Congruence Of Curves Research Articles

Zur differentialthermoanalyse der oxalathydrate der seltenen erdmetalle

Several authors found congruence of curves, obtained by differential thermal analysis of thermal decomposition of oxalate-hydrates of the rare earths. Opinions are divergent regarding the decomposition of the anhydrous oxalates, following the dehydration. Some factors have an important influence on the results, these factors are discussed in detail.

Continuous distributions of dislocations. Ill

The theory of the continuously dislocated crystal is extended. It is shown that the general equations of the theory of Nye are applicable in full only when the lattice curvature (and so also the dislocation density) is small. The correct equations for large curvatures are formulated. The relation between the fundamental node theorem of dislocation theory and certain theorems of the theory of generalized space is considered in some detail. It is also shown that when the lattice lines of the dislocated crystal are treated as a system of independent congruences of curves, the tensor describing the local dislocation density may be expressed in terms of certain invariants connected with the system of congruences. For the rotation case, a relation between the dislocation density and Ricci’s coefficients of rotation is derived.

Generalization of normal curvature of a curve in a Riemannian V n

In the present paper a congruence of curves through points of a hypersurface V n imbedded in a Riemannian V n+1 has been considered. In analogy with the normal curvature of a curve C in V n , the generalized normal curvature of C at any point of it, relative to the curve of the congruence through that point, has been defined as the negative of the resolved part along C, of the derived vector of the unit tangent to the curve of the congruence through the point along C. The concepts of normal curvature of a hypersurface, principal directions, principal curvatures, lines of curvature, conjugate directions, asymptotic directions and asymptotic lines have been generalized and generalizations of several known theorems on the curvature of a hypersurface V n in V n+1 have been obtained.

Variation of Congruences of Curves of an Orthogonal Ennuple in a Riemannian Space

Consider any three congruences of an orthogonal ennuple at a point of a Riemannian space. When one congruence is moved by local and a second congruence is moved by parallel displacement in the direction of the third congruence, the rate of change of cosine of the angle between the first two congruences is well known as Ricci's coefficient of rotation and has been extensively studied. It is the purpose of this note to investigate the corresponding rate of change, when the third congruence is replaced by an arbitrary one, in connection with parallelism and equidistance of congruences as studied by Miss Peters [2; 3].

Equidistance and parallelism of the congruences of curves through points of a subspace

Equidistance and parallelism of the congruences of curves through points of a subspace

On permanent vector-lines in 𝑁 dimensions

Prim and Truesdell have recently given a simple vector proof of Zorawski's condition for the permanence of vector-lines in a moving fluid.' Their argument is necessarily by its form confined to threedimensional space, and it seems of interest to investigate the problem in a Euclidean space of N dimensions, using a slightly different approach. We consider a Euclidean space EN with rectangular cartesians xi. Two vector fields are given, vi(x, t), ci(x, t), where t is a parameter (the time); we shall call v* the primary and ci the secondary vector fields; vi plays the part of velocity. Any vector field defines for given t a congruence of curves in an obvious way, the direction of the curve at each point coinciding with that of the vector field. We are not particularly interested here in the congruence defined by the primary vector field (stream-lines), and we shall use the expression vector-lines to refer exclusively to the congruence defined by the secondary field, that is, those curves which satisfy

Lineal element transformations of space for which normal congruences of curves are converted into normal congruences

Lineal element transformations of space for which normal congruences of curves are converted into normal congruences

