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{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 δβ1⋅⋅⋅βpα1⋅⋅⋅αp 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 α β γ ⋯ = A ( α β ) γ ⋯ + A [ α β ] γ ⋯ {\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... |
Γ α γ β . {\displaystyle \Gamma ^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\gamma \beta }.} === Riemann curvature tensor === If this tensor is defined as R ρ σ μ ν = Γ ρ ν σ , μ − Γ ρ μ σ , ν + Γ ρ μ λ Γ λ ν σ − Γ ρ ν λ Γ λ μ σ , {\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 γ is any smooth strictly monotone parameterization of the trajectory. See also Line element. The inverse matrix gαβ of the metric tensor... |
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vect... |
V of ker ( L ) {\displaystyle \ker(L)} . This is the generalization to linear operators of the row space, or coimage, of a matrix. == Generalization to modules == The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, ra... |
= c0 = 0. === The row space of a matrix === The product Ax can be written in terms of the dot product of vectors as follows: A x = [ a 1 ⋅ x a 2 ⋅ x ⋮ a m ⋅ x ] . {\displaystyle A\mathbf {x} ={\begin{bmatrix}\mathbf {a} _{1}\cdot \mathbf {x} \\\mathbf {a} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {a} _{m}\cdot \mathbf {... |
A v = b − b = 0 {\displaystyle A(\mathbf {u} -\mathbf {v} )=A\mathbf {u} -A\mathbf {v} =\mathbf {b} -\mathbf {b} =\mathbf {0} } Thus, the difference of any two solutions to the equation Ax = b lies in the kernel of A. It follows that any solution to the equation Ax = b can be expressed as the sum of a fixed solution v ... |
(in this case, a line through the origin in R3). Here, the vector (−1,−26,16)T constitutes a basis of the kernel of A. The nullity of A is therefore 1, as it is spanned by a single vector. The following dot products are zero: [ 2 3 5 ] [ − 1 − 26 16 ] = 0 a n d [ − 4 2 3 ] [ − 1 − 26 16 ] = 0 , {\displaystyle {\begin{b... |
the orthogonal projection V → W is the orthogonal complement to W in V. == Computation by Gaussian elimination == A basis of the kernel of a matrix may be computed by Gaussian elimination. For this purpose, given an m × n matrix A, we construct first the row augmented matrix [ A I ] , {\displaystyle {\begin{bmatrix}A\\... |
exists an invertible matrix P {\displaystyle P} such that [ A I ] P = [ B C ] , {\displaystyle {\begin{bmatrix}A\\\hline I\end{bmatrix}}P={\begin{bmatrix}B\\\hline C\end{bmatrix}},} with B {\displaystyle B} in column echelon form. Thus A P = B {\displaystyle AP=B} , I P = C {\displaystyle IP=C} , and A C = B {\displays... |
of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the Lapack library. == See also == == Notes and references == == Bibliography == == ... |
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tens... |
covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative ∇ X , Y 2 Z = ∇ X ∇ Y Z − ∇ ∇ ... |
g ) ≡ ( g ) ; μ = 0 , {\displaystyle \nabla _{\mu }\left({\sqrt {g}}\right)\equiv \left({\sqrt {g}}\right)_{;\mu }=0,} where g = | det ( g μ ν ) | . {\displaystyle g=\left|\det \left(g_{\mu \nu }\right)\right|.} It is sometimes convenient to also define the purely covariant version of the curvature tensor by R σ μ ν ρ ... |
width. The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be t... |
t τ s X − 1 τ t Y − 1 τ s X τ t Y Z | s = t = 0 = ( ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X , Y ] ) Z = R ( X , Y ) Z {\displaystyle \left.{\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}^{-1}\tau _{tY}^{-1}\tau _{sX}\tau _{tY}Z\right|_{s=t=0}=\left(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]}\right)Z=R(X,Y)Z} where R {\dis... |
