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## Main Question or Discussion Point

Hi PF,

I am currently trying to teach myself the rudiments of differential forms, in particular their application to physics, and there's something I'd like to ask.

It seems like diff forms can be used to express all kinds of physics, but the area I haven't been able to figure out is stuff related to functional analysis. Basically, what I want to know is, can we make sense of something like [itex]\delta F / \delta \alpha[/itex] where F is a functional of the differential form alpha, i.e.[itex]F[\alpha]=\int f(\alpha)[/itex]. I've been trying to use the Gateaux derivative, something like [itex]\frac{d}{dt}F[\alpha+t\tau]|_{t=0}[/itex] where tau is a test form (of the same degree as alpha). I think this would be related to the functional derivative by [tex]\int \left(\frac{\delta F}{\delta \alpha}\right)\wedge\tau = \frac{d}{dt}F[\alpha+t\tau]\Big |_{t=0}[/tex],

which seems to nearly work, but not quite.

Now is probably a good time for an example. I've been mainly trying to apply this to electromagnetism. Using diff forms we can express Maxwell's equations as dF=0 and d*F=J, where F is the curvature of something, and F=dA. The action can be written:

[tex]S[A] = \int F \wedge \star F +A \wedge J[/tex]

Since we already know F is exact (because of the Bianchi identity), we just need to derive d*F=J using the principle of least action ([itex]\delta S / \delta A = 0[/itex]). Using the Gateaux derivative I get as far as

[tex]\frac{d}{dt}S[A+t\tau]\Big |_{t=0} = \frac{d}{dt}\left(\int (F+td\tau)\wedge(\star F+t \star d\tau) + (A+t\tau)\wedge J \right)\Big |_{t=0}[/tex]

[tex]=\int F \wedge \star d\tau + d\tau \wedge \star F +\tau \wedge J[/tex]

[tex]=\int F \wedge \star d\tau - \tau \wedge d\star F +\tau \wedge J[/tex]

if we assume boundary terms vanish. The only problem is that first term, I don't know how to get rid of it. So... what's going on? Am I even allowed to use the Gateaux derivative like that? I feel like I'm on the right track, but it would be great to hear from someone else, I need a nudge in the right direction (I'm mainly just learning from wikipedia and stuff). Maybe there's a reason not to send *F to *(F+t dtau)? That would be nice!

Thanks :)

I am currently trying to teach myself the rudiments of differential forms, in particular their application to physics, and there's something I'd like to ask.

It seems like diff forms can be used to express all kinds of physics, but the area I haven't been able to figure out is stuff related to functional analysis. Basically, what I want to know is, can we make sense of something like [itex]\delta F / \delta \alpha[/itex] where F is a functional of the differential form alpha, i.e.[itex]F[\alpha]=\int f(\alpha)[/itex]. I've been trying to use the Gateaux derivative, something like [itex]\frac{d}{dt}F[\alpha+t\tau]|_{t=0}[/itex] where tau is a test form (of the same degree as alpha). I think this would be related to the functional derivative by [tex]\int \left(\frac{\delta F}{\delta \alpha}\right)\wedge\tau = \frac{d}{dt}F[\alpha+t\tau]\Big |_{t=0}[/tex],

which seems to nearly work, but not quite.

Now is probably a good time for an example. I've been mainly trying to apply this to electromagnetism. Using diff forms we can express Maxwell's equations as dF=0 and d*F=J, where F is the curvature of something, and F=dA. The action can be written:

[tex]S[A] = \int F \wedge \star F +A \wedge J[/tex]

Since we already know F is exact (because of the Bianchi identity), we just need to derive d*F=J using the principle of least action ([itex]\delta S / \delta A = 0[/itex]). Using the Gateaux derivative I get as far as

[tex]\frac{d}{dt}S[A+t\tau]\Big |_{t=0} = \frac{d}{dt}\left(\int (F+td\tau)\wedge(\star F+t \star d\tau) + (A+t\tau)\wedge J \right)\Big |_{t=0}[/tex]

[tex]=\int F \wedge \star d\tau + d\tau \wedge \star F +\tau \wedge J[/tex]

[tex]=\int F \wedge \star d\tau - \tau \wedge d\star F +\tau \wedge J[/tex]

if we assume boundary terms vanish. The only problem is that first term, I don't know how to get rid of it. So... what's going on? Am I even allowed to use the Gateaux derivative like that? I feel like I'm on the right track, but it would be great to hear from someone else, I need a nudge in the right direction (I'm mainly just learning from wikipedia and stuff). Maybe there's a reason not to send *F to *(F+t dtau)? That would be nice!

Thanks :)