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differential propositional calculus : appendix 4

(Application)

Contents

0.1 Detail of Calculation for the Difference Map

0.1 Detail of Calculation for the Difference Map

Detail of Calculation for D f = E f + f

	E f\|_{d x d y}
+	f\|_{d x d y}
=	D f\|_{d x d y}

	E f\|_{d x (d y)}
+	f\|_{d x (d y)}
=	D f\|_{d x (d y)}

	E f\|_{(d x) d y}
+	f\|_{(d x) d y}
=	D f\|_{(d x) d y}

	E f\|_{(d x)(d y)}
+	f\|_{(d x)(d y)}
=	D f\|_{(d x)(d y)}

f₀

0 + 0 = 0

0 + 0 = 0

0 + 0 = 0

0 + 0 = 0

f₁

	x y	d x	d y
+	(x)(y)	d x	d y
=	((x,y))	d x	d y

	x (y)	d x	(d y)
+	(x)(y)	d x	(d y)
=	(y)	d x	(d y)

	(x) y	(d x)	d y
+	(x)(y)	(d x)	d y
=	(x)	(d x)	d y

	(x)(y)	(d x)	(d y)
+	(x)(y)	(d x)	(d y)
=	0	(d x)	(d y)

f₂

	x (y)	d x	d y
+	(x) y	d x	d y
=	(x,y)	d x	d y

	x y	d x	(d y)
+	(x) y	d x	(d y)
=	y	d x	(d y)

	(x)(y)	(d x)	d y
+	(x) y	(d x)	d y
=	(x)	(d x)	d y

	(x) y	(d x)	(d y)
+	(x) y	(d x)	(d y)
=	0	(d x)	(d y)

f₄

	(x) y	d x	d y
+	x (y)	d x	d y
=	(x,y)	d x	d y

	(x)(y)	d x	(d y)
+	x (y)	d x	(d y)
=	(y)	d x	(d y)

	x y	(d x)	d y
+	x (y)	(d x)	d y
=	x	(d x)	d y

	x (y)	(d x)	(d y)
+	x (y)	(d x)	(d y)
=	0	(d x)	(d y)

f₈

	(x)(y)	d x	d y
+	x y	d x	d y
=	((x,y))	d x	d y

	(x) y	d x	(d y)
+	x y	d x	(d y)
=	y	d x	(d y)

	x (y)	(d x)	d y
+	x y	(d x)	d y
=	x	(d x)	d y

	x y	(d x)	(d y)
+	x y	(d x)	(d y)
=	0	(d x)	(d y)

f₃

	x	d x	d y
+	(x)	d x	d y
=	1	d x	d y

	x	d x	(d y)
+	(x)	d x	(d y)
=	1	d x	(d y)

	(x)	(d x)	d y
+	(x)	(d x)	d y
=	0	(d x)	d y

	(x)	(d x)	(d y)
+	(x)	(d x)	(d y)
=	0	(d x)	(d y)

f₁₂

	(x)	d x	d y
+	x	d x	d y
=	1	d x	d y

	(x)	d x	(d y)
+	x	d x	(d y)
=	1	d x	(d y)

	x	(d x)	d y
+	x	(d x)	d y
=	0	(d x)	d y

	x	(d x)	(d y)
+	x	(d x)	(d y)
=	0	(d x)	(d y)

f₆

	(x,y)	d x	d y
+	(x,y)	d x	d y
=	0	d x	d y

	((x,y))	d x	(d y)
+	(x,y)	d x	(d y)
=	1	d x	(d y)

	((x,y))	(d x)	d y
+	(x,y)	(d x)	d y
=	1	(d x)	d y

	(x,y)	(d x)	(d y)
+	(x,y)	(d x)	(d y)
=	0	(d x)	(d y)

f₉

	((x,y))	d x	d y
+	((x,y))	d x	d y
=	0	d x	d y

	(x,y)	d x	(d y)
+	((x,y))	d x	(d y)
=	1	d x	(d y)