Ricci curvature tensor is the contraction of the first and third indices of the Riemann tensor. R a b ⏟ Ricci ≡ R c a c b = g c d R c a d b ⏟ Riemann {\displaystyle \underbrace {R_{ab}} _{\text{Ricci}}\equiv R^{c}{}_{acb}=g^{cd}\underbrace {R_{cadb}} _{\text{Riemann}}} == Special cases == === Surfaces === For a two-dim... |
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantu... |
tensor(s) to be differentiated and a line joined from the circle pointing downwards to represent the lower index of the derivative. == Tensor manipulation == The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "identities" of tensor manipulations. For example, ε a . . . ... |
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})} for every permutation σ of the symbols {1, 2, ... |
= 0 ∞ Sym k ( V ) . {\displaystyle \operatorname {Sym} (V)=\bigoplus _{k=0}^{\infty }\operatorname {Sym} ^{k}(V).} == Examples == There are many examples of symmetric tensors. Some include, the metric tensor, g μ ν {\displaystyle g_{\mu \nu }} , the Einstein tensor, G μ ν {\displaystyle G_{\mu \nu }} and the Ricci te... |
of the tensor appearing on the right are often denoted by T ( i 1 i 2 ⋯ i k ) = 1 k ! ∑ σ ∈ S k T i σ 1 i σ 2 ⋯ i σ k {\displaystyle T_{(i_{1}i_{2}\cdots i_{k})}={\frac {1}{k!}}\sum _{\sigma \in {\mathfrak {S}}_{k}}T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma k}}} with parentheses () around the indices being symmetrized... |
vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of inertia. Also see Sylvester's law of inertia. For symmetric tensors of arbi... |
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Co... |
and some numerical algorithms have grown in prominence as technologies like parallel computing have made them practical approaches to scientific problems. == Matrix decompositions == === Partitioned matrices === For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a co... |
− 1 {\displaystyle M_{1},\ldots ,M_{n-1}} to form the product M n − 1 ⋯ M 1 A = U {\displaystyle M_{n-1}\cdots M_{1}A=U} , so that equivalently L = M 1 − 1 ⋯ M n − 1 − 1 {\displaystyle L=M_{1}^{-1}\cdots M_{n-1}^{-1}} .: 147 : 96 === Eigenvalue decomposition === The eigenvalue decomposition of a matrix A m × m {\displa... |
b'=X^{-1}b} and x ′ = X − 1 x {\displaystyle x'=X^{-1}x} , then we have b ′ = Λ x ′ {\displaystyle b'=\Lambda x'} .: 33 This is closely related to the solution to the linear system using the singular value decomposition, because singular values of a matrix are the absolute values of its eigenvalues, which are also equi... |
of equation or least squares optimisation may produce highly inaccurate results. Creating stable algorithms for ill-conditioned problems is a central concern in numerical linear algebra. One example is that the stability of householder triangularization makes it a particularly robust solution method for linear systems,... |
Perl Data Language. Many numerical linear algebra commands in R rely on these more fundamental libraries like LAPACK. More libraries can be found on the List of numerical libraries. == References == == Further reading == Dongarra, Jack; Hammarling, Sven (1990). "Evolution of Numerical Software for Dense Linear Algebra"... |
The following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage. == Dense linear algebra == === General information === === Matrix types and operations === Matrix types (special types like bidiagonal/tridiagonal are... |
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for comp... |
k. The minor of order zero is often defined to be 1. For a square matrix, the zeroth minor is just the determinant of the matrix. Let I = 1 ≤ i 1 < i 2 < ⋯ < i k ≤ m , J = 1 ≤ j 1 < j 2 < ⋯ < j k ≤ n , {\displaystyle {\begin{aligned}I&=1\leq i_{1}<i_{2}<\cdots <i_{k}\leq m,\\[2pt]J&=1\leq j_{1}<j_{2}<\cdots <j_{k}\leq ... |
and cofactors == === Cofactor expansion of the determinant === The cofactors feature prominently in Laplace's formula for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an n × n matrix A = (aij), the determinant of A, denoted det(A), can be written as t... |
