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algebraic expressions are often difficult to solve directly. Instead, students need to learn how to transform them according to certain laws, often to determine an unknown quantity. Some tools to introduce students to the abstract side of algebra rely on concrete models and visualizations of equations, including geomet... |
In mathematics, a quintic function is a function of the form g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , {\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,} where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other ... |
cases the polynomial is reducible. As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A solvable quintic is thus an irreduci... |
p 3 r s 2 + 2000 p r 2 s 2 − 3750 p q s 3 + 825 p 2 q 2 s 2 + 2250 q 2 r s 2 + 108 q 5 s − 27 q 4 r 2 − 630 p q 3 r s + 16 p 3 q 3 s − 4 p 3 q 2 r 2 . {\displaystyle {\begin{aligned}\Delta ={}&-128p^{2}r^{4}+3125s^{4}-72p^{4}qrs+560p^{2}qr^{2}s+16p^{4}r^{3}+256r^{5}+108p^{5}s^{2}\\[4pt]&-1600qr^{3}s+144pq^{2}r^{3}-900p... |
b = 0 is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ℓ and m such that a = 5 ℓ ( 3 ℓ 5 − 4 m ) m 2 + ℓ 10 b = 4 ( 11 ℓ 5 + 2 m ) m 2 + ℓ 10 . {\displaystyle a={\frac {5\ell (3\ell ^{5}-4m)}{m^{2}+\ell ... |
c = 4√5, where φ = 1+√5/2 is the golden ratio. Then the only real solution x = −1.84208... is given by − c x = ( a + c ) 2 ( b − c ) 5 + ( − a + c ) ( b − c ) 2 5 + ( a + c ) ( b + c ) 2 5 − ( − a + c ) 2 ( b + c ) 5 , {\displaystyle -cx={\sqrt[{5}]{(a+c)^{2}(b-c)}}+{\sqrt[{5}]{(-a+c)(b-c)^{2}}}+{\sqrt[{5}]{(a+c)(b+c... |
many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section. Up to the scaling of the variable, there are exactly five solvable quintics of the shape x 5 + a x 2 + b {\displaystyle x^{5}+ax^{2}+b} , which are (where s is a scaling factor): x 5 − 2 s 3 x 2 − s 5 5 {\displaystyle x^{... |
their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric functions. At around the same time, Leopold Kronecker, using group theory, developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Later, Fel... |
4 + c r 3 + d r 2 + e r + f = 0 {\displaystyle ar^{5}+br^{4}+cr^{3}+dr^{2}+er+f=0} with: a = ± ( M S + M E ) , b = + ( M S + M E ) 3 R , c = ± ( M S + M E ) 3 R 2 , d = + ( M E ∓ M E ) R 3 ( thus d = 0 for L 2 ) , e = ± M E 2 R 4 , f = ∓ M E R 5 . {\displaystyle {\begin{aligned}&a=\pm (M_{S}+M_{E}),\\&b=+(M_{S}+M_{E})3... |
"Solving quintics in radicals". In Olav Arnfinn Laudal; Ragni Piene (eds.). The Legacy of Niels Henrik Abel. Berlin. pp. 207–225. ISBN 3-540-43826-2. Archived from the original on January 6, 2005.{{cite book}}: CS1 maint: location missing publisher (link) Tóth, Gábor (2002), Finite Möbius groups, minimal immersions of ... |
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound ... |
= f 1 v 1 + f 2 v 2 + ⋯ + f n v n {\displaystyle f_{\gamma }v^{\gamma }=f_{1}v^{1}+f_{2}v^{2}+\cdots +f_{n}v^{n}} (where vi are the components of v in a particular basis and fi are the components of f in the corresponding dual basis). Since a general mixed dyadic tensor is a linear combination of decomposable tensors o... |
) = ∑ i T i i ( x ) {\displaystyle U(x)=\sum _{i}T_{i}^{i}(x)} Since the role of x is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors. Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and ... |
