental. Note that all of these operations are... |
example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. If V has even dimension then Cl(V, Q) is a central simple algebra over K. If V has even dimension then the even subalgebra Cl[0](V, Q) is a central simple algebra over a quadratic extension of K or a sum of... |
be the set of invertible elements x that stabilize the set of vectors under this action, meaning that for all v in V we have: Ξ± ( x ) v x β 1 β V . {\displaystyle \alpha (x)vx^{-1}\in V.} This formula also defines an action of the Lipschitz group on the vector space V that preserves the quadratic form Q, and so gives a... |
β { Β± 1 } β Spin V β‘ ( K ) β SO V β‘ ( K ) β K Γ / ( K Γ ) 2 . {\displaystyle {\begin{aligned}1\to \{\pm 1\}\to \operatorname {Pin} _{V}(K)&\to \operatorname {O} _{V}(K)\to K^{\times }/\left(K^{\times }\right)^{2},\\1\to \{\pm 1\}\to \operatorname {Spin} _{V}(K)&\to \operatorname {SO} _{V}(K)\to K^{\times }/\left(K^{\ti... |
to the orthogonal group. The image consists of the elements of spinor norm 1 β KΓβ/β(KΓ)2. The kernel consists of the elements +1 and β1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V. In the common case when V is a positive or ... |
β = Β± 1 } . {\displaystyle \mathrm {Pin} _{p,q}=\left\{v_{1}v_{2}\cdots v_{r}\mid \forall i\,\|v_{i}\|=\pm 1\right\}.} Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(p, q). T... |
algebra that has a basis that is generated by the matrices Ξ³0, ..., Ξ³3, called Dirac matrices, which have the property that Ξ³ i Ξ³ j + Ξ³ j Ξ³ i = 2 Ξ· i j , {\displaystyle \gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\eta _{ij},} where Ξ· is the matrix of a quadratic form of signature (1, 3) (or (3, 1) corresponding to t... |
action recognition and classification in computer vision. Rodriguez et al propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier Transform. Based on these vectors acti... |
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates. Examples include the deformation gradient and the first... |
p q u q {\displaystyle v'_{p}=Q_{pq}u_{q}} . Now, writing out in full, u = u q e q {\displaystyle u=u_{q}e_{q}} and also v = v p β² e p β² {\displaystyle v=v'_{p}e'_{p}} . This then requires Q to be of the form Q p q ( e p β² β e q ) {\displaystyle Q_{pq}(e'_{p}\otimes e_{q})} . By definition of tensor product, So we can ... |
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus),... |
one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you ... |
}{}^{\mu '}.} This is not to be confused with van der Waerden notation for spinors, which uses hats and overdots on indices to reflect the chirality of a spinor. === Upper and lower indices === Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (supersc... |
{\displaystyle A_{\alpha }B^{\beta }\rightarrow A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\,.} This summation may occur more than once within a term with a distinct symbol per pair of indices, for example: A Ξ± Ξ³ B Ξ± C Ξ³ Ξ² β‘ β Ξ± β Ξ³ A Ξ± Ξ³ B Ξ± C Ξ³ Ξ² . {\displaystyle A_{\alpha }{}^{\gamma }B^{\alph... |
<\zeta }A_{\alpha \beta \gamma }{}^{\delta \epsilon \cdots \lambda }B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda \mu \nu \cdots \zeta }C^{\mu \nu \cdots \zeta }\end{aligned}}} When using multi-index notation, an underarrow is placed underneath the block of indices: A P β Q β B P Q R β C R = β P β β Q β β ... |
has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents mn equations: each index takes on every value of a specific set of values. For instance, if A Ξ± B Ξ² Ξ³ C Ξ³ Ξ΄ + D Ξ± Ξ² E Ξ΄ = T Ξ± Ξ² Ξ΄ {\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\... |