i_{1}<i_{2}<\ldots <i_{k}\leq n,\\[2pt]J&=1\leq j_{1}<j_{2}<\ldots <j_{k}\leq n,\end{aligned}}} be ordered sequences (in natural order) of indexes (here A is an n × n matrix). Then [ A − 1 ] I , J = ± [ A ] J ′ , I ′ det A , {\displaystyle [\mathbf {A} ^{-1}]_{I,J}=\pm {\frac {[\mathbf {A} ]_{J',I'}}{\det \mathbf {A} }... |
× r minor, while all larger minors are zero. We will use the following notation for minors: if A is an m × n matrix, I is a subset of {1, ..., m} with k elements, and J is a subset of {1, ..., n} with k elements, then we write [A]I, J for the k × k minor of A that corresponds to the rows with index in I and the columns... |
e 2 + 2 e 3 ) ∧ ( 4 e 1 − e 2 + e 3 ) {\displaystyle (\mathbf {e} _{1}+3\mathbf {e} _{2}+2\mathbf {e} _{3})\wedge (4\mathbf {e} _{1}-\mathbf {e} _{2}+\mathbf {e} _{3})} where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and altern... |
In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace N {\displaystyle N} is a vector space obtained by "collapsing" N {\displaystyle N} to zero. The space obtained is called a quotient space and is denoted V / N {\displaystyle V/N} (read " V {\displaystyle V} mod N {\displaystyle N} " or ... |
Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. Note that the points along any one such line wil... |
equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence 0 → U → V → V / U → 0. {\displaystyle 0\to U\to V\to V/U\to 0.\,} If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of... |
=== Generalization to locally convex spaces === The quotient of a locally convex space by a closed subspace is again locally convex. Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈ A} where A is an index set. Let M be a closed subspace, and define semin... |
In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} has two arguments, x {\displaystyle x} and y {\displaystyle y} , in an ordered pair ( x , ... |
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant). A mixed tensor of type or valence ( M N ) {\textstyle {\bino... |
γ ϵ , {\displaystyle T_{\alpha }{}^{\lambda \epsilon }=T_{\alpha \beta \gamma }\,g^{\beta \lambda }\,g^{\gamma \epsilon },} T α β γ = g γ λ T α β λ , {\displaystyle T^{\alpha \beta }{}_{\gamma }=g_{\gamma \lambda }\,T^{\alpha \beta \lambda },} T α λ ϵ = g λ β g ϵ γ T α β γ . {\displaystyle T^{\alpha }{}_{\lambda \epsil... |
In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X , Y {\displaystyle X,Y} , that produces an output vector T ( X , Y ) {\displaystyle T(X,Y)} representing the displacement within a tangent space when the tang... |
{\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.} Here Γ k i j {\displaystyle {\Gamma ^{k}}_{ij}} are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, γ k i j = 0 {\displaystyle \gamma ^{k}{}_{ij}... |
e j − ∇ e j e i − [ e i , e j ] ) {\displaystyle {T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)} are the frame-components of the torsion tensor, as given in the previous definition. It can be easil... |
∇ is a mapping TM × TM → End(TM) defined on vector fields X, Y, and Z by R ( X , Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z . {\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.} For vectors at a point, this definition is independent of how the vectors are extended to vector fiel... |
of slipping or twisting that a plane does when rolling along a surface or higher dimensional affine manifold. For example, consider rolling a plane along a small circle drawn on a sphere. If the plane does not slip or twist, then when the plane is rolled all the way along the circle, it will also trace a circle in the ... |
