User:Jon Awbrey/APPENDICES
Contents
 1 Appendices (Version 1)
 2 Appendices (Version 2)
 2.1 Appendix 1
 2.1.1 Table A1. Propositional Forms on Two Variables
 2.1.2 Table A2. Propositional Forms on Two Variables
 2.1.3 Table A3. Ef Expanded Over Differential Features
 2.1.4 Table A4. Df Expanded Over Differential Features
 2.1.5 Table A5. Ef Expanded Over Ordinary Features
 2.1.6 Table A6. Df Expanded Over Ordinary Features
 2.2 Appendix 2
 2.3 Appendix 3
 2.4 Appendix 4
 2.1 Appendix 1
 3 Appendices (Version 3)
 3.1 Appendix 1. Propositional Forms and Differential Expansions
 3.1.1 Table A1. Propositional Forms on Two Variables
 3.1.2 Table A2. Propositional Forms on Two Variables
 3.1.3 Table A3. Ef Expanded Over Differential Features
 3.1.4 Table A4. Df Expanded Over Differential Features
 3.1.5 Table A5. Ef Expanded Over Ordinary Features
 3.1.6 Table A6. Df Expanded Over Ordinary Features
 3.2 Appendix 2. Differential Forms
 3.2.1 Table A7. Differential Forms Expanded on a Logical Basis
 3.2.2 Table A8. Differential Forms Expanded on an Algebraic Basis
 3.2.3 Table A9. Tangent Proposition as Pointwise Linear Approximation
 3.2.4 Table A10. Taylor Series Expansion Df = df + d^{2}f
 3.2.5 Table A11. Partial Differentials and Relative Differentials
 3.2.6 Table A12. Detail of Calculation for the Difference Map
 3.3 Appendix 3. Computational Details
 3.4 Appendix 4. Source Materials
 3.5 Appendix 5. Various Definitions of the Tangent Vector
 3.1 Appendix 1. Propositional Forms and Differential Expansions
Appendices (Version 1)
Appendix 1A. Operator Maps for the Disjunction “f”
Table A1. Computation of “εf”
Table A2. Computation of “Ef”
Table A3. Computation of “Df” (1)
Table A4. Computation of “Df” (2)
Table A5. Computation of “df”
Table A6. Computation of “rf”
Table A7. Computation Summary for Disjunction

Appendix 1B. Operator Maps for the Equality “g”
Table B1. Computation of “εg”
Table B2. Computation of “Eg”
Table B3. Computation of “Dg” (1)
Table B4. Computation of “Dg” (2)
Table B5. Computation of “dg”
Table B6. Computation of “rg”
Table B7. Computation Summary for Equality

Appendix 2
Table C9. Tangent Proposition as Pointwise Linear Approximation



















Table C10. Taylor Series Expansion Df = df + d^{2}f












Table C11. Partial Differentials and Relative Differentials





































Table C12. Detail of Calculation for Df = Ef + f



 




 




 




 




 




 
Appendix 3
Appendix 4
Appendices (Version 2)
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Appendix 1
Table A1. Propositional Forms on Two Variables








 






Table A2. Propositional Forms on Two Variables



0 0 0 0 






 





 





 





 





 
1 1 1 1 
Table A3. Ef Expanded Over Differential Features



 





 





 





 





 





 
Fixed Point Total : 
Table A4. Df Expanded Over Differential Features





 





 





 





 





 
Table A5. Ef Expanded Over Ordinary Features





 





 





 





 





 
Table A6. Df Expanded Over Ordinary Features





 





 





 





 





 
Appendix 2
Differential Forms
The actions of the difference operator and the tangent operator on the 16 bivariate propositions are shown in Tables A7 and A8.
Table A7 expands the differential forms that result over a logical basis:

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cellbasis, pointbasis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:
and 
Table A8 expands the differential forms that result over an algebraic basis:
This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.
Table A7. Differential Forms Expanded on a Logical Basis




















Table A8. Differential Forms Expanded on an Algebraic Basis




















Appendix 3
Table A9. Tangent Proposition as Pointwise Linear Approximation



















Table A10. Taylor Series Expansion Df = df + d^{2}f








 





 





 





 





 
Table A11. Partial Differentials and Relative Differentials





































Appendix 4
Table A12. Detail of Calculation for the Difference Map



 
































































































Appendices (Version 3)
Appendix 1. Propositional Forms and Differential Expansions
Table A1. Propositional Forms on Two Variables












Table A2. Propositional Forms on Two Variables






























Table A3. Ef Expanded Over Differential Features



 






























Table A4. Df Expanded Over Differential Features



 






























Table A5. Ef Expanded Over Ordinary Features



 






























Table A6. Df Expanded Over Ordinary Features



 




 




 




 




 




 
Appendix 2. Differential Forms
The actions of the difference operator and the tangent operator on the 16 bivariate propositions are shown in Tables A7 and A8.
Table A7 expands the differential forms that result over a logical basis:

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cellbasis, pointbasis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:
and 
Table A8 expands the differential forms that result over an algebraic basis:
This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.
Table A7. Differential Forms Expanded on a Logical Basis


 


 


 


 


 
Table A8. Differential Forms Expanded on an Algebraic Basis


 





 


 


 
Table A9. Tangent Proposition as Pointwise Linear Approximation



















Table A10. Taylor Series Expansion Df = df + d^{2}f






 




 




 




 




 
Table A11. Partial Differentials and Relative Differentials


















Table A12. Detail of Calculation for the Difference Map



 




 




 




 




 




 




 




 




 




 




 




 




 




 




 
Appendix 3. Computational Details
Operator Maps for the Logical Conjunction f_{8}(u, v)
Computation of εf_{8}

Computation of Ef_{8}


Computation of Df_{8}



Computation of df_{8}

Computation of rf_{8}

Computation Summary for Conjunction

Operator Maps for the Logical Equality f_{9}(u, v)
Computation of εf_{9}

Computation of Ef_{9}

Computation of Df_{9}


Computation of df_{9}

Computation of rf_{9}

Computation Summary for Equality

Operator Maps for the Logical Disjunction f_{14}(u, v)
Computation of εf_{14}

Computation of Ef_{14}

Computation of Df_{14}


Computation of df_{14}

Computation of rf_{14}

Computation Summary for Disjunction
