Differentiable Functions and Maps |

**Function of a Real Variable** > see analysis.

**Function on a Manifold**

$ __Def__: The function *f* : *M*
→ \(\mathbb R\) is *k*-differentiable at *x* ∈ *M* if, for
for all (*U*, *φ*) such that *x* ∈ *U*, the function
*f* \(\circ\) *φ*^{−1} is
*k*-differentiable at *φ*(*x*).

**Mapping between Manifolds**

$ __Def__: The map *f* : *M*
→ *N* is *k*-differentiable at *x* ∈ *M* if,
for all (*U*, *φ*) such that *x* ∈ *U* and all
(*V*, *ψ*) such that *f*(*x*) ∈ *V*,
the function *ψ* \(\circ\) *f* \(\circ\)
*φ*^{−1} is *k*-differentiable
at *f*(*x*).

* __Pullback of a function__:
The contravariant functor *: (Man, Mor(Man)) → (Vec, Mor(Vec)), taking
*X* → *E*:= {functions on *X*}, *Y* →
*F*:= {functions on *Y*}, and, if *f* ∈ Hom(*X*, *Y*),
*f* → *f* * ∈ Hom(*F*, *E*), defined by

*f* **g* = *g* \(\circ\) *f* , for
all *g* ∈ *F* .

* __Pullback of a one-form__:
A contravariant functor from Man to Vec, as above, taking *X*
→ T**X*, *Y* → T**Y*, and, if *f*
∈ Hom(*X*, *Y*), *f* → *f* *
Hom(T**Y*, T**X*), defined by (*f* **ω*)*v* =
*ω*(*f' v*) \(\circ\) *f*.

* __Pullback of an r-form__:

(*f* **ω*)(*v*_{1},
*v*_{2}, ..., *v*_{r}):=
*ω*(*f*_{*}*v*_{1},
*f*_{*}*v*_{2},
..., *f*_{*}*v*_{r})
\(\circ\) *f* .

@ __References__: Golubitsky & Guillemin 73 [stable mappings and singularities];
> s.a. harmonic map; Singularities.

**Transformations of a Manifold**

$ __Local pseudogroup__:
Given a point *x*_{0} of a manifold *X*,
a set of transformations of a neighborhood of *x*_{0},
*σ*_{t}:
*N*(*x*_{0}) → *X*,
*t* ∈ I ⊂ \(\mathbb R\), with composition law
*σ*_{t} \(\circ\)
*σ*_{s}
= *σ*_{t+s};
__Generator__: The equation *v*(*x*)
= d*σ*_{t}(*x*)/d*t*
|_{t=0} defines a unique vector field
generating the 1-parameter local pseudogroup.

**Differential of a Function**

$ __Def__: The differential of a
function *f* : *M* → \(\mathbb R\) is the 1-form d*f*
∈ T**M* such that \(\langle\)d*f*, *X*\(\rangle\)
= *Xf*, for any vector field* X* ∈ T*M*.

**Differential of a Mapping**

$ __Def__: Given a mapping
*f* : *X* → *Y* between two differentiable manifolds,
its differential at a point *x* ∈ *X* is the mapping *f'*
or *f*_{*}:
T_{x}*X* →
T_{f(x)}*Y*
given by (*f' v*)(*h*):= *v*(*f***h*),
where *h* is a function on *Y*.

* __Remark__: One can
push forward a vector, but not a vector field, usually.

* __In category language__:
A covariant functor from pointed differentiable manifolds to vector
spaces, that constructs everything for us; It associates
(*X*, *x*) → T_{x}*X*,
(*Y*, *y*) → T_{y}*Y*,
and, for *f* : *X* → *Y* such that *y*
= *f*(*x*), *f* → *f'*.

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send feedback and suggestions to bombelli at olemiss.edu – modified 22 jan 2016