2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} by the (Euclidean) cross product: ∇ e i e j = e i × e j . {\displaystyle \nabla _{e_{i}}e_{j}=e_{i}\times e_{j}.} Consider now the parallel transport of the vector e 2 {\displaystyle e_{2}} along the e 1 {\displaystyle e_{1}} axis, starting at the origin. The parallel vector fi... |
the developed curve γ ~ {\displaystyle {\tilde {\gamma }}} is also a closed loop (so that γ ~ ( 0 ) = γ ~ ( 1 ) {\displaystyle {\tilde {\gamma }}(0)={\tilde {\gamma }}(1)} ). On the other hand, if the torsion is non-zero, then the developed curve may not be closed, so that γ ~ ( 0 ) ≠ γ ~ ( 1 ) {\displaystyle {\tilde {... |
dimensions, with curvature 2-form Ω a b {\displaystyle \Omega _{a}^{b}} and torsion 2-form Θ a = D θ a {\displaystyle \Theta ^{a}=D\theta ^{a}} . Let η a b c {\displaystyle \eta _{abc}} be the skew-symmetric Levi-Civita tensor, and t a = 1 2 η a b c ∧ Ω b c , {\displaystyle t_{a}={\tfrac {1}{2}}\eta _{abc}\wedge \Omega... |
Y ) − Δ ( Y , X ) ) {\displaystyle A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)} Then A ( X , Y ) = 1 2 ( T ( X , Y ) − T ′ ( X , Y ) ) {\displaystyle A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)} is the difference of the torsion tensors. ∇ and ∇′ define the same families of affinely parametrized g... |
(2011) Rolling without slipping interpretation of torsion, URL (version: 2011-01-27). |
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a continuum) rather than as discrete particles. Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes... |
the material body B {\displaystyle {\mathcal {B}}} being modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time t {\displaystyle t} is ... |
upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's third law of motion of conservation of linear momentum and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The ... |
Body forces === Body forces are forces originating from sources outside of the body that act on the volume (or mass) of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body (internal forces) are manifested through the contact forces alone. These f... |
field, and the dislocation theory of metals. Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials. Non-polar materials are then those materials with only moments of forces. In the classical branches of continuum mechanics the developmen... |
throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description. === Lagrangian description === In the Lagrangian description the position and physical properties of the particles are described in terms of the material or ref... |
held constant as it does not change with time. Thus, we have d d t [ P i j … ( X , t ) ] = ∂ ∂ t [ P i j … ( X , t ) ] {\displaystyle {\frac {d}{dt}}[P_{ij\ldots }(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}[P_{ij\ldots }(\mathbf {X} ,t)]} The instantaneous position x {\displaystyle \mathbf {x} } is a property of ... |
X = χ − 1 ( x , t ) {\displaystyle \mathbf {X} =\chi ^{-1}(\mathbf {x} ,t)} which provides a tracing of the particle which now occupies the position x {\displaystyle \mathbf {x} } in the current configuration κ t ( B ) {\displaystyle \kappa _{t}({\mathcal {B}})} to its original position X {\displaystyle \mathbf {X} } i... |
particle P {\displaystyle P} in the undeformed configuration and deformed configuration is called the displacement vector u ( X , t ) = u i e i {\displaystyle \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}} , in the Lagrangian description, or U ( x , t ) = U J E J {\displaystyle \mathbf {U} (\mathbf {x} ,t)=U_{J}\m... |
u ( X , t ) = x ( X , t ) − X or u i = x i − δ i J X J {\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=x_{i}-\delta _{iJ}X_{J}} or in terms of the spatial coordinates as U ( x , t ) = x − X ( x , t ) or U J = δ J i x i − X J {\displaystyle \mathbf {U} ... |
