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


\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:


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




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:


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


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.


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


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…


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 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:  


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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