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sphere ==== The unit sphere in ℝ3 comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. In standard spherical coordinates (θ, φ), with θ the colatitude, the angle measured from the z-axis, and φ the angle from the x-axis in the xy-pl... |
indicatrix, a technique to visualize the metric tensor == Notes == == References == Dodson, C. T. J.; Poston, T. (1991), Tensor geometry, Graduate Texts in Mathematics, vol. 130 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-10514-2, ISBN 978-3-540-52018-4, MR 1223091 Gallot, Sylvestre; Hulin, Domi... |
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 linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism) such that P ∘ P = P {\displaystyle P\circ P=P} . That is, whenever P {\displaystyle P} is applied twice to any vector, it gives the same result as if it were applied onc... |
vector is P [ x y z ] = [ x y 0 ] . {\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.} To see that P {\displaystyle P} is indeed a projection, i.e., P = P 2 {\displaystyle P=P^{2}} , we compute P 2 [ x y z ] = P [ x y 0 ] = [ x y 0 ] = P [ x y z ] . {\displaystyle P^{2}{\begin... |
P x = ( I − P ) x {\displaystyle \mathbf {v} =\mathbf {x} -P\mathbf {x} =\left(I-P\right)\mathbf {x} } , and where u ∈ U , v ∈ V . {\displaystyle \mathbf {u} \in U,\mathbf {v} \in V.} The image and kernel of a projection are complementary, as are P {\displaystyle P} and Q = I − P {\displaystyle Q=I-P} . The operator Q ... |
x ) , P y ⟩ = 0 {\displaystyle \langle P\mathbf {x} ,(\mathbf {y} -P\mathbf {y} )\rangle =\langle (\mathbf {x} -P\mathbf {x} ),P\mathbf {y} \rangle =0} . Equivalently: ⟨ x , P y ⟩ = ⟨ P x , P y ⟩ = ⟨ P x , y ⟩ . {\displaystyle \langle \mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,P\mathbf {y} \rangle =\langl... |
\left\|P\mathbf {v} \right\|\leq \left\|\mathbf {v} \right\|} . For finite-dimensional complex or real vector spaces, the standard inner product can be substituted for ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } . ===== Formulas ===== A simple case occurs when the orthogonal projection is onto a line. If u ... |
{\displaystyle A} is the isometry that embeds U {\displaystyle U} into the underlying vector space. The range of P A {\displaystyle P_{A}} is therefore the final space of A {\displaystyle A} . It is also clear that A A T {\displaystyle AA^{\mathsf {T}}} is the identity operator on U {\displaystyle U} . The orthonormali... |
] − 1 [ A T B T ] = [ A B ] ( [ A T B T ] [ A B ] ) − 1 [ A T B T ] = [ A B ] [ A T A O O B T B ] − 1 [ A T B T ] = A ( A T A ) − 1 A T + B ( B T B ) − 1 B T {\displaystyle {\begin{aligned}I&={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}A&B\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmat... |
has dimension n − k {\displaystyle n-k} . It follows that the orthogonal complement of the kernel has dimension k {\displaystyle k} . Let v 1 , … , v k {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}} form a basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the ... |
T A ) − 1 B T x {\displaystyle P\mathbf {x} =\mathbf {x} _{1}=A\mathbf {w} =A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}\mathbf {x} } for any vector x ∈ V {\displaystyle \mathbf {x} \in V} and hence P = A ( B T A ) − 1 B T {\displaystyle P=A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}} . In the case that P {\displaystyle P} is an... |
(P)} , and is therefore not necessarily equal. === Finding projection with an inner product === Let V {\displaystyle V} be a vector space (in this case a plane) spanned by orthogonal vectors u 1 , u 2 , … , u p {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\dots ,\mathbf {u} _{p}} . Let y {\displaystyle y} be a vect... |