{n} (\mathbf {x} ,t)} be the outward unit normal to the surface ∂ Ω {\displaystyle \partial \Omega } . Let v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} be the flow velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface ∂ Ω {\dis... |
ρ ˙ {\displaystyle {\dot {\rho }}} is the material time derivative of ρ {\displaystyle \rho } , v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} is the particle velocity, v ˙ {\displaystyle {\dot {\mathbf {v} }}} is the material time derivative of v {\displaystyle \mathbf {v} } , σ ( x , t ) {\displaystyle {\bo... |
˙ + ∇ ∘ ⋅ q − ρ 0 s = 0 Balance of Energy. {\displaystyle {\begin{aligned}\rho ~\det({\boldsymbol {F}})-\rho _{0}&=0&&\qquad {\text{Balance of Mass}}\\\rho _{0}~{\ddot {\mathbf {x} }}-{\boldsymbol {\nabla }}_{\circ }\cdot {\boldsymbol {N}}^{T}-\rho _{0}~\mathbf {b} &=0&&\qquad {\text{Balance of Linear Momentum}}\\{\bol... |
for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the... |
{\displaystyle \mathbf {x} } at time t {\displaystyle t} . We then have the Clausius–Duhem inequality in integral form: d d t ( ∫ Ω ρ η dV ) ≥ ∫ ∂ Ω ρ η ( u n − v ⋅ n ) dA − ∫ ∂ Ω q ⋅ n T dA + ∫ Ω ρ s T dV . {\displaystyle {{\cfrac {d}{dt}}\left(\int _{\Omega }\rho ~\eta ~{\text{dV}}\right)\geq \int _{\partial \Omega }... |
Mechanica. 138 (3–4): 155–162. doi:10.1007/BF01291841. S2CID 120320672. Fung, Y. C. (1977). A First Course in Continuum Mechanics (2nd ed.). Prentice-Hall, Inc. ISBN 978-0-13-318311-5. Irgens, Fridtjov (10 January 2008). Continuum Mechanics. Springer Science & Business Media. ISBN 978-3-540-74298-2. Liu, I-Shih (28 May... |
Mathematics of Adiabatic Shear Bands. Cambridge, UK: Cambridge University Press. == External links == "Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity" by Gilles Leborgne, April 7, 2021: "Pa... |
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... |
z 3 = 12 , 3 x 3 + 5 y 3 + 3 z 3 = 34 {\displaystyle {\begin{aligned}x^{3}+y^{3}+z^{3}&=10,\\x^{3}+2y^{3}+z^{3}&=12,\\3x^{3}+5y^{3}+3z^{3}&=34\end{aligned}}} has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus... |
3 y 2 = 4 {\displaystyle {\begin{aligned}x^{2}+y^{2}&=1,\\x^{2}+2y^{2}&=2,\\2x^{2}+3y^{2}&=4\end{aligned}}} is inconsistent because the sum of the first two equations contradicts the third one. == Criteria for consistency == As can be seen from the above examples, consistency versus inconsistency is a different issue f... |
In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and is denoted as Z(R); 'Z' stands for the German word Zentrum, meaning "center". If R is a ring, then R is an associative algebra over its center. Conversely, if R is an as... |
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metr... |
i | {\displaystyle \|x\|=\max _{}|x_{i}|} and define multiplication componentwise: ( x 1 , … , x n ) ( y 1 , … , y n ) = ( x 1 y 1 , … , x n y n ) . {\displaystyle \left(x_{1},\ldots ,x_{n}\right)\left(y_{1},\ldots ,y_{n}\right)=\left(x_{1}y_{1},\ldots ,x_{n}y_{n}\right).} The quaternions form a 4-dimensional real Bana... |
functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric seri... |
{C} } with radius ‖ x ‖ {\displaystyle \|x\|} and center 0 , {\displaystyle 0,} and thus is compact. Moreover, the spectrum σ ( x ) {\displaystyle \sigma (x)} of an element x {\displaystyle x} is non-empty and satisfies the spectral radius formula: sup { | λ | : λ ∈ σ ( x ) } = lim n → ∞ ‖ x n ‖ 1 / n . {\displaystyle ... |
ideals of A {\displaystyle A} and the set Δ ( A ) {\displaystyle \Delta (A)} of all nonzero homomorphisms from A {\displaystyle A} to C . {\displaystyle \mathbb {C} .} The set Δ ( A ) {\displaystyle \Delta (A)} is called the "structure space" or "character space" of A , {\displaystyle A,} and its members "characters". ... |
( x ∗ ) ∗ = x {\displaystyle \left(x^{*}\right)^{*}=x} for all x ∈ A {\displaystyle x\in A} (so the map is an involution). ( x + y ) ∗ = x ∗ + y ∗ {\displaystyle (x+y)^{*}=x^{*}+y^{*}} for all x , y ∈ A . {\displaystyle x,y\in A.} ( λ x ) ∗ = λ ¯ x ∗ {\displaystyle (\lambda x)^{*}={\bar {\lambda }}x^{*}} for every λ ∈ ... |
