A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the parabolas of parabolic cylindrical coordinates about the x-axis, which is then relabeled the z-axis. There are several notational conventions. Whereas is used in this work, Arfken (1970) uses .
The equations for the parabolic coordinates are
(1)
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(2)
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(3)
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where , , and . To solve for , , and , examine
(4)
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(5)
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(6)
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so
(7)
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and
(8)
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(9)
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We therefore have
(10)
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(11)
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(12)
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The scale factors are
(13)
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(14)
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(15)
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The line element is
(16)
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and the volume element is
(17)
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The Laplacian is
(18)
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(19)
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(20)
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The Helmholtz differential equation is separable in parabolic coordinates.