Changes

In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \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.

In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \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 <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:

: d''A'' = &prod;_''i'' d''A''_''i'' = d''A''₁ &times; &hellip; &times; d''A''_''n''.

: <math>\operatorname{d}A = \prod_{i=1}^n \operatorname{d}A_i = \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>

Here, d<font face="lucida calligraphy">A</font>_''i'' is an alphabet of two symbols, d<font face="lucida calligraphy">A</font>_''i''&nbsp;=&nbsp;{(d''a''_''i''),&nbsp;d''a''_''i''}, where (d''a''_''i'') is a symbol with the logical value of "not d''a''_''i''".  Each component d''A''_''i'' has the type '''B''', under the correspondence {(d''a''_''i''),&nbsp;d''a''_''i''} <math>\cong</math> {0,&nbsp;1}.  However, clarity is often served by acknowledging this differential usage with a superficially distinct type '''D''', whose intension may be indicated as follows:

Here, <math>\operatorname{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\operatorname{d}\mathcal{A}_i = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \},</math> where <math>\overline{\operatorname{d}a_i}</math> is a symbol with the logical value of <math>\operatorname{not}\ \operatorname{d}a_i.</math>   Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} \cong \{ 0, 1 \}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:

: '''D''' = {(d''a''_''i''),&nbsp;d''a''_''i''} = {same,&nbsp;different} = {stay,&nbsp;change} = {stop,&nbsp;step}.

: <math>\mathbb{D} = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} = \{ \operatorname{same}, \operatorname{different} \} = \{ \operatorname{stay}, \operatorname{change} \} = \{ \operatorname{stop}, \operatorname{step} \}.</math>

Viewed within a coordinate representation, spaces of type '''B'''^''n'' and '''D'''^''n'' may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

'''An Interlude on the Path'''

'''An Interlude on the Path'''