The purpose of this resource is to carefully examine the Wikipedia article Del in cylindrical and spherical coordinates
The identities are reproduced below, and contributors are encouraged to either:
Verify the identity and place its reference using a five em padding after the equation: {{pad|5em}}verified<ref>reference</ref>
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Transformations between coordinates
edit
w:Cartesian coordinates (x , y , z )w:Cylindrical coordinates (ρ , ϕ , z )w:Spherical coordinates (r , θ , ϕ )w:Parabolic cylindrical coordinates (σ , τ , z )*Asterisk indicates that the title is a link to more discussion
ρ
=
x
2
+
y
2
{\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}}
ϕ
=
arctan
(
y
/
x
)
{\displaystyle \phi =\arctan(y/x)}
z
=
z
{\displaystyle z=z}
verified using mathworld[1]
x
=
ρ
cos
ϕ
{\displaystyle x=\rho \cos \phi }
y
=
ρ
sin
ϕ
{\displaystyle y=\rho \sin \phi }
z
=
z
{\displaystyle z=z}
verified using mathworld[2]
x
=
r
sin
θ
cos
ϕ
{\displaystyle x=r\sin \theta \cos \phi }
y
=
r
sin
θ
sin
ϕ
{\displaystyle y=r\sin \theta \sin \phi }
z
=
r
cos
θ
{\displaystyle z=r\cos \theta }
verified using mathworld[3]
x
=
σ
τ
{\displaystyle x=\sigma \tau }
y
=
1
2
(
τ
2
−
σ
2
)
{\displaystyle y={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)}
z
=
z
{\displaystyle z=z}
--no reference
r
=
x
2
+
y
2
+
z
2
{\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}
θ
=
arctan
(
x
2
+
y
2
/
z
)
{\displaystyle \theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)}
ϕ
=
arctan
(
y
/
x
)
{\displaystyle \phi =\arctan(y/x)}
verified using mathworld[4]
r
=
ρ
2
+
z
2
{\displaystyle r={\sqrt {\rho ^{2}+z^{2}}}}
θ
=
arctan
(
ρ
/
z
)
{\displaystyle \theta =\arctan {(\rho /z)}}
ϕ
=
ϕ
{\displaystyle \phi =\phi }
no reference
ρ
=
r
sin
θ
{\displaystyle \rho =r\sin \theta }
ϕ
=
ϕ
{\displaystyle \phi =\phi }
z
=
r
cos
θ
{\displaystyle z=r\cos \theta }
no reference
ρ
cos
ϕ
=
σ
τ
{\displaystyle \rho \cos \phi =\sigma \tau }
ρ
sin
ϕ
=
1
2
(
τ
2
−
σ
2
)
{\displaystyle \rho \sin \phi ={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)}
z
=
z
{\displaystyle z=z}
no reference
ρ
^
=
x
x
^
+
y
y
^
x
2
+
y
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
Verified, see page linked in title
x
^
=
cos
ϕ
ρ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
ϕ
ρ
^
+
cos
ϕ
ϕ
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\hat {\boldsymbol {\rho }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\hat {\boldsymbol {\rho }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
Verified, see page linked in title
x
^
=
sin
θ
cos
ϕ
r
^
+
cos
θ
cos
ϕ
θ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
θ
sin
ϕ
r
^
+
cos
θ
sin
ϕ
θ
^
+
cos
ϕ
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\hat {\boldsymbol {r}}}+\cos \theta \cos \phi {\hat {\boldsymbol {\theta }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\hat {\boldsymbol {r}}}+\cos \theta \sin \phi {\hat {\boldsymbol {\theta }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\boldsymbol {r}}}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
Verified, see page linked in title
σ
^
=
τ
x
^
−
σ
y
^
τ
2
+
σ
2
τ
^
=
σ
x
^
+
τ
y
^
τ
2
+
σ
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\boldsymbol {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}
r
^
=
x
x
^
+
y
y
^
+
z
z
^
x
2
+
y
2
+
z
2
θ
^
=
x
z
x
^
+
y
z
y
^
−
(
x
2
+
y
2
)
z
^
x
2
+
y
2
x
2
+
y
2
+
z
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}}
Verified, see page linked in title
r
^
=
ρ
ρ
^
+
z
z
^
ρ
2
+
z
2
θ
^
=
z
ρ
^
−
ρ
z
^
ρ
2
+
z
2
ϕ
^
=
ϕ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\end{aligned}}}
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
ϕ
^
=
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}}
Vector and scalar fields
edit
A
{\displaystyle \mathbf {A} }
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}}
A
ρ
ρ
^
+
A
ϕ
ϕ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}+A_{z}{\hat {\mathbf {z} }}}
A
r
r
^
+
A
θ
θ
^
+
A
ϕ
ϕ
^
{\displaystyle A_{r}{\hat {\boldsymbol {r}}}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}}
A
σ
σ
^
+
A
τ
τ
^
+
A
ϕ
z
^
{\displaystyle A_{\sigma }{\hat {\boldsymbol {\sigma }}}+A_{\tau }{\hat {\boldsymbol {\tau }}}+A_{\phi }{\hat {\mathbf {z} }}}
∇
f
{\displaystyle \nabla f}
w:gradient of a scalar field.
