Preliminaries to Studying Non-Euclidean Geometry
From EVOCD
Contents |
Introduction
- Motivation (Why this relevant to ME students)
- Very brief history: Gauss, Riemann, Einstein
- Current Uses: GR, Astronomical Scale Continuum Mechanics, e.g. Neutron Stars, Crystal Plasticity: how continuum techniques can account for discontinuities and incompatibilities, e.g. when describing dislocations, voids, dislcinations. TODO: ref
- Potential (Reasons to pursue this further): Non-Euclidean computational models: could be more accurate as they will be able to naturally handle discontinuities and incompatibilities.
- Challenges: Not part of the typical continuum mechanics curriculum. Few engineers are trained in this. Non-Euclidean geometry is often veiled in highly abstracted mathematical language (tangent bundles, vector fields, Lie groups and lie algebra, differential forms, ... ). While some of these abstractedness is necessary to ensure that we are not overly limiting or overly extending the applicability of the math, it is often more helpful to first grasp the basic concepts intuitively and then the generalization will become more clear.
- Here we will focus on an attempt to intuitively present the most basic abstractions in a way that would hopefully lend itself to better intuitive and not just syntactic understanding.
Preliminary concepts: Metric (like a measuring stick), Co-variant and Contra-variant vectors (different representations of a vector once we have have a metric defined), Geometric connections: (how can we compare vectors at different points in space),
Metric and the Metric Tensor
TODO
Covariant and Contravariant Vector Representations
TODO
Geometric Connections and Co-variant Derivative
TODO
Materials for Further Study
TODO