Category Archives: differential geometry

this studies geometry with calculus

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

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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Filed under cucei math, differential geometry, geometry, math, math analysis, multilinear algebra, topology

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

vector calculus examples

the following hieroglyph

\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} \cos V\cos W\\ \cos V\sin W\\ \sin V\end{array}\right)

represent a mapping \mathbb{R}^2\to\mathbb{R}^3. It is enough to take

-\frac{\pi}{2}< V<\frac{\pi}{2}\quad,\quad 0< W<2\pi

to G.P.S -ing (almost) every point in \mathbb{R}^3 which are at distance one from the origin. That is the sphere.

Now the partial derivatives \partial_1=\frac{\partial\phi}{\partial V} and \partial_2=\frac{\partial\phi}{\partial W} are pictorially as:

These derivative are:

\partial_1=\left(\begin{array}{c}-\sin V\cos W\\ -\sin V\sin W\\ \cos V\end{array}\right) and \partial_2=\left(\begin{array}{c}-\cos V\sin W\\ \cos V\cos W\\ 0\end{array}\right)

An easy calculation give that for the inner products \langle\phi,\partial_1\rangle=\langle\phi,\partial_2\rangle=0, so the \partial_i are orthogonal to the position \phi. Then the product \partial_1\times\partial_2, which is orthogonal to the plane determined by the couple of vectors \{\partial_1,\partial_2\}, hence colinear to the position \phi. In fact, after normalization N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}.

A good exercise is to unfold the same program for the torus by employing the parameterization given by:

\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} (2+\cos V)\cos W\\ (2+\cos V)\sin W\\ \sin V\end{array}\right)


Filed under differential geometry, geometry, math

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

umbrella auf Whitney

These are three level surfaces of the function f(x,y,z)=xy^2+z^2

they are at levels 1,0,-1.

This means that the orange points p on the surface \Sigma in the left graphic, that is, p\in\Sigma=f^{-1}(1) or f(p)=1.

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Filed under 3-manifold, calculus on manifolds, differential geometry, geometry, topology