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
ϕ
ϕ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
ϕ
ϕ
^
{\displaystyle {\partial f \over \partial r}{\hat {\boldsymbol {r}}}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}}
1
σ
2
+
τ
2
∂
f
∂
σ
σ
^
+
1
σ
2
+
τ
2
∂
f
∂
τ
τ
^
+
∂
f
∂
z
z
^
{\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\hat {\boldsymbol {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\hat {\boldsymbol {\tau }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}}
∇
⋅
A
{\displaystyle \nabla \cdot \mathbf {A} }
w:divergence of a vector field
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}}
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
ϕ
∂
ϕ
{\displaystyle {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }}
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
σ
+
∂
(
σ
2
+
τ
2
A
τ
)
∂
τ
)
+
∂
A
z
∂
z
{\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}}
∇
×
A
{\displaystyle \nabla \times \mathbf {A} }
w:curl (mathematics) of A
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle \left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right){\hat {\mathbf {x} }}+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right){\hat {\mathbf {y} }}+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right){\hat {\mathbf {z} }}}
(
1
ρ
∂
A
z
∂
ϕ
−
∂
A
ϕ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
ϕ
^
+
1
ρ
(
∂
(
ρ
A
ϕ
)
∂
ρ
−
∂
A
ρ
∂
ϕ
)
z
^
{\displaystyle \left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\mathbf {z} }}}
1
r
sin
θ
(
∂
∂
θ
(
A
ϕ
sin
θ
)
−
∂
A
θ
∂
ϕ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
ϕ
−
∂
∂
r
(
r
A
ϕ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
ϕ
^
{\displaystyle {\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}}
(
1
σ
2
+
τ
2
∂
A
z
∂
τ
−
∂
A
τ
∂
z
)
σ
^
−
(
1
σ
2
+
τ
2
∂
A
z
∂
σ
−
∂
A
σ
∂
z
)
τ
^
{\displaystyle \left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right){\hat {\boldsymbol {\sigma }}}-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right){\hat {\boldsymbol {\tau }}}}
+
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
τ
−
∂
(
σ
2
+
τ
2
A
τ
)
∂
σ
)
z
^
{\displaystyle +{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right)}{\partial \tau }}-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right)}{\partial \sigma }}\right){\hat {\mathbf {z} }}}
Δ
f
≡
∇
2
f
{\displaystyle \Delta f\equiv \nabla ^{2}f}
w:Laplace operator on a scalar field
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
ϕ
2
+
∂
2
f
∂
z
2
{\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}}
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 {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 \phi ^{2}}}
1
σ
2
+
τ
2
(
∂
2
f
∂
σ
2
+
∂
2
f
∂
τ
2
)
+
∂
2
f
∂
z
2
{\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}}
Δ
A
≡
∇
2
A
{\displaystyle \Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A} }
w:Vector Laplacian of
A
{\displaystyle \mathbf {A} }
Δ
A
x
x
^
+
Δ
A
y
y
^
+
Δ
A
z
z
^
{\displaystyle \Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}{\hat {\mathbf {z} }}}
(
Δ
A
ρ
−
A
ρ
ρ
2
−
2
ρ
2
∂
A
ϕ
∂
ϕ
)
ρ
^
{\displaystyle {\mathopen {}}\left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\mathclose {}}{\hat {\boldsymbol {\rho }}}}
