Talk:Tensors in curvilinear coordinates

Restored content from curvilinear coordinates (early April 2012)
See here. This article can let loose on the tensor calculus + applications while the main article curvilinear coordinates should be the intro to the formalism in vector calculus, which was previously immersed in tensors and genuinely unreadable. See also here. Thanks, F = q(E+v×B) ⇄ ∑ici 11:43, 30 April 2012 (UTC)

Presentation
It would look much neater and reduce the byte count if people would actually stop using LaTeX for characters/very short expressions which are trvial to write in html. Here are very good examples:


 * "In orthogonal curvilinear coordinates of $$3$$ dimensions, where..."


 * "Let $$(\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3)$$ be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal.  However, they are required to be linearly independent.  Then a vector $$\mathbf{v}$$ can be expressed as..."

How much effort does it take to use LaTeX $$(\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3)$$? How much to just type (b1, b2, b3)? F = q(E+v×B) ⇄ ∑ici 00:24, 5 May 2012 (UTC)

Curl of a Vector:
It is indicated that the curl is given by the summation over r,s,t: $$\boldsymbol{\nabla}\times\mathbf{v} = \mathcal{E}^{rst} v_{s|r}~ \mathbf{b}_t$$ with: $$   v_{s|r} = v_{s,r} - \Gamma^i_{sr}~v_i $$

However $$\Gamma^i_{sr} = \Gamma^i_{sr}$$ and $$\mathcal{E}^{rst}=-\mathcal{E}^{srt}$$ hence in the summation, pairs r,s,t,i and s,r,t,i mutually compensate in the term $$\Gamma^i_{sr}~v_i$$.

Moreover $$\mathcal{E}^{rrt}=0$$ hence finally there is no contribution at all for the summation of $$\Gamma^i_{sr}~v_i$$ term and we may simply write: $$\boldsymbol{\nabla}\times\mathbf{v} = \mathcal{E}^{rst} v_{s,r} ~ \mathbf{b}_t$$

By the way, Levi-Civita symbol is introduced in section 3.3 but used in section 1.3 — Preceding unsigned comment added by PBenard (talk • contribs) 07:54, 11 September 2012 (UTC)

Mécanique générale
Mécanique générale 105.157.148.67 (talk) 13:17, 17 June 2022 (UTC)