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, , the coordinated (rectangular) functions:
are linear, so their derivatives,
, are equal themselves i.e the gradients obey
, they are dubbed
and they satisfy duality:
Now, if is a parameterization of the surface,
, and
is a “measure” then, doing calculus in the surface means do calculus to
. Let
.
Let us name the coordinated (rectangular) functions on
.
So by the chain rule we have: that is, in terms of gradients:
or
So for the functions we get
and by the same rule just above
where evaluating at the basis of
give
but since it is known that the generate
, then both:
generate
, the co-tangent space at
,… wanna see a picture?
real multilinear algebra
el álgebra multilineal sobre los números reales, , 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
, entonces uno puede incluir los principios de la geometría diferencial (de curvas y de superficies en
) 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-
), de un conjunto abierto euclídeo.
Otra cosa es la complex multilinear algebra…
Filed under algebra, categoría, Category, cucei math, multilinear algebra, what is math
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 .
We should be using the intuition granted by
The definitions are:
- …
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:
is the –module over the symbols
and over an open set
stay tune…
Filed under algebra, cucei math, multilinear algebra
Vektorraum und seines Dualraum sind wie Hasen
Zwei natürlich vektorräume und seines Dualraum
entstanden wie Hasen, ein Überfluß von Vektoren räume sind ist, wenn wir sein tensorprodukt bedenken. Zum Beispiel, der zwei-Rang sind Räume
,
und
Aber drei-Rang sind und
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
Filed under algebra, differential geometry, multilinear algebra