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>
Contribute to Wikiveristy by linking the title to a discussion and/or proof. Just click the redlink and start the page. If you just came in from Wikipedia , the rules about treating each other with respect and obeying copyright laws remain more or less the same as on Wikipedia, but the definition of what constitutes an acceptable article is a lot looser. Here we call them "resources". Welcome to the wacky world of Wikiversity , where an yt hi ng g oe s
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}
↑ 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 .
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