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.
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.
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 [ edit ] 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}} } 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 [ edit ] 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 [ edit ] 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}} 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.
Morse PM , Feshbach H (1953). Methods of Theoretical Physics, Part I . New York: McGraw-Hill. p. 658. ISBN 0-07-043316-X . LCCN 52011515 . Margenau H , Murphy GM (1956). The Mathematics of Physics and Chemistry . New York: D. van Nostrand. pp. 186 –187. LCCN 55010911 . Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers . New York: McGraw-Hill. p. 181. LCCN 59014456 . ASIN B0000CKZX7. Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs . New York: Springer Verlag. p. 96. LCCN 67025285 . Zwillinger D (1992). Handbook of Integration . Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9 . Same as Morse & Feshbach (1953), substituting u k for ξk . Moon P, Spencer DE (1988). "Parabolic-Cylinder Coordinates (μ, ν, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 21–24 (Table 1.04). ISBN 978-0-387-18430-2 .
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