d
V
=
d
x
d
y
d
z
{\displaystyle \mathrm {d} V=\mathrm {d} x\mathrm {d} y\mathrm {d} z}
∇
→
ψ
=
(
∂
ψ
∂
x
,
∂
ψ
∂
y
,
∂
ψ
∂
z
)
{\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial x}},{\frac {\partial \psi }{\partial y}},{\frac {\partial \psi }{\partial z}}\right)}
∇
→
⋅
v
→
=
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
∂
z
{\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}}
∇
v
{\displaystyle {\nabla }\mathrm {v} }
∇
→
⋅
v
→
=
(
∂
v
z
∂
y
−
∂
v
y
∂
z
,
∂
v
x
∂
z
−
∂
v
z
∂
x
,
∂
v
y
∂
x
−
∂
v
x
∂
y
)
{\displaystyle {\vec {\nabla }}\cdot {\vec {v}}=\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}},{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}},{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)}
Δ
ψ
=
∂
2
ψ
x
∂
x
2
+
∂
2
ψ
y
∂
y
2
+
∂
2
ψ
z
∂
z
2
{\displaystyle \Delta \psi ={\frac {\partial ^{2}\psi _{x}}{\partial x^{2}}}+{\frac {\partial ^{2}\psi _{y}}{\partial y^{2}}}+{\frac {\partial ^{2}\psi _{z}}{\partial z^{2}}}}
The general transformation relations to a new coordinate system
(
u
1
,
u
2
,
u
3
)
{\displaystyle (u_{1},u_{2},u_{3})}
d
V
=
d
u
1
U
1
d
u
2
U
2
d
u
3
U
3
{\displaystyle \mathrm {d} V={\frac {\mathrm {d} u_{1}}{U_{1}}}{\frac {\mathrm {d} u_{2}}{U_{2}}}{\frac {\mathrm {d} u_{3}}{U_{3}}}}
∇
→
ψ
=
(
U
1
∂
ψ
∂
u
1
,
U
2
∂
ψ
∂
u
2
,
U
3
∂
ψ
∂
u
3
)
{\displaystyle {\vec {\nabla }}\psi =\left(U_{1}{\frac {\partial \psi }{\partial u_{1}}},U_{2}{\frac {\partial \psi }{\partial u_{2}}},U_{3}{\frac {\partial \psi }{\partial u_{3}}}\right)}
∇
→
⋅
v
→
=
U
1
U
2
U
3
(
∂
∂
u
1
v
u
1
U
2
U
3
+
∂
∂
u
2
v
u
2
U
1
U
3
+
∂
∂
u
3
v
u
3
U
1
U
2
)
{\displaystyle {\vec {\nabla }}\cdot {\vec {v}}=U_{1}U_{2}U_{3}\left({\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{1}}}{U_{2}U_{3}}}+{\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{2}}}{U_{1}U_{3}}}+{\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{3}}}{U_{1}U_{2}}}\right)}
∇
→
×
v
→
=
(
U
2
U
3
(
∂
∂
u
2
v
u
3
U
3
−
∂
∂
u
3
v
u
2
U
2
)
,
U
1
U
3
(
∂
∂
u
3
v
u
1
U
1
−
∂
∂
u
1
v
u
3
U
3
)
,
U
1
U
2
(
∂
∂
u
1
v
u
2
U
2
−
∂
∂
u
2
v
u
1
U
1
)
)
{\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left(U_{2}U_{3}({\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{3}}}{U_{3}}}-{\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{2}}}{U_{2}}}),U_{1}U_{3}({\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{1}}}{U_{1}}}-{\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{3}}}{U_{3}}}),U_{1}U_{2}({\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{2}}}{U_{2}}}-{\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{1}}}{U_{1}}})\right)}
Δ
ψ
=
U
1
U
2
U
3
[
∂
∂
u
1
(
U
1
U
2
U
3
∂
ψ
∂
u
1
)
+
∂
∂
u
2
(
U
2
U
1
U
3
∂
ψ
∂
u
2
)
+
∂
∂
u
3
(
U
3
U
1
U
2
∂
ψ
∂
u
3
)
]
{\displaystyle \Delta \psi =U_{1}U_{2}U_{3}\left[{\frac {\partial }{\partial u_{1}}}({\frac {U_{1}}{U_{2}U_{3}}}{\frac {\partial \psi }{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {U_{2}}{U_{1}U_{3}}}{\frac {\partial \psi }{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {U_{3}}{U_{1}U_{2}}}{\frac {\partial \psi }{\partial u_{3}}})\right]}