The factor I m ⊕ 0 s {\displaystyle I_{m}\oplus 0_{s}} corresponds to the maximal invariant subspace on which P {\displaystyle P} acts as an orthogonal projection (so that P itself is orthogonal if and only if k = 0 {\displaystyle k=0} ) and the σ i {\displaystyle \sigma _{i}} -blocks correspond to the oblique componen... |
P} with range U {\displaystyle U} and kernel V {\displaystyle V} is continuous. This follows from the closed graph theorem. Suppose xn → x and Pxn → y. One needs to show that P x = y {\displaystyle Px=y} . Since U {\displaystyle U} is closed and {Pxn} ⊂ U, y lies in U {\displaystyle U} , i.e. Py = y. Also, xn − Pxn = (... |
the orthogonal complement of the kernel: that ( ker T ) ⊥ → W {\displaystyle (\ker T)^{\perp }\to W} be an isometry (compare Partial isometry); in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian subme... |
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and t... |
ν {\displaystyle g_{\mu \nu }} are a set of 16 real-valued functions (since the tensor g {\displaystyle g} is a tensor field, which is defined at all points of a spacetime manifold). In order for the metric to be symmetric g μ ν = g ν μ , {\displaystyle g_{\mu \nu }=g_{\nu \mu },} giving 10 independent coefficients. If... |
{\nu }}}}g_{\rho \sigma }=\Lambda ^{\rho }{}_{\bar {\mu }}\,\Lambda ^{\sigma }{}_{\bar {\nu }}\,g_{\rho \sigma }.} == Properties == The metric tensor plays a key role in index manipulation. In index notation, the coefficients g μ ν {\displaystyle g_{\mu \nu }} of the metric tensor g {\displaystyle \mathbf {g} } provide... |
2 θ d ϕ 2 {\displaystyle d\Omega ^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}} is the standard metric on the 2-sphere. === Black hole metrics === The Schwarzschild metric describes an uncharged, non-rotating black hole. There are also metrics that describe rotating and charged black holes. ==== Schwarzschild metric... |
μ {\displaystyle x^{\mu }} for the manifold, the volume form can be written v o l g = ± | det ( g μ ν ) | d x 0 ∧ d x 1 ∧ d x 2 ∧ d x 3 {\displaystyle \mathrm {vol} _{g}=\pm {\sqrt {\left|\det(g_{\mu \nu })\right|}}\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}} where det ( g μ ν ) {\displaystyle \det(g_{\mu \nu })} i... |
R_{\nu \rho }\ {\stackrel {\mathrm {def} }{=}}\ {R^{\mu }}_{\nu \mu \rho }} and the scalar curvature R = d e f g μ ν R μ ν {\displaystyle R\ {\stackrel {\mathrm {def} }{=}}\ g^{\mu \nu }R_{\mu \nu }} relate the metric (and the associated curvature tensors) to the stress–energy tensor T μ ν {\displaystyle T_{\mu \nu }} ... |
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovect... |
\mathbf {b} } (juxtaposed; no symbols, multiplication signs, crosses, dots, etc.) the outer product of two column vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } is denoted and defined as a ⊗ b {\displaystyle \mathbf {a} \otimes \mathbf {b} } or a b T {\displaystyle \mathbf {a} \mathbf {b} ^{\... |
i, j, k, have the representations: i = ( 1 0 0 ) , j = ( 0 1 0 ) , k = ( 0 0 1 ) {\displaystyle {\begin{aligned}\mathbf {i} &={\begin{pmatrix}1\\0\\0\end{pmatrix}},&\mathbf {j} &={\begin{pmatrix}0\\1\\0\end{pmatrix}},&\mathbf {k} &={\begin{pmatrix}0\\0\\1\end{pmatrix}}\end{aligned}}} (which can be transposed), the stan... |
product is: a b = ∑ j = 1 N ∑ i = 1 N a i b j e i e j . {\displaystyle \mathbf {ab} =\sum _{j=1}^{N}\sum _{i=1}^{N}a_{i}b_{j}\mathbf {e} _{i}\mathbf {e} _{j}.} This is known as the nonion form of the dyadic. Their outer/tensor product in matrix form is: a b = a b T = ( a 1 a 2 ⋮ a N ) ( b 1 b 2 ⋯ b N ) = ( a 1 b 1 a 1 ... |
{tr} \left(\mathbf {c} _{j}^{\mathsf {T}}\mathbf {a} _{i}\mathbf {b} _{i}^{\mathsf {T}}\mathbf {d} _{j}\right)\\&=\sum _{i,j}(\mathbf {a} _{i}\cdot \mathbf {c} _{j})(\mathbf {b} _{i}\cdot \mathbf {d} _{j})\\&=\mathbf {A} {}_{\centerdot }^{\centerdot }\mathbf {B} \end{aligned}}} Furthermore, since, A T = ∑ i , j ( a i b... |
