BPST instanton - Revision history

en>A bit iffy: + context in lead sentence

2014-10-08T14:39:33Z

+ context in lead sentence

← Older revision		Revision as of 16:39, 8 October 2014
Line 1:		Line 1:
	~~[[Image:Bispherical coordinates.png\|thumb\|right\|350px\|Illustration of bispherical coordinates, which are obtained~~ by ~~rotating a two-dimensional [[bipolar coordinates\|bipolar coordinate system]] about~~ the ~~axis joining its two foci~~. ~~The foci are located at distance 1 from the vertical ''z''-axis. The red self-intersecting torus~~ is ~~the σ=45° isosurface, the blue sphere is the τ=0.5 isosurface,~~ and ~~the yellow half-plane is the φ=60° isosurface~~. ~~The green half-plane marks the ''x''-''z'' plane, from which φ~~ is ~~measured~~. ~~The black point~~ is ~~located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239).]]~~		She is known by the name of Myrtle Shryock. Hiring is her day job now and she will not alter it anytime soon. California is our beginning location. What I love performing is to gather badges but I've been taking on new things recently.<br><br>My web site; [http://www.videoworld.com/blog/211112 at home std test]

	~~'''Bispherical coordinates'~~'' are a three-dimensional [[orthogonal coordinates\|orthogonal]] [[coordinate system]] that results from rotating the two-dimensional [[bipolar coordinates\|bipolar coordinate system]] about the axis that connects the two foci. Thus, the two [[Focus (geometry)\|foci]] <math>F_{1}</math> and <math>F_{2}</math> in [[bipolar coordinates]] remain points (on ~~the <math>z</math>-axis, the axis of rotation) in the bispherical coordinate system.~~

	~~==Definition==~~
	~~The most common definition of bispherical coordinates <math>(\sigma, \tau, \phi)</math> is~~

	~~:<math>~~
	~~x = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi~~
	~~</math>~~

	~~:<math>~~
	~~y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi~~
	~~</math>~~

	~~:<math>~~
	~~z = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}~~
	~~</math>~~

	where the <math>\sigma</math> coordinate of a point <math>P</math> equals the angle <math>F_{1} P F_{2}</math> and the <math>\tau</math> coordinate equals the [[natural logarithm]] of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to the foci

	~~:<math>~~
	~~\tau = \ln \frac{d_{1}}{d_{2}}~~
	~~</math>~~

	~~===Coordinate surfaces===~~
	~~Surfaces of constant <math>\sigma</math> correspond to intersecting tori of different radii~~

	~~:<math>~~
	~~z^{2} +~~
	~~\left( \sqrt{x^2 + y^2} - a \cot \sigma \right)^2 = \frac{a^2}{\sin^2 \sigma}~~
	~~</math>~~

	~~that all pass through the foci but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting spheres of different radii~~

	~~:<math>~~
	~~\left( x^2 + y^2 \right) +~~
	~~\left( z - a \coth \tau \right)^2 = \frac{a^2}{\sinh^2 \tau}~~
	~~</math>~~

	that surround the foci. The centers of the constant-<math>\tau</math> spheres lie along the <math>z</math>-axis, whereas the constant-<math>\sigma</math> tori are centered in the <math>xy</math> plane.

	~~===Inverse formulae===~~

	~~The formulae for the inverse transformation are:~~

	~~:<math>\sigma = \arccos((R^2-a^2)/Q)</math>~~
	~~:<math>\tau = \operatorname{arsinh}(2 a z/Q)</math>~~
	~~:<math>\phi = \operatorname{atan}(y/x) </math>~~

	~~where <math>R=\sqrt{x^2+y^2+z^2}</math> and <math>Q=\sqrt{(R^2+a^2)^2-(2 a z)^2}~~.<~~/math>~~

	~~===Scale factors===~~

	~~The scale factors for the bispherical coordinates <math>\sigma</math~~> ~~and~~ <~~math~~>~~\tau</math> are equal~~

