Tag Archives: tensor

2010/11/27 · 15:51

covariant derivative of covectors

How do you think that the covariant derivative in $\mathbb{R}^3$ is extended over covector fields defined over a surface $\Phi:{\mathbb{R}}^2\hookrightarrow\Sigma\subset{\mathbb{R}}^3$ ?

We use the Riesz Representation’s Lemma, so if

$dx^k(\quad)=\langle\partial^k,\quad\rangle=\langle g^{sk}\partial_s,\quad\rangle$

then

$\nabla_{\partial_i}dx^k(\quad)=\langle \nabla_{\partial_i}\partial^k,\quad\rangle=\langle -{\Gamma^k}_{is}\partial^s,\quad\rangle$

This implies that we have:

$\nabla_{\partial_i}dx^k=-{\Gamma^k}_{is}dx^s$

This contrast nicely with $\nabla_{\partial_i}\partial_k={\Gamma^s}_{ik}\partial_s$

For a general $w=w_sdx^s$ , we use the Leibniz’s rule to get

$\nabla_{\partial_i}w=({w_s}_{,i}-{w_t\Gamma^t}_{si})dx^s$

and

$\nabla_{\partial_i}(w\otimes\theta)=(\nabla_{\partial_i}w)\otimes\theta+w\otimes\nabla_{\partial_i}\theta$

The proof that $\nabla_{\partial_i}\partial^k=\nabla_{\partial_i}(g^{sk}\partial_s)=-{\Gamma^k}_{is}\partial^s$ is very fun!

You gotta remember firmly that the $\partial^k=g^{sk}\partial_s$ form the reciprocal coordinated basis, still tangent vectors but representing (à la Riesz) the coordinated covectors $dx^k$ .

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Filed under differential geometry, multilinear algebra

Tagged as multilineal, multilinear algebra, tensor, tensors

2010/10/05 · 16:47

cambio de coordenadas, cambio de base y cambio de componentes

dos GPS como exchange informazione inter sus mediciones

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Tagged as coordinates, multilinear algebra, relatividad, tensor

2010/03/06 · 21:03

a GPS for a surface to do calculus on it

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Tagged as 2-manifolds calculus, álgebra multilineal, cálculo en variedades, differential geometry, differential geometry of surfaces, geom, geometría diferencial de superficies, geometry, GPS, graffiti math, linear algebra, tensor

2010/03/04 · 19:53

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:

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

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

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?

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Filed under cucei math, differential geometry, fiber bundle, multilinear algebra

Tagged as cálculo en variedades, cotangent space, geometría diferencial de superficies, Multilinearenalgebra, tensor, vector duality

2010/01/29 · 18:35

multilineal lección six

tensor fields on a surface and its covariant derivatives

the collection $T\Sigma$ of all the tangent spaces $T_p\Sigma$ is called the tangent bundle of the surface, i.e.

$T\Sigma=\bigsqcup_pT_p\Sigma$

A vector field in a surface is a mapping $X:\Sigma\to T\Sigma$ with the condition $p\mapsto X\in T_p\Sigma$ and since $\partial_1,\partial_2$ span $T_p\Sigma$ then

$X=X^s\partial_s$

This construction determines a contravariant tensor field of rank one, which is taken as the base to ask how other tensor fields -of any rank and any variance- vary…

wanna know more?

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Tagged as tensor

2009/11/24 · 16:38

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…

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Filed under algebra, categoría, Category, cucei math, multilinear algebra, what is math

Tagged as álgebra multilineal, cohomology, de Rham complex, differential forms, grassmann, Grassmann algebra, homology, module of smooth functions, multilinear algebra, Multilinearenalgebra, tensor, vector calculus

2009/11/19 · 15:16

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 $n$ .