{i} \cdot \mathbf {k} \\{}+{}&A_{21}\mathbf {j} \cdot \mathbf {i} +A_{22}\mathbf {j} \cdot \mathbf {j} +A_{23}\mathbf {j} \cdot \mathbf {k} \\{}+{}&A_{31}\mathbf {k} \cdot \mathbf {i} +A_{32}\mathbf {k} \cdot \mathbf {j} +A_{33}\mathbf {k} \cdot \mathbf {k} \\[6pt]=\qquad &A_{11}+A_{22}+A_{33}\end{aligned}}} in index n... |
= ( i i + j j + k k ) ⋅ a = i ( i ⋅ a ) + j ( j ⋅ a ) + k ( k ⋅ a ) = i a x + j a y + k a z = a {\displaystyle {\begin{aligned}\mathbf {I} \cdot \mathbf {a} &=(\mathbf {i} \mathbf {i} +\mathbf {j} \mathbf {j} +\mathbf {k} \mathbf {k} )\cdot \mathbf {a} \\&=\mathbf {i} (\mathbf {i} \cdot \mathbf {a} )+\mathbf {j} (\math... |
a I × × A = ( A ⋅ ⋅ I ) I − A T I ⋅ ⋅ ( a b ) = ( I ⋅ a ) ⋅ b = a ⋅ b = t r ( a b ) {\displaystyle {\begin{aligned}\left(\mathbf {a} \times \mathbf {I} \right)\cdot \left(\mathbf {b} \times \mathbf {I} \right)&=\mathbf {ba} -\left(\mathbf {a} \cdot \mathbf {b} \right)\mathbf {I} \\\mathbf {I} {}_{\times }^{\,\centerdot... |
1 1 0 ) ( x y ) = ( − y x ) . {\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}-y\\x\end{pmatrix}}.} For any angle θ, the 2d rotation dyadic for a rotation anti-clockwise in the plane is R = I cos θ + J sin θ = ( i i + j j ) cos θ + ( j i − i j ) sin θ = (... |
is the Lorentz factor. == Related terms == Some authors generalize from the term dyadic to related terms triadic, tetradic and polyadic. == See also == Kronecker product Bivector Polyadic algebra Unit vector Multivector Differential form Quaternions Field (mathematics) == Notes == === Explanatory notes === === Citation... |
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector... |
r ⋅ x + s ⋅ x {\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x} , ( r s ) ⋅ x = r ⋅ ( s ⋅ x ) {\displaystyle (rs)\cdot x=r\cdot (s\cdot x)} , 1 ⋅ x = x . {\displaystyle 1\cdot x=x.} The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for m... |
of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element that satisfies the definition of a line... |
NM). If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on X forms a module over C∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over... |
a homomorphism of R-modules if for any m, n in M and r, s in R, f ( r ⋅ m + s ⋅ n ) = r ⋅ f ( m ) + s ⋅ f ( n ) {\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)} . This, like any homomorphism of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphi... |
on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal. Torsion-free A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A ... |
groups, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod, which is the natural generalization of the module category... |
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f... |
differentiable manifold, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allow... |
an oriented surface, etc.) If M is an oriented m-dimensional manifold, and M′ is the same manifold with opposite orientation and ω is an m-form, then one has: ∫ M ω = − ∫ M ′ ω . {\displaystyle \int _{M}\omega =-\int _{M'}\omega \,.} These conventions correspond to interpreting the integrand as a differential form, int... |
∧ dxj) + b(dxj ∧ dxi) = (a − b) dxi ∧ dxj. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also i... |
f. When generalized to higher forms, if ω = f dxI is a simple k-form, then its exterior derivative dω is a (k + 1)-form defined by taking the differential of the coefficient functions: d ω = ∑ i = 1 n ∂ f ∂ x i d x i ∧ d x I . {\displaystyle d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^... |
f is the partial derivative of f with respect to the jth coordinate vector, i.e., ∂f / ∂xj, where x1, x2, ..., xn are the coordinate vectors in U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1, y2, ..., yn are introduced, then ∂ f ∂ x j = ∑ i = 1 n ∂ y i ∂ x ... |