	~~:<math>~~
	~~h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}~~
	~~</math>~~

	~~whereas the azimuthal scale factor equals~~

	~~:<math>~~
	~~h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma}~~
	~~</math>~~

	~~Thus, the infinitesimal volume element equals~~

	~~:<math>~~
	~~dV = \frac{a^3 \sin \sigma}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi~~
	~~</math>~~

	~~and the Laplacian is given by~~

	~~:<math>~~
	~~\begin{align}~~
	~~\nabla^2 \Phi =~~
	~~\frac{\left( \cosh \tau - \cos\sigma \right)^3}{a^2 \sin \sigma}~~
	~~& \left[~~
	~~\frac{\partial}{\partial \sigma}~~
	~~\left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma}~~
	~~\frac{\partial \Phi}{\partial \sigma}~~
	~~\right) \right. \\[8pt]~~
	~~&{} \quad + \left.~~
	~~\sin \sigma \frac{\partial}{\partial \tau}~~
	~~\left( \frac{1}{\cosh \tau - \cos\sigma}~~
	~~\frac{\partial \Phi}{\partial \tau}~~
	~~\right) +~~
	~~\frac{1}{\sin \sigma \left( \cosh \tau - \cos\sigma \right)}~~
	~~\frac{\partial^2 \Phi}{\partial \phi^2}~~
	~~\right]~~
	~~\end{align}~~
	~~</math>~~

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

	~~==Applications==~~
	~~The classic applications of bispherical coordinates are in solving [[partial differential equations]],~~
	~~e.g., [[Laplace's equation]], for which bispherical coordinates allow a~~
	[[separation of variables]]. However, the [[Helmholtz equation]] is not separable in bispherical coordinates. A typical example would be the [[electric field]] surrounding two conducting spheres of different radii.

	~~==References==~~
	~~{{Empty section\|date=July 2010}}~~

	~~==Bibliography==~~
	*{{cite book \| author = Morse PM, Feshbach H \| year = 1953 \| title = Methods of Theoretical Physics, Part I \| publisher = McGraw-Hill \| location = New York \| pages = 665–666}}
	*{{cite book \| author = Korn GA, Korn TM \|year = 1961 \| title = Mathematical Handbook for Scientists and Engineers \| publisher = McGraw-Hill \| location = New York \| page = 182 \| lccn = 5914456}}
	*{{cite book \| author = Zwillinger D \| year = 1992 \| title = Handbook of Integration \| publisher = Jones and Bartlett \| location = Boston, MA \| isbn = 0-86720-293-9 \| page = 113}}
	*{{cite book \| author = Moon PH, Spencer DE \| year = 1988 \| chapter = Bispherical Coordinates (η, θ, ψ) \| title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions \| edition = corrected 2nd ed., 3rd print \| publisher = Springer Verlag \| location = New York \| isbn = 0-387-02732-7 \| pages = 110–~~112 (Section IV, E4Rx)}}~~

	~~==External links==~~
	*[http://~~mathworld~~.~~wolfram~~.com/~~BisphericalCoordinates.html MathWorld description of bispherical coordinates]~~

	~~{{Orthogonal coordinate systems}}~~

	~~[[Category:Coordinate systems]~~]

en>MuDavid: /* The instanton */ linkfix

2012-06-06T09:34:22Z

The instanton: linkfix

New page

[[Image:Bispherical coordinates.png|thumb|right|350px|Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the axis joining its two foci. The foci are located at distance 1 from the vertical ''z''-axis. The red self-intersecting torus is the σ=45° isosurface, the blue sphere is the τ=0.5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the ''x''-''z'' plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239).]]

'''Bispherical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from rotating the two-dimensional [[bipolar coordinates|bipolar coordinate system]] about the axis that connects the two foci. Thus, the two [[Focus (geometry)|foci]] <math>F_{1}</math> and <math>F_{2}</math> in [[bipolar coordinates]] remain points (on the <math>z</math>-axis, the axis of rotation) in the bispherical coordinate system.