replace Ξ³ (incidentally, the contraction on the Ξ³ index became a tensor product), which is entirely inconsistent for reasons shown next. === Indices are the same in every term === The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation ... |
symmetrizing indices mean there are two indices to permute and sum over: A ( Ξ± Ξ² ) Ξ³ β― = 1 2 ! ( A Ξ± Ξ² Ξ³ β― + A Ξ² Ξ± Ξ³ β― ) {\displaystyle A_{(\alpha \beta )\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }+A_{\beta \alpha \gamma \cdots }\right)} while for three symmetrizing indices, there are three in... |
{1}{p!}}\sum _{\sigma }\operatorname {sgn}(\sigma )A_{\alpha _{\sigma (1)}\cdots \alpha _{\sigma (p)}\alpha _{p+1}\cdots \alpha _{q}}\\={}&\delta _{\alpha _{1}\cdots \alpha _{p}}^{\beta _{1}\dots \beta _{p}}A_{\beta _{1}\cdots \beta _{p}\alpha _{p+1}\cdots \alpha _{q}}\\\end{aligned}}} where δβ1β
β
β
Ξ²pΞ±1β
β
β
Ξ±p is the gene... |
symmetric and antisymmetric parts === Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices: A Ξ± Ξ² Ξ³ β― = A ( Ξ± Ξ² ) Ξ³ β― + A [ Ξ± Ξ² ] Ξ³ β― {\displaystyle A_{\alpha \beta \gamma \cdots }=A_{(\alpha \beta )\gamma \cdots }+A_{[\alpha \beta ]\gamma \cdots }} as can be seen by adding the a... |
a connection is defined. For any tensor field, a semicolon ( ; ) placed before an appended lower (covariant) index indicates covariant differentiation. Less common alternatives to the semicolon include a forward slash ( / ) or in three-dimensional curved space a single vertical bar ( | ). The covariant derivative of a ... |
{\displaystyle (A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots })_{;\epsilon }=A^{\alpha }{}_{\beta \cdots ;\epsilon }B^{\gamma }{}_{\delta \cdots }+A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots ;\epsilon }\,.} ==== Connection types ==== A Koszul connection on the tangent bundle of a differentiab... |
_{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }&-\,X^{\alpha _{1}}{}_{,\gamma }T^{\gamma \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -X^{\alpha _{r}}{}_{,\gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\gamma }{}_{\beta _{1}\cdots \beta _{s}}\\&+\,X^{\gamma }{}_{,\beta _{1}}T^{\alpha _{1}\cdots \alpha... |
Ξ Ξ± Ξ³ Ξ² . {\displaystyle \Gamma ^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\gamma \beta }.} === Riemann curvature tensor === If this tensor is defined as R Ο Ο ΞΌ Ξ½ = Ξ Ο Ξ½ Ο , ΞΌ β Ξ Ο ΞΌ Ο , Ξ½ + Ξ Ο ΞΌ Ξ» Ξ Ξ» Ξ½ Ο β Ξ Ο Ξ½ Ξ» Ξ Ξ» ΞΌ Ο , {\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\Gamma ^{\rho }{}_{\nu \sigma ,\mu }-\... |
{\displaystyle {\text{duration}}=\int _{t_{1}}^{t_{2}}{\sqrt {{\frac {-1}{c^{2}}}g_{\alpha \beta }{\frac {dx^{\alpha }}{d\gamma }}{\frac {dx^{\beta }}{d\gamma }}}}\,d\gamma \,,} where Ξ³ is any smooth strictly monotone parameterization of the trajectory. See also Line element. The inverse matrix gΞ±Ξ² of the metric tensor... |
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes. == Definition == By components, it is defined as follows. ... |
In continuum mechanics, the Cauchy stress tensor (symbol Ο {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. The sec... |
other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body, and it is represented by a field T ( n ) {\displaystyle \mathbf {T} ^{(\mathbf {n} )}} , called the traction vector, defined on the surface S {\displaystyle S} and assumed to depend continuously on the su... |
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