formula (*). More generally, for any smooth functions gi and hi on U, we define the differential 1-form α = Σi gi dhi pointwise by α p = ∑ i g i ( p ) ( d h i ) p {\displaystyle \alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}} for each p ∈ U. Any differential 1-form arises this way, and by using (*) it follows that any diffe... |
i 2 … i k = 1 n f i 1 i 2 … i k d x i 1 ∧ d x i 2 ∧ ⋯ ∧ d x i k {\displaystyle \sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}} for a collection of functions fi1i2⋅⋅⋅ik. Antisymmetry, which was already present for 2-forms, makes it possible to ... |
of M. For example, a differential 1-form α assigns to each point p ∈ M {\displaystyle p\in M} a linear functional αp on T p M {\displaystyle T_{p}M} . In the presence of an inner product on T p M {\displaystyle T_{p}M} (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vect... |
the exterior algebra). The antisymmetry inherent in the exterior algebra means that when α ∧ β is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product α ⊗ β is not alternating. There ... |
to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as the Hodge star operator ⋆ : Ω k ( M ) → ∼ Ω n − k ( M ) {\displaystyle \star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)} and the codifferential δ : Ω k ( M ) → Ω k − 1 ( M ... |
pushforward. For any point p ∈ M and any tangent vector v ∈ TpM, there is a well-defined pushforward vector f∗(v) in Tf(p)N. However, the same is not true of a vector field. If f is not injective, say because q ∈ N has two or more preimages, then the vector field may determine two or more distinct vectors in TqN. If f ... |
of a k-form ω is the composite M → f N → ω ⋀ k T ∗ N ⟶ ⋀ k ( d f ) ∗ ⋀ k T ∗ M . {\displaystyle M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.} Another abstract way to view the pull... |
i k . {\displaystyle f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.} Each exterior derivative dfi can be expanded in terms of dx1, ..., dxm. The resulting k-form can be written using Jacobian matrices: f ∗ ω = ∑ i 1 < ⋯ < i k ∑ j 1 < ⋯ < j k ( ω ... |
ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity. === Integration over chains === Let M be an n-manifold and ω an n-form on M. First, assume that there is a parametrization of M by an open subset of Euclidean space. That is, assume that there exis... |
1 r m i φ i , {\displaystyle c=\sum _{i=1}^{r}m_{i}\varphi _{i},} then the integral of a k-form ω over c is defined to be the sum of the integrals over the terms of c: ∫ c ω = ∑ i = 1 r m i ∫ D φ i ∗ ω . {\displaystyle \int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega .} This approach to defining int... |
theorem is the case where this map is the projection from a product to one of its factors. Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let M and N be two orientable m... |
pure dimensions m and n, and f : M → N is a surjective submersion. Fix orientations of M and N, and give each fiber of f the induced orientation. Let ω be an m-form on M, and let η be an n-form on N that is almost everywhere positive with respect to the orientation of N. Then, for almost every y ∈ N, the form ω / ηy is... |
_{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.} This form depends on the orientation of N but not the choice of ζ. Then the k-form γ is uniquely defined by the property ⟨ γ y , v ⟩ = ∫ f − 1 ( y ) β v , {\displaystyle \langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },} and γ is smooth (D... |
1 d x = 1 {\textstyle \int _{0}^{1}dx=1} , while ∫ 1 0 d x = − ∫ 0 1 d x = − 1 {\textstyle \int _{1}^{0}dx=-\int _{0}^{1}dx=-1} . By contrast, the integral of the measure |dx| on the interval is unambiguously 1 (i.e. the integral of the constant function 1 with respect to this measure is 1). Similarly, under a change o... |