==Definition==
The most common definition of bispherical coordinates <math>(\sigma, \tau, \phi)</math> is

:<math>
x = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi
</math>

:<math>
y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi
</math>

:<math>
z = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}
</math>

where the <math>\sigma</math> coordinate of a point <math>P</math> equals the angle <math>F_{1} P F_{2}</math> and the <math>\tau</math> coordinate equals the [[natural logarithm]] of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to the foci

:<math>
\tau = \ln \frac{d_{1}}{d_{2}}
</math>

===Coordinate surfaces===
Surfaces of constant <math>\sigma</math> correspond to intersecting tori of different radii

:<math>
z^{2} +
\left( \sqrt{x^2 + y^2} - a \cot \sigma \right)^2 = \frac{a^2}{\sin^2 \sigma}
</math>

that all pass through the foci but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting spheres of different radii

:<math>
\left( x^2 + y^2 \right) +
\left( z - a \coth \tau \right)^2 = \frac{a^2}{\sinh^2 \tau}
</math>

that surround the foci. The centers of the constant-<math>\tau</math> spheres lie along the <math>z</math>-axis, whereas the constant-<math>\sigma</math> tori are centered in the <math>xy</math> plane.

===Inverse formulae===

The formulae for the inverse transformation are:

:<math>\sigma = \arccos((R^2-a^2)/Q)</math>
:<math>\tau = \operatorname{arsinh}(2 a z/Q)</math>
:<math>\phi = \operatorname{atan}(y/x) </math>

where <math>R=\sqrt{x^2+y^2+z^2}</math> and <math>Q=\sqrt{(R^2+a^2)^2-(2 a z)^2}.</math>

===Scale factors===

The scale factors for the bispherical coordinates <math>\sigma</math> and <math>\tau</math> are equal

:<math>
h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}
</math>

whereas the azimuthal scale factor equals

:<math>
h_\phi = \frac{a \sin \sigma}{\cosh \tau - \cos\sigma}
</math>

Thus, the infinitesimal volume element equals

:<math>
dV = \frac{a^3 \sin \sigma}{\left( \cosh \tau - \cos\sigma \right)^3} \, d\sigma \, d\tau \, d\phi
</math>

and the Laplacian is given by

:<math>
\begin{align}
\nabla^2 \Phi =
\frac{\left( \cosh \tau - \cos\sigma \right)^3}{a^2 \sin \sigma}
& \left[
\frac{\partial}{\partial \sigma}
\left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma}
\frac{\partial \Phi}{\partial \sigma}
\right) \right. \\[8pt]
&{} \quad + \left.
\sin \sigma \frac{\partial}{\partial \tau}
\left( \frac{1}{\cosh \tau - \cos\sigma}
\frac{\partial \Phi}{\partial \tau}
\right) +
\frac{1}{\sin \sigma \left( \cosh \tau - \cos\sigma \right)}
\frac{\partial^2 \Phi}{\partial \phi^2}
\right]
\end{align}
</math>

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

==Applications==
The classic applications of bispherical coordinates are in solving [[partial differential equations]],
e.g., [[Laplace's equation]], for which bispherical coordinates allow a
[[separation of variables]]. However, the [[Helmholtz equation]] is not separable in bispherical coordinates. A typical example would be the [[electric field]] surrounding two conducting spheres of different radii.

==References==
{{Empty section|date=July 2010}}

==Bibliography==
*{{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | pages = 665–666}}
*{{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | page = 182 | lccn = 5914456}}
*{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 113}}
*{{cite book | author = Moon PH, Spencer DE | year = 1988 | chapter = Bispherical Coordinates (η, θ, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer Verlag | location = New York | isbn = 0-387-02732-7 | pages = 110–112 (Section IV, E4Rx)}}

==External links==
*[http://mathworld.wolfram.com/BisphericalCoordinates.html MathWorld description of bispherical coordinates]

{{Orthogonal coordinate systems}}

[[Category:Coordinate systems]]