Category Archives: differential geometry

this studies geometry with calculus

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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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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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)

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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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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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