summaryrefslogtreecommitdiff
path: root/source/know/concept/orthogonal-curvilinear-coordinates
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/orthogonal-curvilinear-coordinates')
-rw-r--r--source/know/concept/orthogonal-curvilinear-coordinates/index.md12
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: $$\nabla^2 f = \nabla \cdot (\nabla f)$$,