as above. === Currents === The differential form analog of a distribution or generalized function is called a current. The space of k-currents on M is the dual space to an appropriate space of differential k-forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chai... |
theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A. The Yang–Mills field F is then defined by F = d A + A ∧ A . {\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .} In the abelian case, such as electromagnetism, A ∧ A = 0, but this... |
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that... |
\zeta }\\&=\left(\delta _{\alpha }^{\gamma }\delta _{\beta }^{\zeta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }\right)R_{\gamma \zeta }\\&=\left(\delta _{\alpha }^{\gamma }\delta _{\beta }^{\zeta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }\right)\left(\Gamma ^{\epsilon }{}_{\gamma \zeta ,\epsilon }-\Gamma... |
g α [ β , γ ] ϵ = 1 2 ( g α β , γ ϵ − g α γ , β ϵ ) . {\displaystyle g_{\alpha [\beta ,\gamma ]\epsilon }\,={\frac {1}{2}}\left(g_{\alpha \beta ,\gamma \epsilon }-g_{\alpha \gamma ,\beta \epsilon }\right).} == Trace == The trace of the Einstein tensor can be computed by contracting the equation in the definition with t... |
has shown that, in a four-dimensional differentiable manifold, the Einstein tensor is the only tensorial and divergence-free function of the g μ ν {\displaystyle g_{\mu \nu }} and at most their first and second partial derivatives. However, the Einstein field equation is not the only equation which satisfies the three ... |
In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field the... |
The continuity equation ∂ μ M 0 α β μ = 0 , {\displaystyle \partial _{\mu }M_{0}^{\alpha \beta \mu }=0\,,} implies: ∂ μ S α β μ = T β α − T α β ≠ 0 {\displaystyle \partial _{\mu }S^{\alpha \beta \mu }=T^{\beta \alpha }-T^{\alpha \beta }\neq 0} and therefore, the stress–energy tensor is not a symmetric tensor. The quant... |
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities bet... |
are irreversible. Almost all of these physical phenomena ultimately involve systems seeking their lowest energy state in keeping with the principle of minimum energy. As they approach this state, they tend to achieve true thermodynamic equilibrium, at which point there are no longer any driving forces in the system and... |
liquids are present and/or drag is present, the analogy is not valid. Other analogies, such as von Karman's and Prandtl's, usually result in poor relations. The most successful and most widely used analogy is the Chilton and Colburn J-factor analogy. This analogy is based on experimental data for gases and liquids in b... |
reversed. == Mass transfer == When a system contains two or more components whose concentration vary from point to point, there is a natural tendency for mass to be transferred, minimizing any concentration difference within the system. Mass transfer in a system is governed by Fick's first law: 'Diffusion flux from hig... |
transfer. === Derivation === The non-dimensional energy equation for fluid flow in a boundary layer can simplify to the following, when heating from viscous dissipation and heat generation can be neglected: u ∗ ∂ T ∗ ∂ x ∗ + v ∗ ∂ T ∗ ∂ y ∗ = 1 R e L P r ∂ 2 T ∗ ∂ y ∗ 2 {\displaystyle {u^{*}{\frac {\partial T^{*}}{\par... |
h {\displaystyle {h}} is the heat transfer coefficient, and the subscripts s {\displaystyle {s}} and b {\displaystyle {b}} compare the surface and bulk values respectively. For mass transfer at an interface, we can equate Fick's law with Newton's law for convection, yielding: J = D d C d y = h m ( C m − C b ) {\display... |
mass analogy may also break down in cases where the governing equations differ substantially. For instance, situations with substantial contributions from generation terms in the flow, such as bulk heat generation or bulk chemical reactions, may cause solutions to diverge. === Applications of the Heat-Mass Analogy === ... |
