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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 16th 2012
   • (edited Oct 16th 2012)
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   Author: Urs
   Format: MarkdownItexat _[[variational calculus]]_ I have started a section _[In terms of smooth spaces](http://ncatlab.org/nlab/show/variational+calculus#InTermsOfSmoothSpaces)_ where I discuss a bit how for $$ S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} $$ a smooth "functional", namely a smooth map of [[smooth spaces]], its "functional derivative" is simply the plain de Rham differential of smooth functions on smooth spaces $$ \mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,. $$ The notation can still be optimized. But I am running out of energy now. Has been a long day.

   at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for

   $$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}$$

   a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces

   $$\mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$$

   The notation can still be optimized. But I am running out of energy now. Has been a long day.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2012
   • (edited Oct 17th 2012)
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   Author: Urs
   Format: MarkdownItexI made explicit at _[[variational calculus]]_ the "mapping space with non-varying boundary configurations", on which the variational caclulus is defined, as the pullback $$ \array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,, $$

   I made explicit at variational calculus the “mapping space with non-varying boundary configurations”, on which the variational caclulus is defined, as the pullback

   $$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,$$
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2012
   • (edited Oct 17th 2012)
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   Author: Urs
   Format: MarkdownItexHm, I haven't thought about this enough. This means for instance that for every $G$-principal bundle on the boundary configuration whose pullback to the bulk configuration space is equipped with a trivialization, there is a canonical flat $\mathfrak{g}$-valued differential form $$ \omega \colon [\Sigma, X]_{\partial \Sigma} \to \flat_{dR}\mathbf{B}G $$ on the "configuration space with non-varying boundary configurations", induced from the commuting diagram $$ \array{ \flat [\partial \Sigma, X] &\stackrel{\flat \mathbf{c}}{\to}& \flat \mathbf{B}G \\ \downarrow && \downarrow \\ [\partial \Sigma,X] &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G \\ \uparrow && \uparrow \\ [\Sigma, X] &\to& * } $$ under the equivalence $\Omega^1_{flat}(-,\mathfrak{g}) \simeq \flat_{dR}\mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G$. Hmm...

   Hm, I haven’t thought about this enough. This means for instance that for every $G$-principal bundle on the boundary configuration whose pullback to the bulk configuration space is equipped with a trivialization, there is a canonical flat $\mathfrak{g}$-valued differential form

   $$\omega \colon [\Sigma, X]_{\partial \Sigma} \to \flat_{dR}\mathbf{B}G$$

   on the “configuration space with non-varying boundary configurations”, induced from the commuting diagram

   $$\array{ \flat [\partial \Sigma, X] &\stackrel{\flat \mathbf{c}}{\to}& \flat \mathbf{B}G \\ \downarrow && \downarrow \\ [\partial \Sigma,X] &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G \\ \uparrow && \uparrow \\ [\Sigma, X] &\to& * }$$

   under the equivalence $\Omega^1_{flat}(-,\mathfrak{g}) \simeq \flat_{dR}\mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G$.

   Hmm…