*(For simplicity, we will only consider the _{}case)*

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*Def 1: By _{}, we mean the k-tuples (_{}) where _{}.*

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*Def 2: By open set, we mean the usual Euclidean open set.*

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*Basic Ideas: A “manifold” is an object, which is
“similar” to a normal object. Usually, A is “similar” to B if there exists a
“specific” map f from A to B.*

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*Def 3.1: A map f: U _{}, where U_{} are
open, is smooth iff: all its partial derivative exists and continuous.*

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*Ex 1: f(x,y)= _{} where x,y are real numbers. So, f map from _{}R.*

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*Def 3.2: A map f:U _{}is smooth, U_{} iff:*

* for
each x _{}, there exist an open X containing x and F: _{}such that _{}.*

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*Ex 2: f(x)= _{}where x lies in [0,1].*

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*Homework 1.1: if f, g is smooth, g**of is also smooth,
provided that gf is well-defined.** (Answer)*

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*Def 4: A map
f is a diffeomorphism iff f is a
homeomorphism and f, _{}are smooth.*

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*Homework 1.2: Give an example of differomorphism and a
proof.** (Answer)*

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*Def 5: A subset M in _{} is called a smooth
manifold of dim m iff:*

* For each
x in M, there exist a nbhd _{}and a diffeomorphism f mapping from _{}to an open set V in _{}*

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*Graphic example of manifold:*

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* A sphere
A Torus*

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