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Diffstat (limited to 'source/know/concept/orthogonal-curvilinear-coordinates/index.md')
| -rw-r--r-- | source/know/concept/orthogonal-curvilinear-coordinates/index.md | 12 |
1 files changed, 5 insertions, 7 deletions
diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md index c7299ee..669358c 100644 --- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md +++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md @@ -21,7 +21,8 @@ where the coordinate surfaces are always perpendicular. Examples of such orthogonal curvilinear systems include [spherical coordinates](/know/concept/spherical-coordinates/), [cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/). +[cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/), +and (trivially) [Cartesian coordinates](/know/concept/cartesian-coordinates/). @@ -690,12 +691,9 @@ When this index notation is written out in full, it becomes: $$\begin{aligned} \nabla^2 f - = \frac{1}{h_1 h_2 h_3} - \bigg( - \pdv{}{c_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \!\Big) - + \pdv{}{c_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \!\Big) - + \pdv{}{c_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \!\Big) - \bigg) + = \frac{1}{h_1 h_2 h_3} \bigg( \pdv{}{c_1} \Big( \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \Big) + + \pdv{}{c_2} \Big( \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \Big) + + \pdv{}{c_3} \Big( \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \Big) \bigg) \end{aligned}$$ This is trivial to prove: $$\nabla^2 f = \nabla \cdot (\nabla f)$$, |