Problem 1, page 78 of Munkres (Analysis on Manifolds):
Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = \mathbf{0}$ and $ Df(3,-1,2) = \begin{pmatrix} 1 & 2 & 1 \\ 1 & -1 & 1 \end{pmatrix}.$
(a) Show that there is a function $g: B \to \mathbb{R}^2$ of class $C^1$ defined on an open set $B$ in $\mathbb{R}$ such that $f(x,g_1(x),g_2(x)) = \mathbf{0}$ for $x \in B$ and $g(3) = (-1,2)$.
(b) Find $Dg(3)$.
(c) Discuss the problem of solving the equation $f(x,y_1,y_2) = \mathbf{0}$ for an arbitrary pair of unknowns in terms of the third, near the point $(3,-1,2)$.
Here's what I have so far:
Let $b=(-1,2)$ so that $a = (3,-1,2)$. Write $f(x,y_1,y_2)$ with $y = (y_1,y_2)$ then,
a =3, and b = (-1,2)
and determinant partial of $f$ w.r.t partial of $y (3,-1,2) =$ ?
$ \det \begin{pmatrix} \frac{\partial f_1}{\partial y_1}(a,b) & \frac{\partial f_1}{\partial y_2}(a,b) \\ \frac{\partial f_2}{\partial y_1}(a,b) & \frac{\partial f_2}{\partial y_2}(a,b) \end{pmatrix} $
Derivative of partial of f = [partial of f w.r.t x partial of f w.r.t. y] implies Df(3,b) = (Stuck on evaluating this), but I know it is the expression above which I have wrote
and what is partial of f1 w.r.t. y1 (a,b), partial of f2 w.r.t. y2 (a,b), partial of f1 w.r.t. y2 (a,b), and partial of f2 w.r.t. y2(a,b)?
For part b:
Dg(3) = -{partial of f w.r.t y(3,b)]^-1 [partial of f w.r.t. x1] at (3,b) =?
for part c, I thought of taking 2 variables u and v s.t. partial of f w.r.t partial of (u,v) is not equal to zero.
Since $f(a)$ is zero then by the implicit function theorem, there is a neighborhood $B$ of $(-1,2)$ in $\mathbb{R}^2$ and a unique function $g: B \to \mathbb{R}^3$ so that $g(a) = b$.