a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors. For example, matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant. Let Λ α β {\displaystyle \Lambda ^{\alpha }{}_{\bet... |
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks. The strong interaction is one of the fundamental interactions of nature, and the quantum field theory (QFT) to describe it is called quantum chromodynamics (QCD). Quarks i... |
g s [ A α , A β ] {\displaystyle G_{\alpha \beta }=\partial _{\alpha }{\mathcal {A}}_{\beta }-\partial _{\beta }{\mathcal {A}}_{\alpha }\pm ig_{\text{s}}[{\mathcal {A}}_{\alpha },{\mathcal {A}}_{\beta }]} Substituting t a A α a = A α {\displaystyle t_{a}{\mathcal {A}}_{\alpha }^{a}={\mathcal {A}}_{\alpha }} and using t... |
Comparison with the electromagnetic tensor === This almost parallels the electromagnetic field tensor (also denoted F ) in quantum electrodynamics, given by the electromagnetic four-potential A describing a spin-1 photon; F α β = ∂ α A β − ∂ β A α , {\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partia... |
charge four-current is the source of the gluon field strength tensor, analogous to the electromagnetic four-current as the source of the electromagnetic tensor. It is given by j ν = t b j b ν , j b ν = ψ ¯ γ ν t b ψ , {\displaystyle j^{\nu }=t^{b}j_{b}^{\nu }\,,\quad j_{b}^{\nu }={\bar {\psi }}\gamma ^{\nu }t^{b}\psi ,... |
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition ... |
nature of the spaces. A commonly used example is the compact-open topology, e.g. loop space. Also available is the product topology on the space of set theoretic functions (i.e. not necessarily continuous functions) YX. In this context, this topology is also referred to as the topology of pointwise convergence. In alge... |
‖ p = ( ∫ R | f | p ) 1 / p {\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}} is finite S ( R ) {\displaystyle {\mathcal {S}}(\mathbb {R} )} , the Schwartz space of rapidly decreasing smooth functions and its continuous dual, S ′ ( R ) {\displaystyle {\mathcal {S}}'(\mathbb {R} )} tempered distribut... |
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extens... |
tensor, and is usually denoted g. == Tensor rank == A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can be written as a product of tensors of the form T = a ⊗ b ⊗ ⋯ ⊗ d {\displaystyle T=a\otimes b\otimes \cdots \otimes d} where a, b, ..., d are nonzero and i... |
∑ i j T i j k x i y j {\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}} for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known. == Universal property == The space T n m ( V ) {\displaystyle T_{n}^{m}(V)} can be characterized by a universal proper... |
m + n ( V ∗ , … , V ∗ ⏟ m , V , … , V ⏟ n ; F ) . {\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).} Each V in the definition of the tensor ... |
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors. == Definition == A tensor... |
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algeb... |
to the derivative of a map between smooth manifolds and the pushforward operations it defines. The differential is also used to define the dual concept of pullback. Stochastic calculus provides a notion of stochastic differential and an associated calculus for stochastic processes. The integrator in a Stieltjes integra... |
the notation for integrals because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as ∫ f ( x ) d x , {\displaystyle \int f(x)\,dx,} the integral sign (which... |
