SCC is a place where u can get free study material for academic & competitive exams like CTET,SSC,IIT,NDA,Medical exams etc,

Sunday, 10 January 2016

Geometrical Methods of Mathematical Physics

 Geometrical Methods of Mathematical Physics



1.Some Basic Mathematics
2.Differentiable Manifolds And Tensors
3.Lie Derivatives And Lie Groups
4.Differential Forms
5.Applications In Physics

6.Connections For Riemannian Manifolds And Gauge Theories

Basic Algebraic Structures

Structures with only internal operations:
Group ( G, × )
Ring ( R, +, × ) : ( no e, or x-1 )
Field ( F, +, × ) : Ring with e & x-1 except for 0.
Structures with external scalar multiplication:
Module ( M, +,  ×  ;  R )
Algebra ( A, +,  ×  ;  R with e )
Vector space ( V, + ; F )

Prototypes:
 R is a field.
 Rn is a vector space
The Space Rn And Its Topology
Goal:  Extend multi-variable calculus (on En) to curved 
spaces without metric.

Bonus:  vector calculus on E3 in curvilinear 
coordinates
Basic calculus concepts & tools (metric built-in):

Limit, continuity, differentiability, …

  r-ball neighborhood,  δ-ε formulism, …

Integration, …

Essential concept in the absence of metric:
  
Proximity → Topology.

A system U of subsets Ui of a set X defines a topology on X if
A topological space is the minimal structure on which concepts of     neighborhood,   continuity,   compactness,   connectedness can be defined.

Trivial topology:    U = { Æ, X  }
  →  every function on X is dis-continuous
Discrete topology:   U = 2X
  →  every function on X is continuous
Exact choice of topology is usually not very important:
2 topologies are equivalent if there exists an homeomorphism (bi-continuous bijection) between them.
Tools for classification of topologies: 
  topological invariances,  homology,  homotopy, …
To read complete topic click here .............Download 

Share on Google Plus Share on whatsapp

Search

Popular Posts

Facebook

Blogger Tips and TricksLatest Tips For BloggersBlogger Tricks
SCC Education © 2017. Powered by Blogger.

Total Pageviews