This is homework. I just finished a question regarding double integration over the unit sphere involving pullbacks of differential forms to provide context (course is advanced Calculus).
The question is given as:
$$f:D->D, D:= {(x,y): x^2+y+^2<_1}$$
$$g(A)=A+a(A)(A-f(A))$$
My task is to find a(A) algebraically. I just am not quite sure where to even start off. At first I thought there might be a theorem in my book involving a a version of stoke's or green's theorem to apply but the rather specific D has me thinking otherwise. I mean sure I could algebraically rearrange g to get a(A) but that seems rather.. superficial and lacking of meaning...
Edit: i should note that $g(a):D->S1$(Unit circle)
Edit: Important details I left out, probably just wasn't thinking it thoroughly. f is C1 and f(A)!= A for each A element D.
Hint: as $$g(A)-A=a(A)(f(A)−A),$$ taking norms: $$\|g(A)-A\|=|a(A)|\|(f(A)−A)\|$$ and $$|a(A)|=\cdots$$ And the sign of $|a(A)|$ will be...
Also remember that $\|g(A)\|=1$.