# 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

## 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

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

## 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

## situation at some 3D-space

that is, a curve $C$,…

## 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 Reciprocal basis Metric tensor, lenght, area, volumen Bilinear transformations Musical isomorphisms Change of basis Calculus in $\mathbb{R}^n$ Partial derivatives Taylor series Jacobians Chain’s rule Directional derivatives Covariant derivative and Gauss equation Coordinated changes Differential forms with exterior derivatives the $\mathbb{R}^3$ de Rham’s complex Covariant gradient little Stokes’ theorems: Green, Gauss. 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 $\mathbb{R}^3$ all classical surfaces rendered tangent space change of basis vector fields and tensor fields Christoffel’s symbols (connection coefficients) 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

## 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 $f(p)=1$.