Category Archives: multilinear algebra

this studies tensors to apply to geometry

Einstein-Penrose ‘s strong sum convention


the rank one tensors’ basis changes

abstractrelativity017

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wedge product example


When bivectors are defined by

\beta^i\wedge\beta^j:=\beta^i\otimes\beta^j-\beta^j\otimes\beta^i,

so, for two generic covectors

\theta=a\beta^1+b\beta^2+c\beta^3 and \phi=d\beta^1+e\beta^2+f\beta^3,

we have the bivector

\theta\wedge\phi=(bf-ce)\beta^2\wedge\beta^3+(cd-af)\beta^3\wedge\beta^1+(ad-be)\beta^1\wedge\beta^2.

 

Otherwise,

Cf. this with the data \left(\begin{array}{c}a\\b\\c\end{array}\right) and \left(\begin{array}{c}d\\e\\f\end{array}\right) to construct the famous

\left(\begin{array}{c}a\\b\\c\end{array}\right)\times\left(\begin{array}{c}d\\e\\f\end{array}\right)=\left(\begin{array}{c}bf-ce\\cd-af\\ad-be\end{array}\right)

So, nobody should be confused about the uses of the symbol \wedge dans le calcul vectoriel XD

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need more?


¿necesitas o requieres un tema en particular? si es alrededor de álgebra multilineal, anímate a interaccionar. También tenemos topología de dimensiones bajas y más…

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Levi-Civita tensor


to see

\varepsilon^i\wedge\varepsilon^j\wedge\varepsilon^k\wedge\varepsilon^l(e_s,e_t,e_u,e_v)={\varepsilon^{ijkl}}_{stuv}

since we are requiring “canonical” duality, i.e.  covectors, \varepsilon^k:V\to R, do

\varepsilon^k(e_l)={\delta^k}_l.

one uses

\varepsilon^i\!\wedge\!\varepsilon^j\!\wedge\!\varepsilon^k\!\wedge\!\varepsilon^l\!=\!\!\sum_{\sigma\in S_4}\!(\!-1\!)^{\sigma}\!\varepsilon^{\sigma(i)}\!\otimes\!\varepsilon^{\sigma(j)}\!\otimes\!\varepsilon^{\sigma(k)}\!\otimes\!\varepsilon^{\sigma(l)}

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Multilinear Algebra


álgebra multilineal es como un cálculo vectorial dos o álgebra lineal tres

entonces para poder hacer cálculos en otras geometrías, inclusive muy diferentes a \mathbb{R}^n vamos viendo hacia donde tenemos que caminar: ver  (un post previo con estas ideas en mente).

RE-ENGINEERING LINEAR ALGEBRA

RE-ENGINEERING VECTOR CALCULUS

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situation at some 3D-space


situation at some 3D-space

that is, a curve C,…

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multilinear algebra 1, a synoptic view


what is math? let us discuss:

Baby Abstract Multilinear Algebra
Baby Multilinear Algebra  of Inner Product Spaces
Calculus in \mathbb{R}^n
Algebraic Differential Geometry
  • Parameterizations: curves and surfaces
  • Tangent vectors, tangent space, tangent bundle
  • Curves in \mathbb{R}^2 and \mathbb{R}^3 and on surfaces in \mathbb{R}^3
  • Surfaces in \mathbb{R}^3
    1. all classical surfaces rendered
    2. tangent space change of basis
    3. vector fields and tensor fields
    4. Christoffel’s symbols (connection coefficients)
    5. Curvatures (Gaussian, Mean, Principals, Normal and Geodesic)
  • Vector Fields, Covector Fields, Tensor Fields
  • Integration: Gauss-Bonnet, Stokes
Baby Manifolds (topological, differential, analytic, anti-analytic, aritmetic,…)
Examples: Lie groups and Fiber bundles

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