## 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,
…

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