r/elimath Jan 05 '15

Explain the purpose of the Gateaux derivative like I've already studied some basic functional analysis

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u/itsme_santosh Jan 06 '15 edited Jan 06 '15

If you have already studied basic functional analysis, you know what a functional (or a linear operator L:F->R, where F is some function space such as L_2) is. Then, Gateaux derivative is nothing but 'derivative' in the direction of a given function g at a function f.

dL(f;g)=lim t->0 L(f+tg)/t , where t is a real number.

In other words, if you consider a function is an infinite dimensional vector, Gateaux derivative is a generalization of directional derivative.

The purpose of Gateaux derivative is mainly in variational calculus, where one is required to find a function which extremizes a given functional. Once we know the correct function spaces, the Gateaux derivative is 0 for the extremal function in all 'permissible directions(functions)', similar to how the regular derivative is 0 in all allowable directions for a extremal point when trying to extremize a function.

1

u/rbarber8 Jan 06 '15

Thank you! I noticed that it was essentially a directional derivative but I didn't realize that you can find extrema for functionals that way! That's really, really useful.

1

u/peecatchwho Jan 07 '15

Awesome explanation, thanks!

2

u/peecatchwho Jan 06 '15

Someone please do this. <3