f {\displaystyle f} ) is then a function whose value at p {\displaystyle p} (usually denoted d f p {\displaystyle df_{p}} ) is not a number, but a linear map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } . Since a linear map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } i... |
now use the same trick as in the one-dimensional case and think of the expression f ( x 1 , x 2 , … , x n ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n})} as the composite of f {\displaystyle f} with the standard coordinates x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} on R n {\displaystyle \mathbb {R} ^... |
enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a complete inner product space, where the inner product and its associated norm define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete Normed ve... |
: V 1 ∩ V 2 → R {\displaystyle f_{1}*f_{2}\colon U_{1}\cap U_{2}\to \mathbb {R} \sim _{p}g_{1}*g_{2}\colon V_{1}\cap V_{2}\to \mathbb {R} } This shows that the germs at p form an algebra. Define I p {\displaystyle {\mathcal {I}}_{p}} to be the set of all smooth germs vanishing at p and I p 2 {\displaystyle {\mathcal {I... |
differential geometry === A fifth approach to infinitesimals is the method of synthetic differential geometry or smooth infinitesimal analysis. This is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive. The main idea of this approach is to replace the ca... |
∙ , d ∙ ) , {\displaystyle (C_{\bullet },d_{\bullet }),} the maps (or coboundary operators) di are often called differentials. Dually, the boundary operators in a chain complex are sometimes called codifferentials. The properties of the differential also motivate the algebraic notions of a derivation and a differential... |
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the de facto standard low-level routines for linea... |
subroutines used hard-coded loops for their low-level operations. For example, if a subroutine needed to perform a matrix multiplication, then the subroutine would have three nested loops. Linear algebra programs have many common low-level operations (the so-called "kernel" operations, not related to operating systems)... |
things, a generalized matrix-vector multiplication (gemv): y ← α A x + β y {\displaystyle {\boldsymbol {y}}\leftarrow \alpha {\boldsymbol {A}}{\boldsymbol {x}}+\beta {\boldsymbol {y}}} as well as a solver for x in the linear equation T x = y {\displaystyle {\boldsymbol {T}}{\boldsymbol {x}}={\boldsymbol {y}}} with T be... |
matrix additions instead of the conventional four real matrix multiplications and two real matrix additions", an algorithm similar to Strassen algorithm first described by Peter Ungar. == Implementations == Accelerate Apple's framework for macOS and iOS, which includes tuned versions of BLAS and LAPACK. Arm Performance... |
Moreover, uBLAS focuses on correctness of the algorithms using advanced C++ features. === Libraries using BLAS === Armadillo Armadillo is a C++ linear algebra library aiming towards a good balance between speed and ease of use. It employs template classes, and has optional links to BLAS/ATLAS and LAPACK. It is sponsore... |
