1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian - PowerPoint PPT Presentation

1d 2d 3d nd euclidian spherical hyperbolic general riemannian
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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.

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

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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

http://home.iitk.ac.in/~anandh/E-book.htm

SPACE

1D, 2D, 3D…. nD

Euclidian, Spherical, Hyperbolic

General Riemannian


1d 2d 3d nd euclidian spherical hyperbolic general riemannian

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


1d 2d 3d nd euclidian spherical hyperbolic general riemannian

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


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