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Integrals on a moving manifold and geometrical probability

  • Adrian Baddeley (a1)

Abstract

For a manifold which is moving and changing with time, consider some numerical property which at each instant is equal to an integral over the manifold. We derive a general expression for the time rate of change of this integral. Corollaries include a precise general form of Crofton's boundary theorem, de Hoff's interface displacement equations (with some new extensions) and a theorem in fluid mechanics.

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References

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Crofton, M. W. (1885) Probability. In Encyclopaedia Britannica, 9th edn., pp. 768788.
Batchelor, G. K. (1970) An Introduction to Fluid Dynamics. Cambridge University Press.
De Hoff, R. T. (1972) The dynamics of microstructural change. In Treatise on Materials Science. Academic Press, New York.
Federer, H. (1969) Geometric Measure Theory. Springer-Verlag, Berlin.
Flanders, H. (1963) Differential Forms. Academic Press, New York.
Ruben, H. and Reed, W. J. (1973) A more general form of a theorem of Crofton. J. Appl. Prob. 10, 479482.
Spivak, M. (1965) Calculus on Manifolds. Benjamin, New York.

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Integrals on a moving manifold and geometrical probability

  • Adrian Baddeley (a1)

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