Talk:Tensors in curvilinear coordinates

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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) ⇄ ∑_ic_i 11:43, 30 April 2012 (UTC)[reply]

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) ⇄ ∑_ic_i 00:24, 5 May 2012 (UTC)[reply]

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: $v_{s|r}=v_{s,r}-\Gamma _{sr}^{i}~v_{i}$

However $\Gamma _{sr}^{i}=\Gamma _{sr}^{i}$ 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 $\Gamma _{sr}^{i}~v_{i}$ .

Moreover ${\mathcal {E}}^{rrt}=0$ hence finally there is no contribution at all for the summation of $\Gamma _{sr}^{i}~v_{i}$ 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)[reply]

Mécanique générale

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