e.g. a fast implementation of exponential integrators and Magnus integrators that handle long integration periods with many time steps. Here, the matrix exponentiation, the computationally expensive part of the integration, can be implemented in parallel for all time-steps by using Batched BLAS functions. == See also =... |
multiplications (1010 floating point multiply-adds) takes 15.77 seconds on 2.6 GHz processor; BLAS implementation takes 1.32 seconds. An Overview of the Sparse Basic Linear Algebra Subprograms: The New Standard from the BLAS Technical Forum [2] |
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relate... |
position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }}... |
, n ^ ) ] = 1 1 − i sin θ ℏ n ^ ⋅ J − 1 − cos θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\wi... |
}}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentu... |
== We define the Rotation of an operator by requiring that the expectation value of the original operator A ^ {\displaystyle {\widehat {\mathbf {A} }}} with respect to the initial state be equal to the expectation value of the rotated operator with respect to the rotated state, ⟨ ψ ′ | A ′ ^ | ψ ′ ⟩ = ⟨ ψ | A ^ | ψ ⟩ {... |
V ^ j {\displaystyle {U(R)}^{\dagger }{\widehat {V}}_{i}U(R)=\sum _{j}R_{ij}{\widehat {V}}_{j}} Any observable vector quantity of a quantum mechanical system should be invariant of the choice of frame of reference. The transformation of expectation value vector which applies for any wavefunction, ensures the above equa... |
Scalar operators from vector operators ==== If V → {\displaystyle {\vec {V}}} and W → {\displaystyle {\vec {W}}} are two vector operators, the dot product between the two vector operators can be defined as: V → ⋅ W → = ∑ i = 1 3 V i ^ W i ^ {\displaystyle {\vec {V}}\cdot {\vec {W}}=\sum _{i=1}^{3}{\hat {V_{i}}}{\hat {W... |
, V − 1 ] = − ℏ V − 1 [ J + , V + 1 ] = 0 [ J + , V 0 ] = 2 ℏ V + 1 [ J + , V − 1 ] = 2 ℏ V 0 [ J − , V + 1 ] = 2 ℏ V 0 [ J − , V 0 ] = 2 ℏ V − 1 [ J − , V − 1 ] = 0 {\displaystyle {\begin{aligned}\left[J_{z},V_{+1}\right]&=+\hbar V_{+1}\\[1ex]\left[J_{z},V_{0}\right]&=0V_{0}\\[1ex]\left[J_{z},V_{-1}\right]&=-\hbar V_{... |
k ⟩ = e x p ( − i θ ℏ n ^ ⋅ J → ) | j , k ⟩ {\displaystyle U(R)|j,k\rangle =|j,k\rangle -i{\frac {\theta }{\hbar }}{\hat {n}}\cdot {\vec {J}}|j,k\rangle +\sum _{k=2}^{\infty }{\frac {\left(-i{\frac {\theta }{\hbar }}{\hat {n}}\cdot {\vec {J}}\right)^{k}}{k!}}|j,k\rangle =exp\left({-i{\frac {\theta }{\hbar }}{\hat {n}}\... |
j = ∑ α q α ( 3 r α i r α j − r α 2 δ i j ) {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }(3r_{\alpha i}r_{\alpha j}-r_{\alpha }^{2}\delta _{ij})} Components of two tensor vector operators can be multiplied to give another Tensor operator. T i j = V i W j {\displaystyle T_{ij}=V_{i}W_{j}} In general, n number of tens... |
similarly by n vector operators. We observe that the subspace spanned by linear combinations of the rank two tensor components form an invariant subspace, ie. the subspace does not change under rotation since the transformed components itself is a linear combination of the tensor components. However, this subspace is n... |
{\displaystyle {\hat {T}}_{ij}={\hat {V_{i}}}{\hat {W_{j}}}} , the invariant subspaces of { T ^ i j } {\displaystyle \{{\hat {T}}_{ij}\}} formed are represented by: One invariant scalar operator V → ⋅ W → {\displaystyle {\vec {V}}\cdot {\vec {W}}} Three linearly independent components from 1 2 ( V ^ i W ^ j − V ^ j W ^... |
