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Suppose $F:C^1(\Omega, [0,T]) \to C^1(\Omega, [0,T])$ with $F(u) = u_t - f(x, t, u, u_x).$

How do I calculate the Frechet derivative of $F$ at the point $w = f(x,t, 0, 0)t$?

It should be $F'(w, v) = v_t - \frac{\partial f}{\partial z}\bigg|_{w}v - \frac{\partial f}{\partial p}\bigg|_{w}v_x$ apparently.

Maybe another day I can do this but forming the difference and then considering another difference to get out the partial derivative is confusing me!

Thanks

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    Thanks @Thomas. I was a bit confused about how to do it formally (as in satisfying the definition on the wikipedia article) but I'll have another go.2012-06-15

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If you just want the result and already do know that the functional is Frechet differentiable (or just don't care, which is not a good attitude ;-) you only have to calculate the directional derivative $\frac{d}{ds}|_{s=0}F(w+sv) = \frac{d}{ds}|_{s=0}\left\{(w+sv)_t - f(x,t,(w+sv), (w+sv)_x)\right\} = v_t - f_zv -f_p v_x$ (assuming $f=f(x,t,z,p)$). This reduces the question to a one dimensional differentiation task.

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    Thanks. (for some reason I kept thinking you said small $f$ in the comment above.) I see you've assumed $F$ is Frechet differentiable and hence Gateaux differentiable and calculated the directional derivative. But if you didn't know if it was Frechet differentiable how would you do it? (I do care!)2012-06-15