where
U
1
−
1
=
(
∂
x
∂
u
1
)
2
+
(
∂
y
∂
u
1
)
2
+
(
∂
z
∂
u
1
)
2
,
{\displaystyle U_{1}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{1}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{1}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{1}}}\right)^{2}}},}
U
2
−
1
=
(
∂
x
∂
u
2
)
2
+
(
∂
y
∂
u
2
)
2
+
(
∂
z
∂
u
2
)
2
,
{\displaystyle U_{2}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{2}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{2}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{2}}}\right)^{2}}},}
U
3
−
1
=
(
∂
x
∂
u
3
)
2
+
(
∂
y
∂
u
3
)
2
+
(
∂
z
∂
u
3
)
2
,
{\displaystyle U_{3}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{3}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{3}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{3}}}\right)^{2}}},}
v
u
1
=
v
x
U
1
∂
x
∂
u
1
+
v
y
U
1
∂
y
∂
u
1
+
v
z
U
1
∂
z
∂
u
1
{\displaystyle v_{u_{1}}=v_{x}U_{1}{\frac {\partial x}{\partial u_{1}}}+v_{y}U_{1}{\frac {\partial y}{\partial u_{1}}}+v_{z}U_{1}{\frac {\partial z}{\partial u_{1}}}}
v
u
2
=
v
x
U
2
∂
x
∂
u
2
+
v
y
U
2
∂
y
∂
u
2
+
v
z
U
2
∂
z
∂
u
2
{\displaystyle v_{u_{2}}=v_{x}U_{2}{\frac {\partial x}{\partial u_{2}}}+v_{y}U_{2}{\frac {\partial y}{\partial u_{2}}}+v_{z}U_{2}{\frac {\partial z}{\partial u_{2}}}}
v
u
3
=
v
x
U
3
∂
x
∂
u
3
+
v
y
U
3
∂
y
∂
u
3
+
v
z
U
3
∂
z
∂
u
3
{\displaystyle v_{u_{3}}=v_{x}U_{3}{\frac {\partial x}{\partial u_{3}}}+v_{y}U_{3}{\frac {\partial y}{\partial u_{3}}}+v_{z}U_{3}{\frac {\partial z}{\partial u_{3}}}}
The cylindrical coordinates
(
u
1
=
r
,
u
2
=
φ
,
u
3
=
z
)
{\displaystyle (u_{1}=r,u_{2}=\varphi ,u_{3}=z)}
(
x
,
y
,
z
)
=
(
r
cos
φ
,
r
sin
φ
,
z
)
{\displaystyle (x,y,z)=(r\cos \varphi ,r\sin \varphi ,z)}
d
V
=
r
d
r
d
φ
d
z
{\displaystyle \mathrm {d} V=r\ \mathrm {d} r\mathrm {d} \varphi \mathrm {d} z}
∇
→
ψ
=
(
∂
ψ
∂
r
,
1
r
∂
ψ
∂
φ
,
∂
ψ
∂
z
)
{\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial r}},{\frac {1}{r}}{\frac {\partial \psi }{\partial \varphi }},{\frac {\partial \psi }{\partial z}}\right)}
∇
→
⋅
v
→
=
1
r
∂
∂
r
(
r
v
r
)
+
1
r
∂
v
φ
∂
φ
+
∂
v
z
∂
z
{\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {1}{r}}{\frac {\partial }{\partial r}}(rv_{r})+{\frac {1}{r}}{\frac {\partial v_{\varphi }}{\partial \varphi }}+{\frac {\partial v_{z}}{\partial z}}}
∇
→
×
v
→
=
(
1
r
∂
v
z
∂
φ
−
∂
v
ϕ
v
z
,
∂
v
r
∂
z
−
∂
v
z
∂
r
,
1
r
∂
∂
r
(
r
v
ϕ
)
−
1
r
∂
v
r
∂
ϕ
)
{\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left({\frac {1}{r}}{\frac {\partial v_{z}}{\partial \varphi }}-{\frac {\partial v_{\phi }}{v_{z}}},{\frac {\partial v_{r}}{\partial z}}-{\frac {\partial v_{z}}{\partial r}},{\frac {1}{r}}{\frac {\partial }{\partial r}}(rv_{\phi })-{\frac {1}{r}}{\frac {\partial v_{r}}{\partial \phi }}\right)}
