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Mini case an Dierentiable Manifold & brass manias ÷ pt . Mathemhh notebook an " is views f the hyprspke " ÷ smarmily : The spare of k - dim 't liner shspmes of IR " 1 or a ) is called the Gasman in f k - plans in R " lei ) & dented GEHR " ) tr 6th , D or Gln , e) or ... ) frqmmnins shop in : . ttgdm geometry - poet Witf first as Rlmd in conty problems , so . . . . Combinator icsnsesp . Schubert Globo , uhh us developed to aow posts He " How may Has in 3 - spare ktsut 4 glen lines ? " . Di erent geometry - door easels f tongues spas , uhih are gnolnb f Lie gaps , ad f Spas w/ posilm Ricci curate . Applied meth - . sign Attn . blind Sore sgnalm . opfmzath ( liner & qndhlnostninb pt yuan a bass mmnin ... e.g. , symmemezewknpoh . ) . clued random way & polymer moody . etc . Gyi . GHRD B the spare f lnythfhtheo Dm in R ? . . so a gy f RP ¥5 . 6,1N ) is Mph & me gulf G , ( R " ) = RIP " . Gz 1123=6 , R 3 Also , on thik f 6 dR3 as the space f all Imes in Mph . 62112 " to at to be a gyf 5×5 ? . 6 , ¢ ' = ep ' I 5 goal is to build up smef the they needed to talk abut these spaces

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Page 1: Mini Di Manifold - Colorado State Universityclayton/teaching/DiffG... · 2015-04-08 · Mini case an Differentiable Manifold& brass manias pt. Mathemhh notebook an " is views fthe

Mini case an Differentiable Manifold & brass manias

÷pt .

Mathemhh notebookan

"

is views f the hyprspke"

÷smarmily :

Thespare of k - dim 't liner shspmes of IR

" 1 or a ) is called the Gasman in f k - plans in R"

lei ) & dented

GEHR ") tr 6th , D or Gln , e) or ... )

frqmmnins shop in : . ttgdm geometry - poet Witf first as Rlmd in conty problems,

so . . .

. Combinator icsnsesp .

Schubert Globo,uhh us developedto aow posts He " How may

Has in

3 -

sparektsut 4 glen lines ?

"

. Different geometry- door easels f tongues spas ,

uhih are gnolnb f Lie gaps ,ad f Spas

w/ posilm Ricci curate

. Applied meth - . sign Attn. blind Sore sgnalm

. opfmzath ( liner & qndhlnostninb pt yuan a bass mmnin ... e.g. , symmemezewknpoh .)

.clued random way & polymer moody

.etc

.

Gyi . GHRD B the spare f lnythfhtheo Dm in R?. . so a gy f RP ¥5

. 6,1N) is Mph & me gulf G, ( R

" ) = RIP "

. Gz 1123=6 ,R 3

Also,

on thik f 6 dR3 as the space f all Imes in Mph

. 62112" to at to be a gyf 5×5 ?

. 6 ,¢

'

= ep'

I 5

←goal is to buildup smef the they needed to talk abut these spaces

Page 2: Mini Di Manifold - Colorado State Universityclayton/teaching/DiffG... · 2015-04-08 · Mini case an Differentiable Manifold& brass manias pt. Mathemhh notebook an " is views fthe

Df: A diftemtstkmnifld of dims inn is a Hansdorft,send onttde tpdogil space M together wl a family f inject 't

meps Ha : Ua ER"

→ M of gusts Us c- IR"

st. :

a Y UalUD = M

@ Far my xp st. UalUH n Ypl Up) = Wt 0 ,

the sb Uj'

lw) & Ypilw) are gusts in Rn & the

mps yj'

olla & Uj'

olfs are smooth 1 8).

𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀌𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀧𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀻𐀼𐀽𐀾𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁎𐁏𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝𐁞𐁟𐁠𐁡𐁢𐁣𐁤𐁥𐁦𐁧𐁨𐁩𐁪𐁫𐁬𐁭𐁮𐁯𐁰𐁱𐁲𐁳𐁴𐁵𐁶𐁷𐁸𐁹𐁺𐁻𐁼𐁽𐁾𐁿3𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀌𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀧𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀻𐀼𐀽𐀾𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁎𐁏𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝𐁞𐁟𐁠𐁡𐁢𐁣𐁤𐁥𐁦𐁧𐁨𐁩𐁪𐁫𐁬𐁭𐁮𐁯𐁰𐁱𐁲𐁳𐁴𐁵𐁶𐁷𐁸𐁹𐁺𐁻𐁼𐁽𐁾𐁿 the family { 1Ua,Ua)} is mnxinl w

.H .

𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀌𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀧𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀻𐀼𐀽𐀾𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁎𐁏𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝𐁞𐁟𐁠𐁡𐁢𐁣𐁤𐁥𐁦𐁧𐁨𐁩𐁪𐁫𐁬𐁭𐁮𐁯𐁰𐁱𐁲𐁳𐁴𐁵𐁶𐁷𐁸𐁹𐁺𐁻𐁼𐁽𐁾𐁿1𐀀𐀁𐀂𐀃𐀄𐀅𐀆𐀇𐀈𐀉𐀊𐀋𐀌𐀍𐀎𐀏𐀐𐀑𐀒𐀓𐀔𐀕𐀖𐀗𐀘𐀙𐀚𐀛𐀜𐀝𐀞𐀟𐀠𐀡𐀢𐀣𐀤𐀥𐀦𐀧𐀨𐀩𐀪𐀫𐀬𐀭𐀮𐀯𐀰𐀱𐀲𐀳𐀴𐀵𐀶𐀷𐀸𐀹𐀺𐀻𐀼𐀽𐀾𐀿𐁀𐁁𐁂𐁃𐁄𐁅𐁆𐁇𐁈𐁉𐁊𐁋𐁌𐁍𐁎𐁏𐁐𐁑𐁒𐁓𐁔𐁕𐁖𐁗𐁘𐁙𐁚𐁛𐁜𐁝𐁞𐁟𐁠𐁡𐁢𐁣𐁤𐁥𐁦𐁧𐁨𐁩𐁪𐁫𐁬𐁭𐁮𐁯𐁰𐁱𐁲𐁳𐁴𐁵𐁶𐁷𐁸𐁹𐁺𐁻𐁼𐁽𐁾𐁿 &@.

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