	(x,y)	(d x)	d y
+	((x,y))	(d x)	d y
=	1	(d x)	d y

	((x,y))	(d x)	(d y)
+	((x,y))	(d x)	(d y)
=	0	(d x)	(d y)

f₅

	y	d x	d y
+	(y)	d x	d y
=	1	d x	d y

	(y)	d x	(d y)
+	(y)	d x	(d y)
=	0	d x	(d y)

	y	(d x)	d y
+	(y)	(d x)	d y
=	1	(d x)	d y

	(y)	(d x)	(d y)
+	(y)	(d x)	(d y)
=	0	(d x)	(d y)

f₁₀

	(y)	d x	d y
+	y	d x	d y
=	1	d x	d y

	y	d x	(d y)
+	y	d x	(d y)
=	0	d x	(d y)

	(y)	(d x)	d y
+	y	(d x)	d y
=	1	(d x)	d y

	y	(d x)	(d y)
+	y	(d x)	(d y)
=	0	(d x)	(d y)

f₇

	((x)(y))	d x	d y
+	(x y)	d x	d y
=	((x,y))	d x	d y

	((x) y)	d x	(d y)
+	(x y)	d x	(d y)
=	y	d x	(d y)

	(x (y))	(d x)	d y
+	(x y)	(d x)	d y
=	x	(d x)	d y

	(x y)	(d x)	(d y)
+	(x y)	(d x)	(d y)
=	0	(d x)	(d y)

f₁₁

	((x) y)	d x	d y
+	(x (y))	d x	d y
=	(x,y)	d x	d y

	((x)(y))	d x	(d y)
+	(x (y))	d x	(d y)
=	(y)	d x	(d y)

	(x y)	(d x)	d y
+	(x (y))	(d x)	d y
=	x	(d x)	d y

	(x (y))	(d x)	(d y)
+	(x (y))	(d x)	(d y)
=	0	(d x)	(d y)

f₁₃

	(x (y))	d x	d y
+	((x) y)	d x	d y
=	(x,y)	d x	d y

	(x y)	d x	(d y)
+	((x) y)	d x	(d y)
=	y	d x	(d y)

	((x)(y))	(d x)	d y
+	((x) y)	(d x)	d y
=	(x)	(d x)	d y

	((x) y)	(d x)	(d y)
+	((x) y)	(d x)	(d y)
=	0	(d x)	(d y)

f₁₄

	(x y)	d x	d y
+	((x)(y))	d x	d y
=	((x,y))	d x	d y

	(x (y))	d x	(d y)
+	((x)(y))	d x	(d y)
=	(y)	d x	(d y)

	((x) y)	(d x)	d y
+	((x)(y))	(d x)	d y
=	(x)	(d x)	d y

	((x)(y))	(d x)	(d y)
+	((x)(y))	(d x)	(d y)
=	0	(d x)	(d y)

f₁₅

1 + 1 = 0

1 + 1 = 0

1 + 1 = 0

1 + 1 = 0

"differential propositional calculus : appendix 4" is owned by Jon Awbrey.
(view preamble)

See Also: differential logic, minimal negation operator

This object's parent.

This is version 1 of differential propositional calculus : appendix 4, born on 2009-05-26.
Object id is 781, canonical name is DifferentialPropositionalCalculusAppendix4.
Accessed 2099 times total.

Classification:

Physics Classification:	02. (Mathematical methods in physics)
	02.10.Ab (Logic and set theory)
	02.10.Ox (Combinatorics; graph theory)
	02.10.Ud (Linear algebra)
	02.20.-a (Group theory )
	02.30.-f (Function theory, analysis)
	02.40.-k (Geometry, differential geometry, and topology )
	02.40.Yy (Geometric mechanics )
	02.50.Tt (Inference methods)
	02.70.-c (Computational techniques )
	02.70.Bf (Finite-difference methods)
	02.70.Wz (Symbolic computation )

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