+
(
Δ
A
ϕ
−
A
ϕ
ρ
2
+
2
ρ
2
∂
A
ρ
∂
ϕ
)
ϕ
^
{\displaystyle +{\mathopen {}}\left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\mathclose {}}{\hat {\boldsymbol {\phi }}}}
+
Δ
A
z
z
^
{\displaystyle +\Delta A_{z}{\hat {\mathbf {z} }}}
(
Δ
A
r
−
2
A
r
r
2
−
2
r
2
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
−
2
r
2
sin
θ
∂
A
ϕ
∂
ϕ
)
r
^
{\displaystyle \left(\Delta A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}}
+
(
Δ
A
θ
−
A
θ
r
2
sin
2
θ
+
2
r
2
∂
A
r
∂
θ
−
2
cos
θ
r
2
sin
2
θ
∂
A
ϕ
∂
ϕ
)
θ
^
{\displaystyle +\left(\Delta A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\hat {\boldsymbol {\theta }}}}
+
(
Δ
A
ϕ
−
A
ϕ
r
2
sin
2
θ
+
2
r
2
sin
θ
∂
A
r
∂
ϕ
+
2
cos
θ
r
2
sin
2
θ
∂
A
θ
∂
ϕ
)
ϕ
^
{\displaystyle +\left(\Delta A_{\phi }-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {\phi }}}}
(
A
⋅
∇
)
B
{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} }
B A A' [5]
(
A
x
∂
B
x
∂
x
+
A
y
∂
B
x
∂
y
+
A
z
∂
B
x
∂
z
)
x
^
{\displaystyle \left(A_{x}{\frac {\partial B_{x}}{\partial x}}+A_{y}{\frac {\partial B_{x}}{\partial y}}+A_{z}{\frac {\partial B_{x}}{\partial z}}\right){\hat {\mathbf {x} }}}
+
(
A
x
∂
B
y
∂
x
+
A
y
∂
B
y
∂
y
+
A
z
∂
B
y
∂
z
)
y
^
{\displaystyle +\left(A_{x}{\frac {\partial B_{y}}{\partial x}}+A_{y}{\frac {\partial B_{y}}{\partial y}}+A_{z}{\frac {\partial B_{y}}{\partial z}}\right){\hat {\mathbf {y} }}}
+
(
A
x
∂
B
z
∂
x
+
A
y
∂
B
z
∂
y
+
A
z
∂
B
z
∂
z
)
z
^
{\displaystyle +\left(A_{x}{\frac {\partial B_{z}}{\partial x}}+A_{y}{\frac {\partial B_{z}}{\partial y}}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right){\hat {\mathbf {z} }}}
(
A
ρ
∂
B
ρ
∂
ρ
+
A
ϕ
ρ
∂
B
ρ
∂
ϕ
+
A
z
∂
B
ρ
∂
z
−
A
ϕ
B
ϕ
ρ
)
ρ
^
{\displaystyle \left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \phi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\phi }B_{\phi }}{\rho }}\right){\hat {\boldsymbol {\rho }}}}
+
(
A
ρ
∂
B
ϕ
∂
ρ
+
A
ϕ
ρ
∂
B
ϕ
∂
ϕ
+
A
z
∂
B
ϕ
∂
z
+
A
ϕ
B
ρ
ρ
)
ϕ
^
{\displaystyle +\left(A_{\rho }{\frac {\partial B_{\phi }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\phi }}{\partial \phi }}+A_{z}{\frac {\partial B_{\phi }}{\partial z}}+{\frac {A_{\phi }B_{\rho }}{\rho }}\right){\hat {\boldsymbol {\phi }}}}
+
(
A
ρ
∂
B
z
∂
ρ
+
A
ϕ
ρ
∂
B
z
∂
ϕ
+
A
z
∂
B
z
∂
z
)
z
^
{\displaystyle +\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{z}}{\partial \phi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right){\hat {\mathbf {z} }}}
(
A
r
∂
B
r
∂
r
+
A
θ
r
∂
B
r
∂
θ
+
A
ϕ
r
sin
θ
∂
B
r
∂
ϕ
−
A
θ
B
θ
+
A
ϕ
B
ϕ
r
)
r
^
{\displaystyle \left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \phi }}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}\right){\hat {\boldsymbol {r}}}}
+
(
A
r
∂
B
θ
∂
r
+
A
θ
r
∂
B
θ
∂
θ
+
A
ϕ
r
sin
θ
∂
B
θ
∂
ϕ
+
A
θ
B
r
r
−
A
ϕ
B
ϕ
cot
θ
r
)
θ
^
{\displaystyle +\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \phi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}\right){\hat {\boldsymbol {\theta }}}}
+
(
A
r
∂
B
ϕ
∂
r
+
A
θ
r
∂
B
ϕ
∂
θ
+
A
ϕ
r
sin
θ
∂
B
ϕ
∂
ϕ
+
A
ϕ
B
r
r
+
A
ϕ
B
θ
cot
θ
r
)
ϕ
^
