Category Archives: multilinear algebra

this studies tensors to apply to geometry

Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes



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wedge product example

When bivectors are defined by


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




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


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


Filed under algebra, cucei math, differential geometry, math analysis, mathematics, multilinear algebra, what is math, word algebra

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


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


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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Filed under differential geometry, fiber bundle, geometry, multilinear algebra

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).



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situation at some 3D-space

situation at some 3D-space

that is, a curve C,…

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Filed under 3-manifold, 3-manifolds, calculus on manifolds, differential geometry, fiber bundle, geometry, low dimensional topology, math, multilinear algebra, topology

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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Filed under algebra, calculus on manifolds, categoría, differential equations, differential geometry, fiber bundle, geometry, math analysis, maths, multilinear algebra, topology, what is math