( i W ^ j ) − T i j ( 0 ) {\displaystyle {\widehat {T}}_{ij}^{(2)}={\tfrac {1}{2}}\left({\widehat {V}}_{i}{\widehat {W}}_{j}+{\widehat {V}}_{j}{\widehat {W}}_{i}\right)-{\tfrac {1}{3}}{\widehat {V}}_{k}{\widehat {W}}_{k}\delta _{ij}={\widehat {V}}_{(i}{\widehat {W}}_{j)}-T_{ij}^{(0)}} In general cartesian tensors of ra... |
of n ^ = x ^ ± i y ^ {\displaystyle {\hat {n}}={\hat {x}}\pm i{\hat {y}}} or n ^ = z ^ {\displaystyle {\hat {n}}={\hat {z}}} , we get: [ J ± , T ^ m ( j ) ] = ℏ ( j ∓ m ) ( j ± m + 1 ) T ^ m ± 1 ( j ) [ J z , T ^ m ( j ) ] = ℏ m T ^ m ( j ) {\displaystyle {\begin{aligned}\left[J_{\pm },{\widehat {T}}_{m}^{(j)}\right]&=... |
higher order spherical tensor operators. In general, spherical tensor operators can be constructed from two perspectives. One way is to specify how spherical tensors transform under a physical rotation - a group theoretical definition. A rotated angular momentum eigenstate can be decomposed into a linear combination of... |
R ) = ∑ q ′ D ( R ) q q ′ ( 2 ) ∗ T ^ q ′ ( 2 ) {\displaystyle {U(R)}^{\dagger }{\widehat {T}}_{q}^{(2)}U(R)=\sum _{q'}{{D(R)}_{qq'}^{(2)}}^{*}{\widehat {T}}_{q'}^{(2)}} ==== Using Spherical Harmonics ==== Define an operator by its spectrum: Υ l m | r ⟩ = r l Y l m ( θ , ϕ ) | r ⟩ = Υ l m ( r → ) | r ⟩ {\displaystyle \... |
(−1)±q will satisfy the commutation relations. The above choice of phase has the advantages of being real and that the tensor product of two commuting Hermitian operators is still Hermitian. Some authors define it with a different sign on q, without the k, or use only the floor of k. == Angular momentum and spherical h... |
are still many non-vanishing matrix elements to be calculated. A great simplification can be achieved by expressing the components of r, not with respect to the Cartesian basis, but with respect to the spherical basis. First we define, r q = e ^ q ⋅ r {\displaystyle r_{q}={\hat {\mathbf {e} }}_{q}\cdot \mathbf {r} } Ne... |
1 m q ⟩ {\displaystyle \langle \ell 'm'|\ell 1mq\rangle } The radial integral is independent of the three magnetic quantum numbers (m′, q, m), and the trick we have just used does not help us to evaluate it. But it is only one integral, and after it has been done, all the other integrals can be evaluated just by comput... |
978-0-306-47123-0. K.T. Hecht (2000). Quantum mechanics. Graduate texts in contemporary physics. Springer. ISBN 978-0-387-989-198. ==== Condensed matter physics ==== J.A. Mettes; J.B. Keith; R.B. McClurg (2002). "Molecular Crystal Global Phase Diagrams:I Method of Construction" (PDF). B.Henderson, R.H. Bartram (2005). ... |
In mathematics and theoretical physics, a tensor is antisymmetric or alternating on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant. For example, T i j k … = − T j i k … = T... |
d e f . {\displaystyle {\begin{aligned}M_{[ab]}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}} where δ a b … c d … {\displaystyle \delta _{ab\dots }^{cd\dots }} is the generalized Kronecker delta, and the Einstein summation convention is in use. M... |
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravita... |
}{}_{\nu }=T^{\mu \alpha }g_{\alpha \nu }.} This article uses the spacelike sign convention (− + + +) for the metric signature. == Conservation law == === In special relativity === The stress–energy tensor is the conserved Noether current associated with spacetime translations. The divergence of the non-gravitational s... |
