өңдеу
Үш өлшемді
q
1
,
q
2
,
q
3
{\displaystyle q_{1},\ q_{2},\ q_{3}}
қисықсызықты координаттар үшін:
Δ
f
(
q
1
,
q
2
,
q
3
)
=
div
grad
f
(
q
1
,
q
2
,
q
3
)
=
{\displaystyle \Delta f(q_{1},\ q_{2},\ q_{3})=\operatorname {div} \,\operatorname {grad} \,f(q_{1},\ q_{2},\ q_{3})=}
=
1
H
1
H
2
H
3
[
∂
∂
q
1
(
H
2
H
3
H
1
∂
f
∂
q
1
)
+
∂
∂
q
2
(
H
1
H
3
H
2
∂
f
∂
q
2
)
+
∂
∂
q
3
(
H
1
H
2
H
3
∂
f
∂
q
3
)
]
,
{\displaystyle ={\frac {1}{H_{1}H_{2}H_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {H_{2}H_{3}}{H_{1}}}{\frac {\partial f}{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {H_{1}H_{3}}{H_{2}}}{\frac {\partial f}{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {H_{1}H_{2}}{H_{3}}}{\frac {\partial f}{\partial q_{3}}}\right)\right],}
мұндағы
H
i
{\displaystyle H_{i}\ }
Ламе коэффициенттері . Цилиндрлік координаттарда түзуден тыс
r
=
0
{\displaystyle \ r=0}
Δ
f
=
1
r
∂
∂
r
(
r
∂
f
∂
r
)
+
∂
2
f
∂
z
2
+
1
r
2
∂
2
f
∂
φ
2
{\displaystyle \Delta f={1 \over r}{\partial \over \partial r}\left(r{\partial f \over \partial r}\right)+{\partial ^{2}f \over \partial z^{2}}+{1 \over r^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}}
Сфералық координаттар бас нүктеден тыс (үш өлшемді кеңістікте):
Δ
f
=
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
{\displaystyle \Delta f={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}
немесе
Δ
f
=
1
r
∂
2
∂
r
2
(
r
f
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
.
{\displaystyle \Delta f={1 \over r}{\partial ^{2} \over \partial r^{2}}\left(rf\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}.}
Егер
f
=
f
(
r
)
{\displaystyle \ f=f(r)}
Δ
f
=
d
2
f
d
r
2
+
n
−
1
r
d
f
d
r
.
{\displaystyle \Delta f={d^{2}f \over dr^{2}}+{n-1 \over r}{df \over dr}.}
Параболалық координаттарда (үш өлшемді кеңістікте) бас нүктеден тыс:
Δ
f
=
1
σ
2
+
τ
2
[
1
σ
∂
∂
σ
(
σ
∂
f
∂
σ
)
+
1
τ
∂
∂
τ
(
τ
∂
f
∂
τ
)
]
+
1
σ
2
τ
2
∂
2
f
∂
φ
2
{\displaystyle \Delta f={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left[{\frac {1}{\sigma }}{\frac {\partial }{\partial \sigma }}\left(\sigma {\frac {\partial f}{\partial \sigma }}\right)+{\frac {1}{\tau }}{\frac {\partial }{\partial \tau }}\left(\tau {\frac {\partial f}{\partial \tau }}\right)\right]+{\frac {1}{\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}}
Параболалық цилиндр координаттарында бас нүктеден тыс:
Δ
F
(
u
,
v
,
z
)
=
1
c
2
(
u
2
+
v
2
)
[
∂
2
F
∂
u
2
+
∂
2
F
∂
v
2
]
+
∂
2
F
∂
z
2
{\displaystyle \Delta F(u,v,z)={\frac {1}{c^{2}(u^{2}+v^{2})}}\left[{\frac {\partial ^{2}F}{\partial u^{2}}}+{\frac {\partial ^{2}F}{\partial v^{2}}}\right]+{\frac {\partial ^{2}F}{\partial z^{2}}}}
![{\displaystyle ={\frac {1}{H_{1}H_{2}H_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {H_{2}H_{3}}{H_{1}}}{\frac {\partial f}{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {H_{1}H_{3}}{H_{2}}}{\frac {\partial f}{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {H_{1}H_{2}}{H_{3}}}{\frac {\partial f}{\partial q_{3}}}\right)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20976c3be6b574fb20939f284033ac2c05f005ac)
![{\displaystyle \Delta f={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left[{\frac {1}{\sigma }}{\frac {\partial }{\partial \sigma }}\left(\sigma {\frac {\partial f}{\partial \sigma }}\right)+{\frac {1}{\tau }}{\frac {\partial }{\partial \tau }}\left(\tau {\frac {\partial f}{\partial \tau }}\right)\right]+{\frac {1}{\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70817e419f2c0c845d8e553d32111fb9d54f2971)
![{\displaystyle \Delta F(u,v,z)={\frac {1}{c^{2}(u^{2}+v^{2})}}\left[{\frac {\partial ^{2}F}{\partial u^{2}}}+{\frac {\partial ^{2}F}{\partial v^{2}}}\right]+{\frac {\partial ^{2}F}{\partial z^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d0ec7ea68177c4df0a25ece5f32a2c8de2e9e3)
