# Tag Archives: tensors

## Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes

Filed under math, mathematics, multilinear algebra, word algebra

## 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 $dx^k$.

Filed under differential geometry, multilinear algebra

## 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… : )

Filed under math

## 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

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

## covariant and contravariant

Filed under differential geometry, multilinear algebra

## 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