Length-scale where the effects of gravity, i.e. One example is using the tools of continuum mechanics at astronomical Situations when the assumptions of Euclidean geometry no longer apply. In most cases, weĪssume Euclidean geometry for the continuum, but there are several Study stress and strain fields within a material by treating that
4 Affine Connection and Covariant DerivativeĬontinuum mechanics applies the tools of differential geometry to. 3.3 Relationship to the Contravariant and Covariant Representaiton. 3.2 Relationship to the Inner Product (Dot Product) of Vectors. 2.3 Covariant Representation of a Point. 2.2 Contravariant Representation of a Point. 2 Contravariant and Covariant Representation of Points, Reciprocal Basis. This is the purpose of the affine connection. So, before one can add, subtract or compute the inner products of an arbitrary set of vectors, there will need to be a way to specify an equivalent set of vectors that are all attached to the same point in space. Whereas in Euclidean geometry a vector has only a length and direction, but no position, in Riemannian geometry, however, a vector cannot be disassociated from its position in space. The term affine pertains to the ability to preserve parallel relationships. Affine connection defines how vectors at different points in space can be compared to each other. geodesic), the dot product of vectors and what it means for lines to be perpendicular. Once that choice is made, it further gives rise to the definition of straight line (a.k.a. A metric is a choice that one can attribute to a mathematical space. In Riemannian geometry, the metric is specified by the metric tensor. In general the metric can vary from one point in space to another. 
NON EUCLIDEAN GEOMETRY EXAMPLES HOW TO
The metric specifies how to compute distances between points in space.Contravariant and covariant representations are different representations of a given vector or point in space with respect to a given set of basis vectors and a reciprocal set of basis vectors.
Here is a brief overview of the basic concepts described in later sections: The goal is to provide the necessary vocabulary that would enable the reader to be better equipped for understanding continuum mechanics research that makes use of Riemannian geometry.
This article is an attempt to briefly introduce some of the most basic abstractions of differential geometry such as contravariant versus covariant representations of vectors and points, metric and metric tensor and affine connection.
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