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Encyclopedia > Kelvin transform

The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions. In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations and complex analysis. ...


In order to define the Kelvin transform f* of a function f, it is necessary to first consider the concept of inversion in a sphere in Rn as follows.


It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin.


Given a fixed sphere S(0,R) with centre 0 and radius R, the inversion of a point x in Rn is defined to be

A useful effect of this inversion is that the origin 0 is the image of , and is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa.


The Kelvin transform of a function is then defined by:


If D is an open subset of Rn which does not contain 0, then for any function f defined on D, the Kelvin transform f* of f with respect to the sphere S(0,R) is

One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result:

Let D be an open subset in Rn which does not contain the origin 0. Then a function u is harmonic, subharmonic or superharmonic in D if and only if the Kelvin transform u* with respect to the sphere S(0,R) is harmonic, subharmonic or superharmonic in D*.

See also

William Thomson, 1st Baron Kelvin, GCVO, OM, PC, PRS FRSE (26 June 1824 – 17 December 1907) was an Irish-Scottish mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. ... In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from...

References

  • J. L. Doob (2001). Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag. ISBN 3-540-41206-9.
  • L. L. Helms (1975). Introduction to potential theory. R. E. Krieger. ISBN 0-882-75224-3.
  • O. D. Kellogg (1953). Foundations of potential theory. Dover. ISBN 0-486-60144-7.

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The establishment of Kelvin Grove as a distinct district began with the opening on Kelvin Grove School in 1893, its name being drawn from the name of a sawmill that had opened in 1879 in the vicinity of the Kaimanawa Street-Bryce Place intersection.
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