of this is 0 = ∫ ∂ N ξ μ T ν μ − g d 3 s ν . {\displaystyle 0=\int _{\partial N}\xi ^{\mu }T^{\nu }{}_{\mu }{\sqrt {-g}}\ \mathrm {d} ^{3}s_{\nu }\,.} == In special relativity == In special relativity, the stress–energy tensor contains information about the energy and momentum densities of a given system, in addition t... |
}{\mathcal {L}}=\partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\phi _{\alpha }\right]} Now, in flat space, one can write d ν L = ∂ μ [ δ ν μ L ] {\textstyle d_{\nu }{\mathcal {L}}=\partial _{\mu }[\delta _{\nu }^{\mu }{\mathcal {L}}]} . Doing this and ... |
(\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\phi _{\alpha }-\delta _{\mu }^{\mu }{\mathcal {L}}.} Since δ μ μ = 4 {\displaystyle \delta _{\mu }^{\mu }=4} , T μ μ = ∂ L ∂ ( ∂ μ ϕ α ) ∂ μ ϕ α − 4 L . {\displaystyle T^{\mu }{}_{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\p... |
{x} _{\text{p}}(t)} is: T α β ( x , t ) = m v α ( t ) v β ( t ) 1 − ( v / c ) 2 δ ( x − x p ( t ) ) = E c 2 v α ( t ) v β ( t ) δ ( x − x p ( t ) ) {\displaystyle T^{\alpha \beta }(\mathbf {x} ,t)={\frac {m\,v^{\alpha }(t)v^{\beta }(t)}{\sqrt {1-(v/c)^{2}}}}\;\,\delta \left(\mathbf {x} -\mathbf {x} _{\text{p}}(t)\right... |
0 0 0 p 0 0 0 0 p ) . {\displaystyle T^{\alpha \beta }=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).} === Electromagnetic stress–energy tensor === The Hilbert stress–energy tensor of a source-free electromagnetic field is T μ ν = 1 μ 0 ( F μ α g α β F ν β − 1 4 g μ ν F δ γ F δ γ ) {\d... |
2 ∂ L m a t t e r ∂ g μ ν + g μ ν L m a t t e r , {\displaystyle T_{\mu \nu }={\frac {-2}{\sqrt {-g}}}{\frac {\delta S_{\mathrm {matter} }}{\delta g^{\mu \nu }}}={\frac {-2}{\sqrt {-g}}}{\frac {\partial \left({\sqrt {-g}}{\mathcal {L}}_{\mathrm {matter} }\right)}{\partial g^{\mu \nu }}}=-2{\frac {\partial {\mathcal {L}... |
In linear algebra, linear transformations can be represented by matrices. If T {\displaystyle T} is a linear transformation mapping R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} and x {\displaystyle \mathbf {x} } is a column vector with n {\displaystyle n} entries, then there exists an m ... |
process (suppose that n = 2 in this case) reveals that: T ( x ) = 5 x = 5 I x = [ 5 0 0 5 ] x {\displaystyle T(\mathbf {x} )=5\mathbf {x} =5I\mathbf {x} ={\begin{bmatrix}5&0\\0&5\end{bmatrix}}\mathbf {x} } The matrix representation of vectors and operators depends on the chosen basis; a similar matrix will result from ... |
elements, a i , j {\displaystyle a_{i,j}} , of j-th column of the matrix A. === Eigenbasis and diagonal matrix === Yet, there is a special basis for an operator in which the components form a diagonal matrix and, thus, multiplication complexity reduces to n. Being diagonal means that all coefficients a i , j {\displays... |
an angle θ counterclockwise (positive direction) about the origin the functional form is x ′ = x cos θ − y sin θ {\displaystyle x'=x\cos \theta -y\sin \theta } and y ′ = x sin θ + y cos θ {\displaystyle y'=x\sin \theta +y\cos \theta } . Written in matrix form, this becomes: [ x ′ y ′ ] = [ cos θ − sin θ sin... |
2 ] {\displaystyle \mathbf {A} ={\frac {1}{\lVert \mathbf {u} \rVert ^{2}}}{\begin{bmatrix}u_{x}^{2}&u_{x}u_{y}\\u_{x}u_{y}&u_{y}^{2}\end{bmatrix}}} As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also li... |
