Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / F (Multivariable Calculus)

The inverse function theorem says that a continuously differentiable map on an open set, with invertible derivative at a point, restricts to a $C^1$ diffeomorphism between neighborhoods of that point and its image. The inverse derivative is the inverse matrix of the original derivative at the corresponding preimage.

Put $s=1+x+y\ne 0$. Differentiating each coordinate givesThe determinant never vanishes on $\Omega$, so the theorem applies at every point. The image point $(1/3,1/3)$ has preimage $(1,1)$: the equations $x/s=y/s=1/3$ imply $x=y=1$. Therefore

For fixed $x,y$, the map $z\mapsto z^3+z+xy-1$ is strictly increasing because its derivative is $3z^2+1>0$, and it tends to opposite infinities at the two ends of the real line. The intermediate value theorem gives exactly one root, defining $g(x,y)$.

For $F(x,y,z)=z^3+z+xy-1$, $F_z=3z^2+1$ never vanishes. The implicit function theorem supplies a $C^1$ solution locally around every $(x,y)$. Uniqueness makes all these local functions agree on overlaps, proving $g\in C^1({\mathbb{R}}^2)$. Implicit differentiation givesAt $(1,1)$ the root is zero, so $Dg(1,1)=(-1,-1)$.