Variational Calculus help: minimising a function

[pamparana]

Hello everyone,

Edit: title should read gradient of a functional. Oh well...

I need a bit of help regarding variational calculus. I am not really very good at advanced calculus and only took basic calculus in high school and am having a bit of difficulty with a particular problem.

I have a function which is as follows:

$$X(\Psi) = \int(T-S(U(\nabla(\Psi)))^{2}$$

I am, of course, ultimately working in discrete space. Here T is an image and $$S(U(\nabla(\Psi)))$$ is also an image where $$U(\nabla(\Psi))$$ represents a transformation or mapping between different coordinates (here $$\nabla$$ denotes the spatial gradient of the variable $$\Psi$$). Now what I need to do is calculate the analytical derivative of this function wrt to $$\Psi$$.

I am guessing the traditional definition of the limit can be extended to functions like this. Now, I am quite stuck and have no idea how to make any further progress.

$$\lim_{h \to 0} \frac{X(\Psi+h)-X(\Psi)}{h}$$

I am guessing the next step would be to substitute but really could not see any way to make further progress. I must admit my calculus is very rusty and I have been stumped with this for days now. I would really appreciate any help anyone can give me on this.

Happy 2011 ahead to everyone.

Luca

 

[Sourabh N]

Try Taylor series.

 

[pamparana]

I am pretty sure that there is a better analytical way to do this than using Taylor series and finite methods and using numerical methods at the very last stage when I discretise the whole thing...

I think I might have made a mistake in the variational formulation as well...Will edit the post once I figure it out...