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Erschienen in:
2017 | OriginalPaper | Buchkapitel
5. Applications of Tensors and Differential Geometry
verfasst von : Hung Nguyen-Schäfer, Jan-Philip Schmidt
Erschienen in: Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers
Verlag: Springer Berlin Heidelberg
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Abstract
Nabla operator is a linear map of an arbitrary tensor into an image tensor in N-dimensional curvilinear coordinates. The Nabla operator can be usually defined in N-dimensional Cartesian coordinates {xi} using Einstein summation convention asAccording to Eq. 2.12, the relation between the bases of Cartesian and general curvilinear coordinates can be written asMultiplying Eq. (5.2) by giej, one obtains the basis of Cartesian coordinates expressed in the curvilinear coordinate basis.Using chain rule of coordinate transformation, the Nabla operator in the general curvilinear coordinates {ui} results from Eq. (5.3) [1, 2].Thus, the Nabla operator can be written in the curvilinear coordinates {ui}using Einstein summation convention.
$$ \nabla \equiv {\mathbf{e}}^i\frac{\partial }{\partial {x}^i}\ \mathrm{f}\mathrm{o}\mathrm{r}\ i = 1,2,\dots, N $$
$$ {\mathbf{g}}_i=\frac{\partial \mathbf{r}}{\partial {u}^i}=\frac{\partial \mathbf{r}}{\partial {x}^j}\frac{\partial {x}^j}{\partial {u}^i}={\mathbf{e}}_j\frac{\partial {x}^j}{\partial {u}^i} $$
$$ {\mathbf{e}}^j={\mathbf{g}}^i\frac{\partial {x}^j}{\partial {u}^i} $$
$$ \begin{array}{c}\nabla \equiv {\mathbf{e}}^i\left(\frac{\partial }{\partial {u}^j}\frac{\partial {u}^j}{\partial {x}^i}\right)={\mathbf{g}}^k\frac{\partial {x}^i}{\partial {u}^k}\left(\frac{\partial }{\partial {u}^j}\frac{\partial {u}^j}{\partial {x}^i}\right)\\ {}={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left(\frac{\partial {x}^i}{\partial {u}^k}\frac{\partial {u}^j}{\partial {x}^i}\right)={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left(\frac{\partial {u}^j}{\partial {u}^k}\right)\\ {}={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left({\delta}_k^j\right)={\mathbf{g}}^k\frac{\partial }{\partial {u}^k}\end{array} $$
$$ \nabla \equiv {\mathbf{g}}^i\frac{\partial }{\partial {u}^i}={\mathbf{g}}^i{\nabla}_i\ \mathrm{f}\mathrm{o}\mathrm{r}\ i = 1,2,\dots, N $$