is a unit vector): [ x ′ y ′ z ′ 1 ] = [ 1 − 2 a 2 − 2 a b − 2 a c − 2 a d − 2 a b 1 − 2 b 2 − 2 b c − 2 b d − 2 a c − 2 b c 1 − 2 c 2 − 2 c d 0 0 0 1 ] [ x y z 1 ] {\displaystyle {\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}}={\begin{bmatrix}1-2a^{2}&-2ab&-2ac&-2ad\\-2ab&1-2b^{2}&-2bc&-2bd\\-2ac&-2bc&1-2c^{2}&-2cd\\0&0&0... |
expressed with matrix multiplication. The functional form x ′ = x + t x ; y ′ = y + t y {\displaystyle x'=x+t_{x};y'=y+t_{y}} becomes: [ x ′ y ′ 1 ] = [ 1 0 t x 0 1 t y 0 0 1 ] [ x y 1 ] . {\displaystyle {\begin{bmatrix}x'\\y'\\1\end{bmatrix}}={\begin{bmatrix}1&0&t_{x}\\0&1&t_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\... |
However, this is not true when using perspective projections. === Perspective projection === Another type of transformation, of importance in 3D computer graphics, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection... |
Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics. == Vector and tensor algebra in three-dimensional curvilinear coor... |
, Z 3 ) {\displaystyle Z^{\acute {i}}=g^{\acute {i}}(Z^{1},Z^{2},Z^{3})} for i ´ = 1 , 2 , 3 {\displaystyle {\acute {i}}=1,2,3} and we can write the free equations more compactly as Z i ´ = Z i ´ ( Z 1 , Z 2 , Z 3 ) = Z i ´ ( Z i ) {\displaystyle Z^{\acute {i}}=Z^{\acute {i}}(Z^{1},Z^{2},Z^{3})=Z^{\acute {i}}(Z^{i})} f... |
{1}}^{3}&J_{\acute {2}}^{3}&J_{\acute {3}}^{3}\end{pmatrix}}={\begin{pmatrix}{\partial {Z^{1}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{2}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{2}} \over \partial {Z^{\... |
; b i = g i j b j ; b i = g i j b j {\displaystyle v^{i}=g^{ik}~v_{k}~;~~v_{i}=g_{ik}~v^{k}~;~~\mathbf {b} ^{i}=g^{ij}~\mathbf {b} _{j}~;~~\mathbf {b} _{i}=g_{ij}~\mathbf {b} ^{j}} The components of a vector are related by: 30–32 v ⋅ b i = v k b k ⋅ b i = v k δ k i = v i {\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i... |
k {\displaystyle {\mathcal {E}}^{ijk}={\cfrac {1}{J}}~\varepsilon ^{ijk}={\cfrac {1}{\sqrt {g}}}~\varepsilon ^{ijk}} === Vector operations === ==== Identity map ==== The identity map I {\displaystyle \mathbf {I} } defined by I ⋅ v = v {\displaystyle \mathbf {I} \cdot \mathbf {v} =\mathbf {v} } can be shown to be:: 39 I... |
{x} }{\partial q^{m}}}\times {\frac {\partial \mathbf {x} }{\partial q^{n}}}={\frac {\partial (x_{p}\mathbf {e} _{p})}{\partial q^{m}}}\times {\frac {\partial (x_{q}\mathbf {e} _{q})}{\partial q^{n}}}={\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}\mathbf {e} _{p}\times \mathbf {e} _{q}... |
of two second-order tensors U = S ⋅ T {\displaystyle {\boldsymbol {U}}={\boldsymbol {S}}\cdot {\boldsymbol {T}}} can be expressed in curvilinear coordinates as U i j b i ⊗ b j = S i k T . j k b i ⊗ b j = S i . k T k j b i ⊗ b j {\displaystyle U_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=S_{ik}T_{.j}^{k}\mathbf {b} ^{... |
the determinant, [ b 1 , b 2 , b 3 ] = det F [ e 1 , e 2 , e 3 ] . {\displaystyle \left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\det {\boldsymbol {F}}\left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]~.} Since [ e 1 , e 2 , e 3 ] = 1 {\displaystyle \left[\mathbf {e} _{1},\mathbf {e} _{2},... |
g ( b i × b j ) {\displaystyle \varepsilon _{ijk}~\mathbf {b} ^{k}={\cfrac {1}{J}}(\mathbf {b} _{i}\times \mathbf {b} _{j})={\cfrac {1}{\sqrt {g}}}(\mathbf {b} _{i}\times \mathbf {b} _{j})} where ε i j k {\displaystyle \varepsilon _{ijk}} is the usual permutation symbol. We have not identified an explicit expression fo... |
