Kedjeregeln av functionals

This excercise named (b) i need help with.

I have tried and beginned with calculating deltaK(y,n). The uppermost picture. The description is picture 2 and exercise is picture 3.

How to do it?

You are probably expected to start from the general definition of the Gateaux variation (the one with $δ$ F) rather than the ”particular case” δJ.

Start with

$\frac{d}{d ε} G (K (y + ε η), J (y + ε η)) |_{ε = 0}$

and use ordinary chain rule from two-variable calculus.

OK I got K and J in the wrong order.

Svara