Changes

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.

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 <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:

Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\operatorname{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:

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

: <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}A_i</math> is a set of two differential propositions, <math>\operatorname{d}A_i = \{(\operatorname{d}a_i), \operatorname{d}a_i\},</math> where <math>(\operatorname{d}a_i)</math> is a proposition with the logical value of "<math>\mbox{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>\{(\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}.

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