that such a mapping and its inverse exist and are continuous, we can write : 55 x = φ ( q 1 , q 2 , q 3 ) ; q i = ψ i ( x ) = [ φ − 1 ( x ) ] i {\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})~;~~q^{i}=\psi ^{i}(\mathbf {x} )=[{\boldsymbol {\varphi }}^{-1}(\mathbf {x} )]^{i}} The fields ψ i ( x )... |
c i = ∂ f ∂ q i c i {\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f_{\varphi }(q^{1}+\alpha ~c^{1},q^{2}+\alpha ~c^{2},q^{3}+\alpha ~c^{3}){\biggr |}_{\alpha =0}={\cfrac {\partial f_{\varphi }}{\partial q^{i}}}~c^{i}={\cfrac {\partial f}{\partial q^{i}}}~c... |
basis, b i {\displaystyle \mathbf {b} ^{i}} . All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x {\displaystyle \mathbf {x} } . Since c {\displaystyle \mathbf {c} } is arbitrary, we can write ∇ f ( x ) = ∂ ... |
i ⋅ b k ) , j + ( b j ⋅ b k ) , i − ( b i ⋅ b j ) , k ] {\displaystyle \Gamma _{ijk}={\frac {1}{2}}(g_{ik,j}+g_{jk,i}-g_{ij,k})={\frac {1}{2}}[(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}+(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}-(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}]} ==== Christoffel symbols of the secon... |
= b i g i i {\displaystyle {\hat {\mathbf {b} }}^{i}={\cfrac {\mathbf {b} ^{i}}{\sqrt {g^{ii}}}}} is the normalized contravariant basis vector. === Second-order tensor field === The gradient of a second order tensor field can similarly be expressed as ∇ S = ∂ S ∂ q i ⊗ b i {\displaystyle {\boldsymbol {\nabla }}{\boldsy... |
}}\mathbf {v} )} In terms of components with respect to a curvilinear basis ∇ ⋅ v = ∂ v i ∂ q i + Γ ℓ i i v ℓ = [ ∂ v i ∂ q j − Γ j i ℓ v ℓ ] g i j {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }=\left[{\cfrac {\partial v_{i}}{\partial q^... |
g m i ∂ g i m ∂ q ℓ v ℓ = ∂ v i ∂ q i + 1 2 g ∂ g ∂ q ℓ v ℓ {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1}{2g}}~{\frac {\partial g}{\partial g_{mi}}}~{\frac {\partial g_{im}}{\partial q^{\ell }}}~v^{\ell }={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1... |
Therefore, ∇ 2 φ = 1 g ∂ ∂ q i ( g l i ∂ φ ∂ q l g ) {\displaystyle \nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}\left(g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~{\sqrt {g}}\right)} === Curl of a vector field === The curl of a vector field v {\displaystyle \mathbf {v} } in cova... |
three-dimensional orthogonal curvilinear coordinates ( q 1 , q 2 , q 3 ) {\displaystyle (q^{1},q^{2},q^{3})} as d x = ∑ i = 1 3 ∑ j = 1 3 ( ∂ x i ∂ q j e i ) d q j {\displaystyle \mathrm {d} \mathbf {x} =\sum _{i=1}^{3}\sum _{j=1}^{3}\left({\cfrac {\partial x_{i}}{\partial q^{j}}}~\mathbf {e} _{i}\right)\mathrm {d} q^{... |
) {\displaystyle \mathbf {b} _{r}=(\cos \theta ,\sin \theta )} , b θ = ( − r sin θ , r cos θ ) {\displaystyle \mathbf {b} _{\theta }=(-r\sin \theta ,r\cos \theta )} . The normalized basis vectors are e r = ( cos θ , sin θ ) {\displaystyle \mathbf {e} _{r}=(\cos \theta ,\sin \theta )} , e θ = ( − sin θ , cos ... |
g_{ij}=0} when i ≠ j {\displaystyle i\neq j} , we have | ∂ x ∂ t | = ∑ i g i i ( ∂ q i ∂ t ) 2 = ∑ i h i 2 ( ∂ q i ∂ t ) 2 {\displaystyle \left|{\partial \mathbf {x} \over \partial t}\right|={\sqrt {\sum _{i}g_{ii}~\left({\cfrac {\partial q^{i}}{\partial t}}\right)^{2}}}={\sqrt {\sum _{i}h_{i}^{2}~\left({\cfrac {\parti... |
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