Coordinate surfaces of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to z =2. These surfaces intersect at the point P (shown as a black sphere), which has Cartesian coordinates roughly (2, -1.5, 2).In mathematics , parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the
perpendicular
z
{\displaystyle z}
-direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.
Basic definition
Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate. The parabolic cylindrical coordinates (σ , τ , z ) are defined in terms of the Cartesian coordinates (x , y , z ) by:
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{aligned}x&=\sigma \tau \\y&={\frac {1}{2))\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned))}
The surfaces of constant σ form confocal parabolic cylinders
2
y
=
x
2
σ
2
−
σ
2
{\displaystyle 2y={\frac {x^{2)){\sigma ^{2))}-\sigma ^{2))
that open towards +y , whereas the surfaces of constant τ form confocal parabolic cylinders
2
y
=
−
x
2
τ
2
+
τ
2
{\displaystyle 2y=-{\frac {x^{2)){\tau ^{2))}+\tau ^{2))
that open in the opposite direction, i.e., towards −y . The foci of all these parabolic cylinders are located along the line defined by x = y = 0 . The radius r has a simple formula as well
r
=
x
2
+
y
2
=
1
2
(
σ
2
+
τ
2
)
{\displaystyle r={\sqrt {x^{2}+y^{2))}={\frac {1}{2))\left(\sigma ^{2}+\tau ^{2}\right)}
that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics ; for further details, see the Laplace–Runge–Lenz vector article.
Scale factors
The scale factors for the parabolic cylindrical coordinates σ and τ are:
h
σ
=
h
τ
=
σ
2
+
τ
2
h
z
=
1
{\displaystyle {\begin{aligned}h_{\sigma }&=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2))}\\h_{z}&=1\end{aligned))}
Differential elements
The infinitesimal element of volume is
d
V
=
h
σ
h
τ
h
z
d
σ
d
τ
d
z
=
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
{\displaystyle dV=h_{\sigma }h_{\tau }h_{z}d\sigma d\tau dz=(\sigma ^{2}+\tau ^{2})d\sigma \,d\tau \,dz}
The differential displacement is given by:
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 \,{\boldsymbol {\hat {\sigma ))}+{\sqrt {\sigma ^{2}+\tau ^{2))}\,d\tau \,{\boldsymbol {\hat {\tau ))}+dz\,\mathbf {\hat {z)) }
The differential normal area is given by:
d
S
=
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 d\mathbf {S} ={\sqrt {\sigma ^{2}+\tau ^{2))}\,d\tau \,dz{\boldsymbol {\hat {\sigma ))}+{\sqrt {\sigma ^{2}+\tau ^{2))}\,d\sigma \,dz{\boldsymbol {\hat {\tau ))}+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \mathbf {\hat {z)) }
Del
Let f be a scalar field. The gradient is given by
∇
f
=
1
σ
2
+
τ
2
∂
f
∂
σ
σ
^
+
1
σ
2
+
τ
2
∂
f
∂
τ
τ
^
+
∂
f
∂
z
z
^
{\displaystyle \nabla f={\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2)))){\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma ))}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2)))){\partial f \over \partial \tau }{\boldsymbol {\hat {\tau ))}+{\partial f \over \partial z}\mathbf {\hat {z)) }
The Laplacian is given by
∇
2
f
=
1
σ
2
+
τ
2
(
∂
2
f
∂
σ
2
+
∂
2
f
∂
τ
2
)
+
∂
2
f
∂
z
2
{\displaystyle \nabla ^{2}f={\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))))
Let A be a vector field of the form:
A
=
A
σ
σ
^
+
A
τ
τ
^
+
A
z
z
^
{\displaystyle \mathbf {A} =A_{\sigma }{\boldsymbol {\hat {\sigma ))}+A_{\tau }{\boldsymbol {\hat {\tau ))}+A_{z}\mathbf {\hat {z)) }
The divergence is given by
∇
⋅
A
=
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
σ
+
∂
(
σ
2
+
τ
2
A
τ
)
∂
τ
)
+
∂
A
z
∂
z
{\displaystyle \nabla \cdot \mathbf {A} ={\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))
The curl is given by
∇
×
A
=
(
1
σ
2
+
τ
2
∂
A
z
∂
τ
−
∂
A
τ
∂
z
)
σ
^
−
(
1
σ
2
+
τ
2
∂
A
z
∂
σ
−
∂
A
σ
∂
z
)
τ
^
+
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
τ
)
∂
σ
−
∂
(
σ
2
+
τ
2
A
σ
)
∂
τ
)
z
^
{\displaystyle \nabla \times \mathbf {A} =\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2)))){\frac {\partial A_{z)){\partial \tau ))-{\frac {\partial A_{\tau )){\partial z))\right){\boldsymbol {\hat {\sigma ))}-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2)))){\frac {\partial A_{z)){\partial \sigma ))-{\frac {\partial A_{\sigma )){\partial z))\right){\boldsymbol {\hat {\tau ))}+{\frac {1}{\sigma ^{2}+\tau ^{2))}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2))}A_{\tau }\right)}{\partial \sigma ))-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2))}A_{\sigma }\right)}{\partial \tau ))\right)\mathbf {\hat {z)) }
Other differential operators can be expressed in the coordinates (σ , τ ) by substituting the scale factors into the general formulae found in orthogonal coordinates .
