1.Solve the equation $f(x,y,z):= x^{4}+2x\cos(y) + \sin (z) = 0$ in a neighbourhood of $p=(0,0,0)$ for z in the form : $z=g(x,y)$
2. Give the maximal domain for $g$ and the neighbourhood of $p$ and
3.calculate the differential of $g$. Check the connection between the theorem of implicit function for the differential of $g$ and the differential of $f$.
I solve like this:
- $z= \arcsin(-x^{4}-2x\cos(y)$ ,
we can see that the maximal domain must be : $Do_{x}=[-1,1]$ and $Do_{y}=[\frac{1}{2}(3\pi) , \frac{1}{2}(7\pi)]$ . How does one calculate the neighbourhood of $p$?
$\displaystyle Dg(x,y)= \begin{pmatrix} \frac{-2(2x^{3}+\cos(y))}{(1-(x^{4}+2x\cos(y))^{2})^{1/2}} & \frac{2x\sin(y)}{(1-(x^{4}+2x\cos(y))^{1/2}} \end{pmatrix}$
how can I show the connection, if put $Df(x,y,g(x,y))$ then i get $0$ and that is not even invertible.
Thanks for any efforts to explain.