Δ
ψ
=
∂
2
ψ
∂
r
2
+
1
r
∂
ψ
∂
r
+
1
r
2
∂
2
ψ
∂
φ
2
+
∂
2
ψ
∂
z
2
{\displaystyle \Delta \psi ={\frac {\partial ^{2}\psi }{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \varphi ^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}}
Spherical coordinates in most of the physics conventions. Notice that this is different than the mathematics convention in which
θ
{\displaystyle \theta }
φ
{\displaystyle \varphi }
The spherical polar coordinates
(
u
1
=
r
,
u
2
=
θ
,
u
3
=
φ
)
{\displaystyle (u_{1}=r,u_{2}=\theta ,u_{3}=\varphi )}
R
3
{\displaystyle \mathrm {R} ^{3}}
(
x
,
y
,
z
)
=
(
r
sin
θ
cos
φ
,
r
sin
θ
sin
φ
,
r
cos
θ
)
{\displaystyle (x,y,z)=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )}
U
1
−
1
=
1
,
U
2
−
1
=
r
,
U
3
−
1
=
r
sin
θ
{\displaystyle U_{1}^{-1}=1,\ U_{2}^{-1}=r,\ U_{3}^{-1}=r\sin \theta }
d
V
=
r
2
sin
θ
d
r
d
φ
d
θ
{\displaystyle \mathrm {d} V=r^{2}\sin \theta \ \mathrm {d} r\mathrm {d} \varphi \mathrm {d} \theta }
∇
→
ψ
=
(
∂
ψ
∂
r
,
1
r
∂
ψ
∂
θ
,
1
r
sin
θ
∂
ψ
∂
φ
)
{\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial r}},{\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }},{\frac {1}{r\sin \theta }}{\frac {\partial \psi }{\partial \varphi }}\right)}
∇
→
⋅
v
→
=
1
r
2
∂
∂
r
(
r
2
v
r
)
+
1
r
sin
θ
∂
∂
θ
(
v
θ
sin
θ
)
+
1
r
sin
θ
∂
v
φ
∂
φ
{\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}v_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(v_{\theta }\sin \theta )+{\frac {1}{r\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}}
∇
→
×
v
→
=
(
1
r
sin
θ
(
∂
∂
θ
(
sin
θ
v
φ
)
−
∂
v
θ
∂
φ
)
,
1
r
sin
θ
(
∂
v
r
∂
φ
−
sin
θ
∂
∂
r
(
r
v
φ
)
)
,
1
r
(
∂
∂
r
(
r
v
θ
)
−
∂
v
r
∂
θ
)
)
{\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left({\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}(\sin \theta v_{\varphi })-{\frac {\partial v_{\theta }}{\partial \varphi }}\right),{\frac {1}{r\sin \theta }}\left({\frac {\partial v_{r}}{\partial \varphi }}-\sin \theta {\frac {\partial }{\partial r}}(rv_{\varphi })\right),{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rv_{\theta })-{\frac {\partial v_{r}}{\partial \theta }}\right)\right)}
Δ
ψ
=
1
r
2
∂
∂
r
(
r
2
∂
ψ
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
ψ
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
ψ
∂
φ
2
{\displaystyle \Delta \psi ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial \psi }{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}\psi }{\partial \varphi ^{2}}}}
![{\displaystyle \Delta \psi =U_{1}U_{2}U_{3}\left[{\frac {\partial }{\partial u_{1}}}({\frac {U_{1}}{U_{2}U_{3}}}{\frac {\partial \psi }{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {U_{2}}{U_{1}U_{3}}}{\frac {\partial \psi }{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {U_{3}}{U_{1}U_{2}}}{\frac {\partial \psi }{\partial u_{3}}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee0cb28a690df3c54792d953cee9613efe8c809)
