Differential geometry

Recognition of non-Euclidean geometries creates the new problem of describing the position and motion of finite objects in curved space, a problem which does not occur in Euclidean spaces that extend uniformly to inhnity. An object such as the distance between two points. 

This problem was addressed by Gauss for two-dimensional surfaces in Euclidean space and later extended by Riemann to general rr-dimensional non-Euclidean spaces. The procedure is of interest here as it provides the facility to investigate the gravitational field in general relativity. The vital assumption is that in the limit of an infinitesimally small object simple Euclidean geometry would apply, suggesting that the methods of infinitesimal 

The concept of the derivative of a function in calculus is essentially the same as the tangent line or slope of a curve, and the integral of a function can be geometrically interpreted as the area under a curve. 

According to the ordinary meaning of partial differentiation, the partial derivatives of a position vector 

The differential of r, representing displacement in any given direction, is given by the scalar product 

The curvilinear coordinate s will be given such that it represents the arc length and therefore, just as in the case of the Lame parameters for the cylindrical shell in Section 6.2.2, the following is required  

The unit tangent vector e s) is given by the corresponding derivative of the cross-sectional position So s). Further on, the unit normal vector e (s) is defined orthogonal to the unit axis vector in parallel to the undeformed reference line and to the unit tangent vector es s). Thus, the moving trihedral is given as 

With Eqs. (7.16) and (7.18), this relations allows us to express the radius R s) in terms of the cross-sectional position  

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M. Spivak. A Comprehensive Introduction to Differential Geometry, Vol. III. Berkeley Publish or Perish, 1979. 

Y. Talpaert, Differential Geometry With Applications to Mechanics and Physics (2001)... 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

Method Based on the First and Second Fundamental Forms of the Differential Geometry... 

The mean, Gaussian, and principal curvatures are well-defined quantities from the standpoint of the differential geometry. If the surface is given by the... 

In this method the local curvatures are calculated by using the first and second fundamental forms of differential geometry [7]. The surface is parameterized near the point of interest (POI) as p(u, v) (see Fig. 32). The coordinates (u, v) are set arbitrary on the surface in such a way that POI is located at p(u, v) = (0,0). The first and the second forms of the differential geometry are expressed as... 

The computation of the curvatures from the bulk field spatial derivatives of the field sharp peaks at the phase interface. This is a common situation in the late-stage kinetics of the phase separating/ordering process, when the order parameter is saturated and the domains are separated by thin walls. Here, to calculate the curvatures, we propose a much more accurate method. It is based on the observation that the local curvatures are quantities that can be inferred solely from the shape of the interface, without appealing to the properties of the bulk field [Pg.212]

Vol. 1481 D. Ferns, U. Pinkall, U. Simon, B. Wegner (Eds.), Global Differential Geometry and Global Analysis. Proceedings, 1991. VIII, 283 pages. 1991. 

In the last years, the control based on differential geometry has emerged as a powerful tool to deal with a great variety of dynamic nonlinear systems [11, 15, 26], This control approach allows the transformation of a nonlinear system into a partially or totally linear one, by means of a nonlinear state transformation, which is obtained from directional derivatives of the output. It is important to remark that geometric control differs totally from the linear approximation of dynamics by calculation of the Jacobian. 

Before focusing in the controller design, it is important to review some basic concepts of the geometric control theory. The control tools based in differential geometry are proposed for those nonlinear dynamical systems called affine systems. So, let s star by its definition. 

During my undergraduate years, 1935-1939, in Honors Mathematics and Physics at the University of Toronto, increasingly, I became interested in mathematical physics, picking up some elementary quantum mechanics and relativity. My first encounter with Einstein s general relativity theory (GRT) was in the substantial treatise of Levi-Civita on differential geometry, which ends with a 150-page introduction to GRT. This is a beautiful theory, which I presented in lectures from 1950 in Toronto because it had become the dominant orthodoxy everyone should know ... 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

Kuchar, K. (1976) Kinematics of tensor fields in hyperspace, ll. Journal of Mathematical Physics. 17(5) 792—800. (Differential geometry in hyperspace is used to investigate kinematical relationships between hypersurface projections of spacetime tensor fields in a Riemannian spacetime.)... 

Equation (2.1.3a) has been studied extensively in different mathematical and physical contexts ranging from differential geometry to reaction-diffusion, electrokinetics, colloid stability, theory of polyelectrolytes, etc. In... 

Such a reader might find relief in differential geometry, the mathematical study of multiple coordinate systems. There are many excellent standard texts, such as Isham s book [I] for a gentle introduction to some basic concepts of differential geometry, try [Si]. A text that discusses covariant and contravariant tensors is Spivak s introduction to differential geometry [Sp, Volume I, Chapter 4]. For a quick introduction aimed at physical calculations, try Joshi s book [Jos]. 

Each of the groups we introduce in this text is a Lie group. We give the formal definition in terms of manifolds however, readers unfamiliar with differential geometry may think of a manifold as analogous to a nicely parametrized surface embedded in R. More to the point for our purposes, a manifold is a set on which differentiability is well defined. Since all the mamfolds we will consider are nicely parameterized, we can define differentiability in terms of the parameters. 

Students of differential geometry may recognize this set as the tangent space to the man-... 

The projective space P(C2) has many names. In mathematical texts it is often called one-dimensional complex projective space, denoted CP (Students of complex differential geometry may recognize that the space PCC ) is onedimensional as a complex manifold loosely speaking, this means that around any point of (C ) there is a neighborhood that looks like an open subset of C, and these neighborhoods overlap in a reasonable way.) In physics the space appears as the state space of a spin-1/2 particle. In computer science, it is known as a qubit (pronounced cue-hit ), for reasons we will explain in Section 10.2. In this text we will use the name qubit because CP has mathematical connotations we wish to avoid. 

Our proof uses some differential geometry from Appendix B. One can replace the theory by a concrete calculation of the derivative of [Pg.321]

The proof, which requires a knowledge of differential geometry beyond the prerequisites of the text, is in Appendix B. 

Exercise 10.2 (For students of differential geometry) Show that for any natural number n, the set P(C + ) is a real manifold of dimension 2n. 

In this appendix we prove Proposition 10.6 from Section 10.4, which states that the irreducible projective unitary Lie group representations of SO(3) are in one-to-one correspondence with the irreducible (linear) unitary Lie group representations of St/ (2). The proof requires some techniques from topology and differential geometry. 

I] Isham, C.J., Modem Differential Geometry for Physicists, Second Edition, World Scientific, Singapore, 1999. 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

Vol. 1255 Differential Geometry and Differential Equations. Proceedings, 1985. Edited by C. Gu, M. Berger and R.L. Bryant. XII, 243 pages. 19S7. 

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