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 = ± x 2 {\displaystyle y=\pm x^{2}} (or parametrically, x ( t ) = t , y ( t ) = ± 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 ) × ( 0 , 1 ) {\displaystyle (0,1)\times (0,1)} . Furthermore, the functions defined for constant x {\displaystyle x} ... |
0} . If f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a continuous function at point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} , then | f | {\displaystyle |f|} is also continuous at the same point. If f : R n → 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 ( ∇ {\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 and physics, vector is a term that refers to quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces. Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as d... |
do not apply. More generally, when some data can be represented naturally by vectors, they are often called vectors even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples. Rotation vector, a Euclidean vector whose direction is that of the axis of a rotatio... |
Vector potential, a vector field whose curl is a given vector field Vector flow, a set of closely related concepts of the flow determined by a vector field === See also === Ricci calculus Vector Analysis, a textbook on vector calculus by Wilson, first published in 1901, which did much to standardize the notation and vo... |
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form, α, that is the exterior derivative of another differential form β, i.e. α = dβ. Thus, an exact form is in the image of d, and... |
uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative " d θ {\displaystyle d\theta } ". The upshot is that d θ {\displaystyle d\theta } is a one-form on R 2 ∖ { 0 } {\displaystyle \mathbb {R} ^{2}\smallsetminus \{0\}} that is not act... |
y ) d y {\displaystyle \alpha =f(x,y)\,dx+g(x,y)\,dy} that are of real interest. The formula for the exterior derivative d {\displaystyle d} here is d α = ( g x − f y ) d x ∧ d y {\displaystyle d\alpha =(g_{x}-f_{y})\,dx\wedge dy} where the subscripts denote partial derivatives. Therefore the condition for α {\displays... |
that ζ − η = d β {\displaystyle \zeta -\eta =d\beta } then one says that ζ and η are cohomologous to each other. Exact forms are sometimes said to be cohomologous to zero. The set of all forms cohomologous to a given form (and thus to each other) is called a de Rham cohomology class; the general study of such classes i... |
the current two-form I := j 1 ( x 1 , x 2 , x 3 ) d x 2 ∧ d x 3 + j 2 ( x 1 , x 2 , x 3 ) d x 3 ∧ d x 1 + j 3 ( x 1 , x 2 , x 3 ) d x 1 ∧ d x 2 . {\displaystyle \mathbf {I} :=j_{1}(x_{1},x_{2},x_{3})\,{\rm {d}}x_{2}\wedge {\rm {d}}x_{3}+j_{2}(x_{1},x_{2},x_{3})\,{\rm {d}}x_{3}\wedge {\rm {d}}x_{1}+j_{3}(x_{1},x_{2},x_{... |
one can already guess that E → {\displaystyle {\vec {E}}} and B → , {\displaystyle {\vec {B}},} ρ {\displaystyle \rho } and j → , {\displaystyle {\vec {j}},} φ {\displaystyle \varphi } and A → {\displaystyle {\vec {A}}} can be unified to quantities with six rsp. four nontrivial components, which is the basis of the rel... |
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 α γ B α C γ β ≡ ∑ α ∑ γ A α γ B α C γ β . {\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 ⇁ Q ⇁ B P Q R ⇁ C R = ∑ P ⇁ ∑ Q ⇁ ∑ ... |
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 α B β γ C γ δ + D α β E δ = T α β δ {\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\... |
replace γ (incidentally, the contraction on the γ 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 ( α β ) γ ⋯ = 1 2 ! ( A α β γ ⋯ + A β α γ ⋯ ) {\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... |
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