1 / 4

1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

MATERIALS SCIENCE & ENGINEERING . Part of . A Learner’s Guide. AN INTRODUCTORY E-BOOK. Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh.

nijole
Télécharger la présentation

1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm SPACE 1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian

  2. SPACE • A simplistic point of view is that space is a perfect inert vacuum and matter resides in that space. • The real picture is more complicated, with actual space deviating from the ‘smooth picture’ at the very small lengthscales. Space at these lengthscales is teeming with virtual particles which fleetingly come into existence. • There are unsolved questions regarding the number of dimensions and curvature of space we live in → though it is assumed to be 3D Euclidean locally* → At the scale of the universe it is understood to be non-Euclidean • Many of the theories of physics which describe nature (e.g. the string theories) require higher dimensions (10 or more in some of them!). Dimensions higher than 3 are supposed to be ‘compactified’ • Apart from the possibility of higher spatial dimensions, in Einstein’s description of gravity, the 3D space is intertwined with time into a 4D ‘space-time unit’. • In some theories, the structure of glasses and quasicrystals are described by hyper-dimensional constructs. * refer next slide

  3. SPACE continued… • Gaussian curvature is the product of two orthogonal curvatures • Mean curvature is the average of two orthogonal curvatures • Space can be:Euclidean (flat) → Zero (0) Gaussian Curvature  Spherical → Positive (+) Gaussian Curvature  Hyperbolic → Negative () Gaussian Curvature • If for a Hyperbolic surface the mean curvature is zero then the surface is called a Minimal Surface Emphasis though it is assumed that space is: 3D → in general it can be nD Euclidean → it can be Non-Euclidean with local variations in the curvature of space

More Related