Vector and Tensor Theorems on Curved Surfaces

Abstract

Some unfamiliar results are described, with examples of their application in mechanics.

Keywords

differential surface geometry and flow meteorology plasticity

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References

Ericksen, J. L. Tensor fields, in Handbuch der Physik, III/1, pp. 794-858, ed. S. Flügge, Springer, Berlin, 1960.

Spiegel, M. R. Vector Analysis and an Introduction to Tensor Analysis, Schaum, New York , 1959.

Republished as Applications of Tensor Analysis, Dover, New York , 1957.

Gurtin, M. E. Configurational Forces as Basic Concepts of Continuum Physics, Springer, Berlin , 2000.

Cullen, M. J. P. , Douglas, R. J. , Roulstone, I. and Sewell, M. J. Generalised semi-geostrophic theory on a sphere. Joint Centre for Mesoscale Meteorology Internal Report No. 138, Reading University, UK , submitted for publication, 2003.

Hoskins, B. J. The geostrophic momentum approximation and the semi-geostrophic equations . Journal of Atmospheric Sciences, 32, 233-242 (1975).

Chynoweth, S. and Sewell, M. J. Dual variables in semi-geostrophic theory . Proceedings of the Royal Society of London A, 424, 155-186 (1989).

Roulstone, I. and Sewell, M. J. The mathematical structure of theories of semi-geostrophic type . Philosophical Transactions of the Royal Society of London A, 355, 2489-2517 (1997).

Sewell, M. J. A survey of plastic buckling, in Stability, pp. 85-197, ed. H. Leipholz , University of Waterloo Press, Canada , 1972.

10.

Hill, R. Generalized constitutive relations for incremental deformation of metal crystals by multi-slip . Journal of the Mechanics and Physics of Solids, 14, 95-102 (1966).

11.

Sewell, M. J. Maximum and Minimum Principles. A Unified Approach, with Applications, Cambridge University Press, Cambridge , 1987.

12.

Sewell, M. J. Some applications of transformation theory in mechanics, in Large-Scale Atmosphere-Ocean Dynamics II. Geometric Methods and Models, pp. 143-223, ed. J. Norbury and I. Roulstone , Cambridge University Press, Cambridge , 2002.

13.

White, A. A. A view of the equations of meteorological dynamics and various approximations, in Large-Scale Atmosphere-Ocean Dynamics I. Analytical Methods and Numerical Models, pp. 1-100, ed. J. Norbury and I. Roulstone , Cambridge University Press, Cambridge , 2002.

14.

Mirsky, L. An Introduction to Linear Algebra, Clarendon, Oxford , 1955.

15.

Green, A. E. and Zerna, W. Theoretical Elasticity, Clarendon, Oxford , 1954.

16.

Synge, J. L. Classical dynamics, in Handbuch der Physik, III/1, pp. 1-225, ed. S. Flügge, Springer, Berlin, 1960.