Contributions to Tensor Analysis

Citation

Guy, William Thomas (1951) Contributions to Tensor Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/XC95-GY46. https://resolver.caltech.edu/CaltechETD:etd-10022006-144129

Abstract

This thesis treats two separate problems. The first concerns the transverse vibrations of a beam and of a thin rectangular flat plate. These vibrations are associated with a function space which has the properties of a generalized "Riemannian" function space. The geodesics of this space are shown to play a role analogous to that played by the geodesics of the configuration space in the classical treatment of the finite dimensional case. Part I is introductory and treats a few aspects of the vibrations of beams with various end conditions under a change of parameter. Part II develops the integro-differential equation for the thin rectangular flat plate. The associated function space and its geodesics are then studied in some detail. The space is found to be not one of constant Riemannian curvature. An example is worked out to illustrate the ideas, and an extension is suggested. The second problem (part III) considers the equations of motion of hydrodynamics of viscous flow with moving axes. Use is made of the space of a kinetic metric introduced by McVittie, who considered non-viscous flow only. The Newtonian equations are obtained by taking certain approximations. The equations of motion in terms of the vorticity tensor are developed. Two examples are discussed illustrating the theory, one concerning instability necessary for tropical cyclones.

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