{\displaystyle +\left(A_{r}{\frac {\partial B_{\phi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\phi }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\phi }}{\partial \phi }}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}\right){\hat {\boldsymbol {\phi }}}}
d
l
=
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}}
d
l
=
d
ρ
ρ
^
+
ρ
d
ϕ
ϕ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\phi \,{\hat {\boldsymbol {\phi }}}+dz\,{\hat {\mathbf {z} }}}
d
l
=
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
ϕ
ϕ
^
{\displaystyle d\mathbf {l} =dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\phi \,{\hat {\boldsymbol {\phi }}}}
d
l
=
σ
2
+
τ
2
d
σ
σ
^
+
σ
2
+
τ
2
d
τ
τ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\hat {\boldsymbol {\tau }}}+dz\,{\hat {\mathbf {z} }}}
Differential normal area
d
S
{\displaystyle d\mathbf {S} }
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle dy\,dz{\hat {\mathbf {x} }}+dx\,dz{\hat {\mathbf {y} }}+dx\,dy{\hat {\mathbf {z} }}}
ρ
d
ϕ
d
z
ρ
^
+
d
ρ
d
z
ϕ
^
+
ρ
d
ρ
d
ϕ
z
^
{\displaystyle \rho \,d\phi \,dz{\hat {\boldsymbol {\rho }}}+d\rho \,dz{\hat {\boldsymbol {\phi }}}+\rho \,d\rho \,d\phi {\hat {\mathbf {z} }}}
r
2
sin
θ
d
θ
d
ϕ
r
^
+
r
sin
θ
d
r
d
ϕ
θ
^
+
r
d
r
d
θ
ϕ
^
{\displaystyle r^{2}\sin \theta \,d\theta \,d\phi {\hat {\mathbf {r} }}+r\sin \theta \,dr\,d\phi {\hat {\boldsymbol {\theta }}}+r\,dr\,d\theta {\hat {\boldsymbol {\phi }}}}
σ
2
+
τ
2
d
τ
d
z
σ
^
+
σ
2
+
τ
2
d
σ
d
z
τ
^
+
(
σ
2
+
τ
2
)
d
σ
d
τ
z
^
{\displaystyle {\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz{\hat {\boldsymbol {\tau }}}+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau {\hat {\mathbf {z} }}}
d
V
=
d
x
d
y
d
z
{\displaystyle dV=dx\,dy\,dz}
verified[6]
d
V
=
ρ
d
ρ
d
ϕ
d
z
{\displaystyle dV=\rho \,d\rho \,d\phi \,dz}
verified[7]
d
V
=
r
2
sin
θ
d
r
d
θ
d
ϕ
{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi }
verified[8]
d
V
=
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
{\displaystyle dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz}
Non-trivial calculation rules:
div
grad
f
≡
∇
⋅
∇
f
=
∇
2
f
≡
Δ
f
{\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f}
curl
grad
f
≡
∇
×
∇
f
=
0
{\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
div
curl
A
≡
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
curl
curl
A
≡
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
Lagrange's formula for del)
Δ
(
f
g
)
=
f
Δ
g
+
2
∇
f
⋅
∇
g
+
g
Δ
f
{\displaystyle \Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f}
References
edit
↑ http://mathworld.wolfram.com/CylindricalCoordinates.html ↑ http://mathworld.wolfram.com/CylindricalCoordinates.html ↑ http://mathworld.wolfram.com/SphericalCoordinates.html ↑ http://mathworld.wolfram.com/SphericalCoordinates.html ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Mathworld ↑ James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
↑
James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
↑
James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
[1]
[2]
↑
Weisstein, Eric W. "Convective Operator" . Mathworld . Retrieved 23 March 2011 .
↑
Huba J.D. (1994). "NRL Plasma Formulary revised" (PDF) . Office of Naval Research. Retrieved 11 June 2014 .
Backup copy from Wikipedia
edit
