Tag Archives: tensors

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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mathoverflow cucei cimat


I would like to add that the grasping of the fundamental sense for these objects and properties, are implanted around the generalization of calculus: differential forms and its applications…

the phrase was awarded with a Nice-Answer badge, which supports the fight for the differential form formalism… : )

link

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simu exam comment…


The inversion of a matrix.

For a change of basis in \mathbb{R}^3:

\eta_1=\varepsilon_1

\eta_2=\varepsilon_1+2\varepsilon_2

\eta_3=\varepsilon_2-\varepsilon_3

we have as a change-of-basis-matrix:

\left(\begin{array}{ccc}1&1&0\\ 0&2&1\\ 0&0&-1\end{array}\right)

and by solving for:

\varepsilon_1=\eta_1

\varepsilon_2=-\frac{1}{2}\eta_1+\frac{1}{2}\eta_2

\varepsilon_2=-\frac{1}{2}\eta_1+\frac{1}{2}\eta_2-\eta_3

then we get as an inverse of that matrix:

\left(\begin{array}{ccc}1&-1/2&-1/2\\ 0&1/2&1/2\\ 0&0&-1\end{array}\right)

All that inside a recent exam did at the dept of maths

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Filed under algebra, mathematics, multilinear algebra, what is mathematics

covariant and contravariant


Covariant and Contravariant description of tensors from Spiegel's book

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demostración del lema de representación de Riesz de un covector en un espacio vectorial euclídeo


En un espacio vectorial euclídeo V tenemos una manera fácil de representar la base dual de una base arbitraria en V mismo… ¿quieres ver la demostración? entre las cosas que se manejan en esta demostración están las famosas leyes de subir y bajar los índices de las bases involucradas (la de inicio y su recíproca)  y de los componentes de un mismo vector, en  estas diferentes bases 

¿Quieres ver la demostración?

sigue esta . . . liga

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Filed under algebra, cucei math, multilinear algebra