Changes

In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.

In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.

Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A} = \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{a_1, \ldots, a_n\},</math> that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math>  (Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)

Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A}</math> <math>=\!</math> <math>\{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}</math> <math>=\!</math> <math>\{a_1, \ldots, a_n\},</math> that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math>  (Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)

A ''tangent space'' to <math>A\!</math> at a point <math>x,\!</math> written <math>\operatorname{T}_x(A),</math> takes the form <math>\operatorname{d}A</math> <math>=\!</math> <math>\langle \operatorname{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math>  Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

A ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> frequently denoted <math>\operatorname{T}_x(A),</math> takes the form <math>\operatorname{d}A</math> <math>=\!</math> <math>\langle \operatorname{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math>  Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

Proceeding as we did before with the base space ''A'', we can analyze the individual tangent space at a point of ''A'' as a product of distinct and independent factors:

Proceeding as we did before with the base space ''A'', we can analyze the individual tangent space at a point of ''A'' as a product of distinct and independent factors: