# Tag Archives: Multilinearenalgebra

## simuexam de multi 2011a

Filed under algebra, differential geometry, math, multilinear algebra

## tangent space duality of a surface

let us describe how the euclidean duality is carried to the tangent bundle of a surface.

We know how euclidean duality is: For each euclidean space, $\mathbb{R}^n$, the coordinated (rectangular)  functions: $x^i(p)=p^i$ are linear, so their derivatives, $Jx^i$, are equal themselves i.e the gradients obey $Jx^i=x^i$, they are dubbed $dx^i$ and they satisfy duality:

$dx^i(e_k)={\delta^i}_k$

Now, if $\Phi:\Omega\to\mathbb{R}^3$  is a parameterization of the surface, $\Sigma\subset\mathbb{R}^3$, and $f:\Sigma\to\mathbb{R}$ is a “measure” then, doing calculus in the surface means do calculus to $f\circ\Phi$. Let $g=f\circ\Phi$.

Let us name $\xi^1,\xi^2,\xi^3$ the coordinated (rectangular) functions on $\mathbb{R}^3$.

So by the chain rule we have: $Jg=Jf\cdot J\Phi$ that is, in terms of gradients:

${\rm grad}(g)={\rm grad}(f)\!\cdot\!J\Phi$

or

$[\begin{array}{cc}\frac{\partial g}{\partial x^1}&\frac{\partial g}{\partial x^2}\end{array}]=[\begin{array}{ccc}\frac{\partial f}{\partial\xi^1}&\frac{\partial f}{\partial\xi^2}&\frac{\partial f}{\partial\xi^3}\end{array}]\!\cdot\!\left(\begin{array}{cc}\frac{\partial \xi^1}{\partial x^1}&\frac{\partial \xi^1}{\partial x^2}\\\frac{\partial \xi^2}{\partial x^1}&\frac{\partial \xi^2}{\partial x^2}\\\frac{\partial \xi^3}{\partial x^1}&\frac{\partial \xi^3}{\partial x^2}\end{array}\right)$

So for the functions $u^i=x^i\circ\Phi^{-1}$ we get $x^i=u^i\circ\Phi$ and by the same rule just above

$dx^i=du^i\!\cdot\!J\Phi$

where evaluating at the basis $e_1,e_2$ of $\mathbb{R}^2$ give

$du^i(J\Phi e_k)=dx^i(e_k)={\delta^i}_k$

but since it is known that the $J\Phi e_1,J\Phi e_2$ generate $T_p\Sigma$, then both: $du^1,du^2$ generate $T_p\Sigma^*$, the co-tangent space at $p\in\Sigma$,… wanna see a picture?

## real multilinear algebra

el álgebra multilineal sobre los números reales, $\mathbb{R}$, incluye a las formas diferenciales euclideas, ahí uno estudia la amalgama producida por el álgebra lineal y el cálculo en varias variables. Pero además si uno dispone del lenguage elemental del álgebra tensorial de espacios vectoriales sobre los reales, es decir la categoría ${\rm{Vect}}_{\mathbb{R}}$, entonces uno puede incluir los principios de la geometría diferencial (de curvas y de superficies en $\mathbb{R}^3$) para obtener un curso realmente útil y moderno. En el mero corazón de esta teoría está el complejo de de Rham que permite construir los módulos cohomológicos del álgebra de Grassmann (módulo-$C^{\infty}$), de un conjunto abierto euclídeo.

Otra cosa es la complex multilinear algebra…

## wedge complements in finite dimension

in the next counting experiment we are going  to calculate the dimensions of some linear subspaces of the Grassmann algebra of a vector space of dimension $n$.

We should  be using the intuition granted by $\Lambda(\mathbb{R}^n)$

The definitions are:

• $C_{dx}=\{\alpha\mid \alpha\wedge dx=0\}$
• $M_{dx}={C_{dx}}^{\top}$
• $C_{dx\wedge dy}=\{\alpha\mid \alpha\wedge dx\wedge dy=0\}$
• $M_{dx\wedge dy}={C_{dx\wedge dy}}^{\top}$
• $C_{dx\wedge dy\wedge dz}$
• $M_{dx\wedge dy\wedge dz}$

doesn’t anybody know the name of the result?…  ‘cuz if it hasn’t, I will claim mine : )

Meanwhile, let me refrain the definition  that says:

$\Lambda(\Omega)$

is the $C^{\infty}(\Omega)$module over the symbols $dx^1,dx^2,...,dx^n$ and over an open set $\Omega\subseteq\mathbb{R}^n$

stay tune…

## Vektorraum und seines Dualraum sind wie Hasen

Zwei natürlich vektorräume $V$ und seines Dualraum $V^*$ entstanden wie Hasen, ein Überfluß von Vektoren räume sind ist, wenn wir sein tensorprodukt bedenken. Zum Beispiel, der zwei-Rang sind  Räume

$V\otimes V$, $V\otimes V^*$ und $V^*\otimes V^*$

Aber drei-Rang  sind  $V\otimes V\otimes V,V\otimes V\otimes V^*,V\otimes V^*\otimes V^*$ und $V^*\otimes V^*\otimes V^*$ und so weiter…

Eine gute Übung soll feststellen, welches ihre Basis und zu ist, bestimmen, wie die Bauteile für jedes Element in einem besonderen räume ändern, wenn wir uns die Basis ändern