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can b... |
with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.) It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom,... |
∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c) The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself. After the two-element Boolea... |
∪ (union) and ∧ := ∩ (intersection). If R is an arbitrary ring then its set of central idempotents, which is the set A = { e ∈ R : e 2 = e and e x = x e for all x ∈ R } , {\displaystyle A=\left\{e\in R:e^{2}=e{\text{ and }}ex=xe\;{\text{ for all }}\;x\in R\right\},} becomes a Boolean algebra when its operations are def... |
Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring. More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Bo... |
representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all clopen sets in some (compact totally disconnected Hausdorff) topological space. == Axiomatics == The first axiomatization of Boolean lattices/algebras in general was given by the English philosoph... |
Works cited === Davey, B.A.; Priestley, H.A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. Cohn, Paul M. (2003), Basic Algebra: Groups, Rings, and Fields, Springer, pp. 51, 70–81, ISBN 9781852335878 Givant, Steven; Halmos, Paul (2009), Introduction to Boolean ... |
Wolfram Demonstrations Project, 2007. Burris, Stanley N.; Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2. Weisstein, Eric W. "Boolean Algebra". MathWorld. |
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.... |
+g^{n-1})=1-g^{n},} shows that an element g of finite order n > 1 induces a zero divisor 1 − g in R. The zero divisor problem asks whether this is the only obstruction; in other words, Given a field K and a torsion-free group G, is it true that K[G] contains no zero divisors? No counterexamples are known, but the probl... |
Let ϕ : M → N {\displaystyle \phi :M\to N} be a smooth map between smooth manifolds M {\displaystyle M} and N {\displaystyle N} . Then there is an associated linear map from the space of 1-forms on N {\displaystyle N} (the linear space of sections of the cotangent bundle) to the space of 1-forms on M {\displaystyle M} ... |
sheaf of smooth functions on M {\displaystyle M} .) More generally, if f : N → A {\displaystyle f:N\to A} is a smooth map from N {\displaystyle N} to any other manifold A {\displaystyle A} , then ( ϕ ∗ f ) ( x ) = f ( ϕ ( x ) ) {\displaystyle (\phi ^{*}f)(x)=f(\phi (x))} is a smooth map from M {\displaystyle M} to A {\... |
Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself: Φ : V → W , Φ ∗ : W ∗ → V ∗ . {\displaystyle \Phi \colon V\rightarrow W,\qquad \Phi ^{*}\colon W^{*}\rightarrow V^{*}.} From a tensorial point of view, it is natural to try to extend the notion of pullback to ten... |
for x {\displaystyle x} in M {\displaystyle M} and X {\displaystyle X} in T x M {\displaystyle T_{x}M} . == Pullback of (covariant) tensor fields == The construction of the previous section generalizes immediately to tensor bundles of rank ( 0 , s ) {\displaystyle (0,s)} for any natural number s {\displaystyle s} : a (... |
{\displaystyle X_{j}} in T x M {\displaystyle T_{x}M} . The pullback of differential forms has two properties which make it extremely useful. It is compatible with the wedge product in the sense that for differential forms α {\displaystyle \alpha } and β {\displaystyle \beta } on N {\displaystyle N} , ϕ ∗ ( α ∧ β ) = ϕ... |
== Pullback and Lie derivative == See Lie derivative. By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on M {\displaystyle M} , and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained. == Pullback of co... |
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