View source for Notes:An introduction to manifolds - Loring W. Tu

==Chapter 1==
===Section 1: Smooth functions on Euclidean space===
====1.1: {{M|C^\infty}} vs Analytic functions====
[[Example:A smooth function that is not real analytic]]
====1.2: Taylor's theorem with remainder====
===Section 2: Tangent vectors in {{M|\mathbb{R}^n}} as Derivations===
* {{MM|D_vf\eq\lim_{t\rightarrow 0}\left(\frac{f(c(t))-f(p)}{t}\right)\eq\frac{d}{dt}f(c(t))\Big\vert_{t\eq 0} }}
====2.2: Germs of functions====
* '''[[Equivalence relation]]''' yeah yeah yeah
* We define an equivalence relation on the {{M|C^\infty}} functions defined in some neighbourhood of {{M|p\in\mathbb{R}^n}}.
* Consider the set of all [[ordered pairs]], {{M|(f,U)}}, where {{M|U}} is an [[open neighbourhood]]<ref group="Unsure">Page 12, first paragraph - he says neighbourhood, is that my neighbourhood or is that my open neighbourhood</ref> and {{M|f:U\rightarrow\mathbb{R} }} is a {{M|C^\infty}} ({{AKA}}: [[Smooth function]]).
** We define {{M|(f,U)\sim(g,V)}} if:
*** There exists an open set: {{M|W\subseteq U\cap V}} with {{M|p\in W}} such that:
**** {{M|f\big\vert_W\eq g\big\vert_W}}
* '''Germ: ''' The [[equivalence class]] of {{M|(f,U)}} is called the '''Germ''' of {{M|f}} at {{M|p}}
* We write the set of all germs of {{M|C^\infty}} functions on {{M|\mathbb{R}^n}} at {{M|p}} as:
** {{M|C^\infty_p(\mathbb{R}^n)}} or simply {{M|C^\infty_p}} if the space is obvious.
=====Example=====
Define:
* {{MM|f:(\mathbb{R}-\{1\})\rightarrow\mathbb{R} }} by {{M|f:x\mapsto\dfrac{1}{1-x} }} and
* {{M|g:(0,1)\rightarrow\mathbb{R} }}<ref group="Note">{{M|(-1,1):\eq\{x\in\mathbb{R}\ \vert\ -1 < x < 1\} }} - an [[open interval]] of {{M|\mathbb{R} }}</ref> by {{MM|g:x\mapsto 1+\sum^\infty_{n\eq 1}x^n}}

Then {{M|f}} and {{M|g}} have the same germ at any point {{M|p\in (-1,1)}}
{{Explore topic|grade=B|msg=This is an important concept}}


Then we just hit the concept of an [[algebra]]
===Section 3: The exterior algebra of multicovectors===
====3.3: Multilinear functions====
* {{M|k}}-linear function. A [[multilinear]] function: {{M|f:V^k\rightarrow\mathbb{R} }}. 
* '''''Future: '' Permutation action: ''' Let {{M|f}} be {{M|k}}-linear and let {{M|\sigma\in S_k}} - the [[symmetric group]] on {{M|k}} symbols. Then:
** {{M|(\sigma f)(v_1,\ldots,v_k):\eq f(v_{\sigma(1)},\ldots,v_{\sigma(k)})}}
* '''Symmetric: ''' {{M|\forall\sigma\in S_k[\sigma f\eq f]}}
* '''Alternating: ''' {{M|\forall\sigma\in S_k[\sigma f\eq\text{Sign}(\sigma)f]}}
=====Notations=====
* {{M|L_k(V)}} - all {{M|k}}-linear functions
* {{M|A_k(V)}} - all ''alternating'' {{M|k}}-linear functions.
'''Lemma 3.11: '''
* If {{M|\sigma,\tau\in S_k}} and {{M|f}} is {{M|k}}-linear then:
** {{M|\tau(\sigma f)\eq(\tau\sigma)f}}
====3.5: The symmetrising and alternating operators====
Let {{M|f\in L_k(V)}}, then:
* {{MM|Sf:\eq \sum_{\sigma \in S_k} \sigma f}}
* {{MM|Af:\eq \sum_{\sigma \in S_k} \text{Sign}(\sigma)\sigma f}}

'''Lemma 3.14: '''
* If {{M|f\in L_k(V)}} is an alternating {{M|k}}-linear function already then:
** {{M|Af\eq (k!)f}}
====3.6: The tensor product====
Let {{M|f\in L_k(V)}} and {{M|g\in L_\ell(V)}}, then their ''tensor product'' is a {{M|(k+\ell)}}-linear function, {{M|f\otimes g}} defined as follows:
** {{M|(f\otimes g)(v_1,\ldots,v_{k+\ell}):\eq f(v_1,\ldots,v_k)g(v_{k+1},\ldots,v_{k+\ell})}}
====3.7: The wedge product====
Let {{M|f\in A_k(V)}} and {{M|g\in A_\ell(V)}}, the wedge product is a product that is alternating also:
* {{M|f\wedge g:\eq \frac{1}{k!\ell !}A(f\otimes g)}}, or explicitly:
* {{MM|f\wedge g(v_1,\ldots,v_{k+\ell})\eq\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots, v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})}}
This is obviously alternating.


Suppose that {{M|f(v_1,v_2)g(\text{whatever})}} is a term, then so is {{M|-f(v_2,v_1)g(\text{whatever})}} say too. 

Remember {{M|f}} is alternating by definition, that means:
* {{M|f(v_2,v_1)\eq -f(v_1,v_2)}}

So we really have {{M|2f(v_1,v_2)g(\text{whatever})}} in the term. There are a lot of redundancies.

'''Definition: '''
* A permutation, {{M|\sigma\in S_{k+\ell} }} is a {{M|(k,\ell)}}-shuffle if:
** {{M|\sigma(1)<\sigma(2)<\cdots<\sigma(k-1)<\sigma(k)}} and {{M|\sigma(k+1)<\sigma(k+2)<\cdots<\sigma(k+\ell-1)<\sigma(k+\ell)}}

Now we may re-write {{M|f\wedge g}} as:
* {{MM|(f\wedge g)\eq\sum_{\sigma\ :\ (k,\ell)\text{-shuffle} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots,v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})}}

{{Caveat|All of this is VERY informal... there needs to be proof.... but I'll go along}}
===Section 4: Differential Forms on {{M|\mathbb{R}^n}}===
====4.1: Differential {{M|1}}-forms and the differential of a function====
* '''Cotangent space: ''' to {{M|\mathbb{R}^n}} at {{M|p\in\mathbb{R}^n}} is denoted by {{M|T^*_p(\mathbb{R}^n)}} is defined to be:
** {{M|(T_p(\mathbb{R}^n))^*}} - or {{M|(T_p(\mathbb{R}^n))^\vee}} in this author's notation - of the tangent space {{M|T_p(\mathbb{R}^n)}}

==Notes==
<references group="Note"/>
==Unsures==
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