Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T21:47:50.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrals on a moving manifold and geometrical probability

Published online by Cambridge University Press:  01 July 2016

Adrian Baddeley
Adrian Baddeley*
Affiliation:
The Australian National University
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

GEOMETRICAL PROBABILITYCROFTON'S THEOREMINTERFACE DISPLACEMENT EQUATIONSSMOOTHLY CHANGING MANIFOLDVELOCITYBOUNDARY VELOCITYBOUNDARY EXPANSION VELOCITYDIFFERENTIAL FORM
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 588 - 603
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Crofton, M. W. (1885) Probability. In Encyclopaedia Britannica, 9th edn., pp. 768–788.Google Scholar
Batchelor, G. K. (1970) An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
De Hoff, R. T. (1972) The dynamics of microstructural change. In Treatise on Materials Science. Academic Press, New York.Google Scholar
Federer, H. (1969) Geometric Measure Theory. Springer-Verlag, Berlin.Google Scholar
Flanders, H. (1963) Differential Forms. Academic Press, New York.Google Scholar
Ruben, H. and Reed, W. J. (1973) A more general form of a theorem of Crofton. J. Appl. Prob. 10, 479–482.Google Scholar
Spivak, M. (1965) Calculus on Manifolds. Benjamin, New York.Google Scholar