Relationship to other coordinate systems
Relationship to cylindrical coordinates (ρ , φ , z ) :
ρ
cos
φ
=
σ
τ
ρ
sin
φ
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{aligned}\rho \cos \varphi &=\sigma \tau \\\rho \sin \varphi &={\frac {1}{2))\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned))}
Parabolic unit vectors expressed in terms of Cartesian unit vectors:
σ
^
=
τ
x
^
−
σ
y
^
τ
2
+
σ
2
τ
^
=
σ
x
^
+
τ
y
^
τ
2
+
σ
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\boldsymbol {\hat {\sigma ))}&={\frac {\tau {\hat {\mathbf {x} ))-\sigma {\hat {\mathbf {y} ))}{\sqrt {\tau ^{2}+\sigma ^{2))))\\{\boldsymbol {\hat {\tau ))}&={\frac {\sigma {\hat {\mathbf {x} ))+\tau {\hat {\mathbf {y} ))}{\sqrt {\tau ^{2}+\sigma ^{2))))\\\mathbf {\hat {z)) &=\mathbf {\hat {z)) \end{aligned))}
Parabolic cylinder harmonics
Since all of the surfaces of constant σ , τ and z are conicoids , Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables , a separated solution to Laplace's equation may be written:
V
=
S
(
σ
)
T
(
τ
)
Z
(
z
)
{\displaystyle V=S(\sigma )T(\tau )Z(z)}
and Laplace's equation, divided by V , is written:
1
σ
2
+
τ
2
[
S
¨
S
+
T
¨
T
]
+
Z
¨
Z
=
0
{\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2))}\left[{\frac {\ddot {S)){S))+{\frac {\ddot {T)){T))\right]+{\frac {\ddot {Z)){Z))=0}
Since the Z equation is separate from the rest, we may write
Z
¨
Z
=
−
m
2
{\displaystyle {\frac {\ddot {Z)){Z))=-m^{2))
where m is constant. Z (z ) has the solution:
Z
m
(
z
)
=
A
1
e
i
m
z
+
A
2
e
−
i
m
z
{\displaystyle Z_{m}(z)=A_{1}\,e^{imz}+A_{2}\,e^{-imz))
Substituting −m 2 for
Z
¨
/
Z
{\displaystyle {\ddot {Z))/Z}
, Laplace's equation may now be written:
[
S
¨
S
+
T
¨
T
]
=
m
2
(
σ
2
+
τ
2
)
{\displaystyle \left[{\frac {\ddot {S)){S))+{\frac {\ddot {T)){T))\right]=m^{2}(\sigma ^{2}+\tau ^{2})}
We may now separate the S and T functions and introduce another constant n 2 to obtain:
S
¨
−
(
m
2
σ
2
+
n
2
)
S
=
0
{\displaystyle {\ddot {S))-(m^{2}\sigma ^{2}+n^{2})S=0}
T
¨
−
(
m
2
τ
2
−
n
2
)
T
=
0
{\displaystyle {\ddot {T))-(m^{2}\tau ^{2}-n^{2})T=0}
The solutions to these equations are the parabolic cylinder functions
S
m
n
(
σ
)
=
A
3
y
1
(
n
2
/
2
m
,
σ
2
m
)
+
A
4
y
2
(
n
2
/
2
m
,
σ
2
m
)
{\displaystyle S_{mn}(\sigma )=A_{3}y_{1}(n^{2}/2m,\sigma {\sqrt {2m)))+A_{4}y_{2}(n^{2}/2m,\sigma {\sqrt {2m)))}
T
m
n
(
τ
)
=
A
5
y
1
(
n
2
/
2
m
,
i
τ
2
m
)
+
A
6
y
2
(
n
2
/
2
m
,
i
τ
2
m
)
{\displaystyle T_{mn}(\tau )=A_{5}y_{1}(n^{2}/2m,i\tau {\sqrt {2m)))+A_{6}y_{2}(n^{2}/2m,i\tau {\sqrt {2m)))}
The parabolic cylinder harmonics for (m , n ) are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:
V
(
σ
,
τ
,
z
)
=
∑
m
,
n
A
m
n
S
m
n
T
m
n
Z
m
{\displaystyle V(\sigma ,\tau ,z)=\sum _{m,n}A_{mn}S_{mn}T_{mn}Z_{m))
Applications
The classic applications of parabolic cylindrical coordinates are in solving partial differential equations , e.g., Laplace's equation or the Helmholtz equation , for which such coordinates allow a separation of variables . A typical example would be the electric field surrounding a flat semi-infinite conducting plate.