Parallelism and Equidistance of Congruences of Curves of Orthogonal Ennuples

Parallelism and Equidistance of Congruences of Curves of Orthogonal Ennuples

The geometry of Riemannian spaces

The primary purpose of this paper is to expose, in as simple and clear a form as is possible, the fundamentals of the geometric structure of a Riemannian space. It is a general truth that the methods which pierce most deeply into the heart of a geometric theory are invariant methods, that is, methods which are independent of the choice of the coordinates in terms of which the theory is expressed analytically. In the case of Riemannian geometry, these are the methods of tensor analysis. As important, perhaps, as the use of invariant methods is the expression of the analytic theory, so far as possible, in terms of invariant quantities alone. For it is in this form that the theory becomes most illuminating and suggestive. But, in ordinary tensor analysis, the components of a tensor are not invariants. A first step toward our goal will be, then, to introduce for Riemannian geometry an intrinsic tensor analysis, that is, a form of tensor analysis in which the components of all tensors are invariants. Any theory of the geometry of a Riemannian space presupposes that the space is referred to a certain ennuple of congruences of curves. In the ordinary theory, this ennuple consists of the parametric curves. In the intrinsic theory, it is an ennuple E, whose choice, as will presently be evident, is entirely arbitrary. The ordinary components of a tensor, that is, the components in the ordinary theory, are referred to the differentials of the coordinates xi pertaining to the ennuple of the parametric curves. The intrinsic components are referred to the differentials of arc, dsi, of the curves of the ennuple E. There is no need in the intrinsic theory of actual coordinates pertaining to the ennuple E; the differentials of arc dsi suffice. Accordingly, E can be chosen arbitrarily; it does not have to be a parametric ennuple, that is, an ennuple with which coordinates can be associated. It may be, and we shall ordinarily take it to be, an ennuple of general type, and hence, of course, not necessarily orthogonal. Intrinsic components, referred to E, of the covariant derivative of a tensor are made possible by the introduction of invariant Christoffel symbols in

Congruences of curves in the geometry of paths

Congruences of curves in the geometry of paths

Some New Theorems in Geometry of a Surface

The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

Some New Theorems in Geometry of a Surface

The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

Normal congruences of curves in Riemann space

Normal congruences of curves in Riemann space

On triply orthogonal congruences

1. A vector field is defined by the value of a vector a at each point of the field. This vector a is a function of p, the vector to the point from some given origin. If a is taken as tangent to a curve at each of the points of the space considered, these tangents will envelop a congruence of curves, the vector lines of the field of a. The tensor or length of a is not determinable from the congruence, the congruence depending only upon the unit vector Ua. The congruence itself however determines many properties of the field, which are of course purely geometric properties, and in physical phenomena. are not dependent on physical facts except through the law by which Ua is determined. It is purposed to consider some of these properties which belong to the congruences determined by three unit vectors which are everywhere mutually perpendicular. These will generally be indicated by a, ,3, y; but in case they are the tangent, principal normal, and binormal of a curve, we will use v with a proper subscript to indicate the principal normal, and pi with the same subscript to indicate the binormal. For instance, if we are discussing a curve of the a congruence the normal will be vP, and the binormal A. The algebra used throughout is Quaternions, all the usual symbols of that algebra being introduced with no explanation. However it should be noted that such an expression as aj3 is always a quaternion product. The symbol 4 and the symbol 0 are throughout linear vector operators,in the case of 4 related to certain vectors which are subscripts of 4). For instance we define for any vector 7: X,7 = S ( ) V * X7. The occurrence of such operators is due to the fact that if the point under discussion at the end of p is displaced to p + dp, the vector X becomes X + d7 where d7 -SdpV* = ) dp.

Equitangential congruences of curves in space

Equitangential congruences of curves in space

Congruences of Curves

Congruences of Curves

Congruences of curves

where u and v denote the parameters determining the curves and t is the parameter which determines the points on the curve. By this method of definition the equations of a congruence of curves are given a form similar to that usually adopted in the discussions of rectilinear congruences, so that one proceeds naturally to a generalization of some of the properties of the latter. At the same time the curves, as above defined, are looked upon as the intersections of surfaces, namely, those defined by (1), when u and v respectively are constants. In ?? 1, 2 we show how the theorems of DARBOUX can be established when the congruence is defined in this manner. Several examples are given to illustrate the different theorems. In ? 3 we consider the problem of determiining a function 4 (u, v ) such that the tangents to the curves of the congruence at the points of intersection with the surface, defined by (1) after t has been replaced by 4,, shall form a normal congruence, or in the second place the ruled surfaces u = const., v = const. of this congruence of tangents shall be developable. It is shown that for every congruence a function exists which furnishes a solution to the first of these problems, and the condition is found which the functions/, f2, f3 must satisfy in order that 4 may be constant. However, there does not always exist a function such that the second condition is satisfied. But when the curves t = const.,