Tag Archives: tensor

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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cambio de coordenadas, cambio de base y cambio de componentes


dos GPS como exchange informazione inter sus mediciones

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a GPS for a surface to do calculus on it


 

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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?

 

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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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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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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…

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