Two-dimensional parabolic coordinates ${\displaystyle (\sigma ,\tau )}$  are defined by the equations, in terms of Cartesian coordinates:

${\displaystyle x=\sigma \tau }$ 

${\displaystyle y={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)}$ 

The curves of constant ${\displaystyle \sigma }$  form confocal parabolae

${\displaystyle 2y={\frac {x^{2}}{\sigma ^{2}}}-\sigma ^{2}}$ 

that open upwards (i.e., towards ${\displaystyle +y}$ ), whereas the curves of constant ${\displaystyle \tau }$  form confocal parabolae

${\displaystyle 2y=-{\frac {x^{2}}{\tau ^{2}}}+\tau ^{2}}$ 

that open downwards (i.e., towards ${\displaystyle -y}$ ). The foci of all these parabolae are located at the origin.

The Cartesian coordinates ${\displaystyle x}$  and ${\displaystyle y}$  can be converted to parabolic coordinates by:

${\displaystyle \sigma =\operatorname {sign} (x){\sqrt {{\sqrt {x^{2}+y^{2}}}-y}}}$ 

${\displaystyle \tau ={\sqrt {{\sqrt {x^{2}+y^{2}}}+y}}}$ 

The scale factors for the parabolic coordinates ${\displaystyle (\sigma ,\tau )}$  are equal

${\displaystyle h_{\sigma }=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$ 

Hence, the infinitesimal element of area is

${\displaystyle dA=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau }$ 

and the Laplacian equals

${\displaystyle \nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial \sigma ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \tau ^{2}}}\right)}$ 

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$  and ${\displaystyle \nabla \times \mathbf {F} }$  can be expressed in the coordinates ${\displaystyle (\sigma ,\tau )}$  by substituting the scale factors into the general formulae found in orthogonal coordinates.

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the ${\displaystyle z}$ -direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

${\displaystyle x=\sigma \tau \cos \varphi }$ 

${\displaystyle y=\sigma \tau \sin \varphi }$ 

${\displaystyle z={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)}$ 

where the parabolae are now aligned with the ${\displaystyle z}$ -axis, about which the rotation was carried out. Hence, the azimuthal angle ${\displaystyle \phi }$  is defined

${\displaystyle \tan \varphi ={\frac {y}{x}}}$ 

The surfaces of constant ${\displaystyle \sigma }$  form confocal paraboloids

${\displaystyle 2z={\frac {x^{2}+y^{2}}{\sigma ^{2}}}-\sigma ^{2}}$ 

that open upwards (i.e., towards ${\displaystyle +z}$ ) whereas the surfaces of constant ${\displaystyle \tau }$  form confocal paraboloids

${\displaystyle 2z=-{\frac {x^{2}+y^{2}}{\tau ^{2}}}+\tau ^{2}}$ 

that open downwards (i.e., towards ${\displaystyle -z}$ ). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

${\displaystyle g_{ij}={\begin{bmatrix}\sigma ^{2}+\tau ^{2}&0&0\\0&\sigma ^{2}+\tau ^{2}&0\\0&0&\sigma ^{2}\tau ^{2}\end{bmatrix}}}$ 

The three dimensional scale factors are:

${\displaystyle h_{\sigma }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$ 

${\displaystyle h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$ 

${\displaystyle h_{\varphi }=\sigma \tau }$ 

It is seen that the scale factors ${\displaystyle h_{\sigma }}$  and ${\displaystyle h_{\tau }}$  are the same as in the two-dimensional case. The infinitesimal volume element is then

${\displaystyle dV=h_{\sigma }h_{\tau }h_{\varphi }\,d\sigma \,d\tau \,d\varphi =\sigma \tau \left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \,d\varphi }$ 

and the Laplacian is given by

${\displaystyle \nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left[{\frac {1}{\sigma }}{\frac {\partial }{\partial \sigma }}\left(\sigma {\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {1}{\tau }}{\frac {\partial }{\partial \tau }}\left(\tau {\frac {\partial \Phi }{\partial \tau }}\right)\right]+{\frac {1}{\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \varphi ^{2}}}}$ 

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$  and ${\displaystyle \nabla \times \mathbf {F} }$  can be expressed in the coordinates ${\displaystyle (\sigma ,\tau ,\phi )}$  by substituting the scale factors into the general formulae found in orthogonal coordinates.

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 660. ISBN 0-07-043316-X. LCCN 52011515.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 185–186. LCCN 55010911.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 180. 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 ξ_k.
• Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 34–36 (Table 1.08). ISBN 978-0-387-18430-2.

• "Parabolic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• MathWorld description of parabolic coordinates