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Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems (view source)
Revision as of 14:38, 22 January 2008
, 14:38, 22 January 2008→Logical Tangent Vectors: markup
Discuss variation in portrayal of ''v'' in d''f''(''u'', ''v''):
Discuss variation in portrayal of ''v'' in d''f''(''u'', ''v''):
:{| cellpadding="4"
| 1.
| as ordinary vector in second component of product space '''B'''<sup>''n''</sup> × '''B'''<sup>''n''</sup>,
|-
| 2.
| as tangent vector map : ('''B'''<sup>''n''</sup> → '''B''') → '''B''', dual to '''B'''<sup>''n''</sup> ?
|-
| 3.
| as tangent vector map : ('''D'''<sup>''n''</sup> → '''B''') → '''B''', dual to '''D'''<sup>''n''</sup> ?
|}
Discuss differential as map : T(''U'') = ''U''T → '''B'''.
Discuss differential as map : T(''U'') = ''U''T → '''B'''.
It helps to introduce some notation:
It helps to introduce some notation:
:{| cellpadding="4"
| Let
| '''R'''
| =
| {real values}
|
|-
| Let
| '''B'''
| =
| {boolean values}
| = {0, 1} = {false, true}.
|-
| Let
| ''X''
| =
| '''R'''<sup>''n''</sup>,
| ''f'' : '''R'''<sup>''n''</sup> → '''R'''.
|-
| Let
| ''U''
| =
| '''B'''<sup>''n''</sup>,
| ''p'' : '''B'''<sup>''n''</sup> → '''B'''.
|}
In these terms, analogies of the following form are being explored:
In these terms, analogies of the following form are being explored: