"A coordinate system then is a map from a region of the differentiable manifold to the set of n-tuples. If we choose a point, the map determines an n-tuple of coordinates of the point. Hence, we have the coordinate map and the coordinates as components, or values taken by the set of coordinate functions that constitute the map. When two coordinate systems overlap, it is required that the functionsexpressing the coordinate transformation are continuous and have continuousderivatives up to some order, appropriate to achieve some specific purpose." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"A differentiable manifold endowed with an Euclidean connection has a metric structure superimposed on an affine structure. The affine structure allows for a concept of autoparallels, or lines of constant direction. The metric structure gives rise to a concept of stationary length between two points, not necessarily maximum or minimum length." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"A map is an application from an arbitrary set to another arbitrary set. It can thus be used in all cases where the more specific terms of function and functional also apply. The domain of maps called functions also is arbitrary, but the range or codomain must be at least a ring. Finally, a functional maps an algebra of functions to another algebra, or to itself in particular." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"A pair of region and coordinate assignment is called a chart. The region. in the chart of Cartesian coordinates in the plane is the whole plane. And the region in the chart for the system of polar coordinates is the plane punctured at the chosen origin of the system." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"[...] affine curvature can be viewed as being about transporting a vector around a closed curve and returning to the point of departure with a different vector. Torsion, on the other hand, is about representing in affine space small curves of general affinely connected differentiable manifolds. That the paths have to be very small for integration of vector-valued integrals to mean anything at all is seldom if ever considered in books on differential geometry, more interested in formal aspects than on geometric interpretations. Or theyhave it wrong for failing to refer to very small paths." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"An affine space is a space of objects called points such that, after arbitrarily taking a point to play the role of zero, we can establish a one-to-one correspondence between its points and the vectors of a vector space, called its associated vector space. If the vector space is Euclidean, the affine space is called Euclideanspace (i.e. without 'vector' between the words 'Euclidean' and 'space'). The study of affine and Euclidean space is geometry." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"[...] a point separates algebra from geometry. In algebra there is a zero; in geometry, there is not any but we introduce it arbitrarily in order to bring algebra into it. There are additional subtle differences. They are related to what we have just said. They have to do with what is it in algebra that algebraists and geometers consider most relevant. Grossly speaking, it is matrices and linear equations for algebraists. And it is groups for geometers." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"Charts play the role of providing the arena that supports differentiation in a continuous set of points. The purpose of having different charts and not just one is to be able to adequately cover all points in the set, one small piece at the time. We then endow the set with additional structure, materialized through a system of connection equations and a system of equations of structure, the latter speaking of whether the former is integrable or not." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"In Riemann’s and Cartan’s theories, a surface is a differentiable manifold of dimension two, meaning that we need two independent coordinates to label its points. But, as we shall see in an appendix, the theory of differentiable manifolds of dimension two and the theory of surfaces developed before Riemann do not coincide. Thus, whereas we may speak of the torsion of a differentiable 2-manifold, we may not speak of the torsion of a surface in the theory of curves and surfaces of that time." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"The integrands of the integrals on a differential manifold (on its curves, surfaces, etc.; all of them oriented) constitute a structure called an algebra. To be more specific, it is a graded algebra. [...] The subspaces constituted by all the elements of a graded algebra thathave the same grade constitute modules (generalization of the concept of vector space)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"In Euclidean vector spaces, there is the concept of orthonormal bases. But the non-orthonormal bases are as legitimate as those which are. On the other hand, in the geometry of Euclidean spaces, orthonormal bases have a legitimacy that arbitrary bases do not. Why? Because in geometry the role of groups becomes paramount. Euclidean geometry is the study of figures and properties that are invariant under the Euclidean group, the basic figures being the orthonormal bases at all points. Affine geometry is the study of figures and properties invariant under the affine group, the basic figures being all vector bases at all points." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"In the tensor algebra constructed upon some vector space (or over some module), the vectors (or the elements of the module) are considered as tensors of grade or rank one. The vectors that one finds in the vector calculus are called tangent vectors. They constitute so called tangent vector spaces (one at each point of the manifold), though it is difficult to realize in Euclidean spaces why should one call them tangent. This will later become evident. Elements of the space that results from tensor product of copies of a tangent vector space are called tangent tensors." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"It is the tensorial composition law in the structure that makes tensors be tensors. Thus, elements of a structure may look like tensors but not be so because its tensor product by a similar object does not yield another similar one. Example: the tensor product of two so called skew-symmetric (also called antisymmetric) tensors is not skew-symmetric in general. The skew-symmetric product in this case is the exterior product. Their algebra is exterior algebra and its elements should be called multivectors." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"One sometimes finds in the literature the statement that a tensor transforms in such and such way under a coordinate transformation. Tensors do not transform. They are invariants. Their components do, under changes of section of the frame bundle." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"The most familiar manifold, however, is the space-time manifold, which has 4 dimensions. It is described by a time coordinate and three spatial coordinates. In addition to being a differentiable manifold, space-time has much more additional structure. It is at the level of this additional structure, which will be the subject of later chapters, that the space-time of Newtonian physics differs from the space-time of special relativity and from the space-times of Einstein’s theory of gravity (also called general relativity)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"There is a great virtue in calculus of differential forms. It is appropriate for differentiation in general spaces known as differentiable manifolds, which, in general, lack a metric and a connection or rule to compare vectors at different points. We must, however, have a concept of continuity in the space in question, which eliminates automatically discrete sets of points. The set must be such that regions of the same can be represented unequivocally by open sets of n-tuples of real numbers (it could also be complex numbers [...]). The openness of the set has to do with the behavior at the borders of the region." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"To summarize, affine torsion is about representing on affine space itself (i.e. on the corresponding Klein space) tiny closed curves of an affinely connected manifold and 'quantifying' its failure to close in the form of a vector, which one obtains through integration of the torsion. Let it not be forgotten that the representation of closed curves fails to close even if the torsion is zero but the affine curvature is not. But the reason for the failure to close is not the same one when the torsion is zero as when the affine curvature is zero. In general, both torsion and curvature will contribute to the failure to close." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"Vector calculus is horrible for several reasons. One of them is that its curl is based on the vector product. So, we do not have a curl in other dimensions. Another reason is that it uses tangent vectors where it should use differential forms. One more is that it often uses more structure than needed to solve a problem, say a metric structure. Still another one is that one can do so little with it that it has to be complemented with all the other calculi that we also think of replacing with differential forms. The Kähler calculus - based on Clifford algebra of differential forms - replaces tangent vectors and tangent-valued operators with differential forms whose coefficients are respectively functions and operators." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"We say that a set of quantities transforms vectorially under a group of transformations of the bases of a vector space if, under an element of the group, they transform like the components of vectors, whether contravariant or covariant. We say that some quantity is a scalar if it is an invariant under the transformations of a group. A quantity (respectively, a set of quantities) may be scalar (respectively vectorial) under the transformation of a group, while not being so under the transformations of another group." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
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