Here's the problem:
Let $F:R^4 \to R^2, g:R^2\to R^2, h:R\to R$ all differentiable, such that:
$g(x_0,y_0) = (2,3), h(x_0) = 1, h(y_0) = 4$.
Suppose that $F(h(x),g(x,y),h(y)) = (0,0), \forall(x,y) \in R^2$.
The following Jacobian Matrices are given:
$JF(1,2,3,4) = \bigg[\matrix{1\\-2}\matrix{7\\0}\matrix{0\\-5}\matrix{2\\5}\bigg], Jh(x_0) = \big[\matrix{14}\big], Jh(y_0) = \big[\matrix{10}\big]$.
Calculate the jacobian matrix of $g$ in the point $(x_0,y_0)$.
What I Know
I'm really lost here… I know how to calculate a Jacobian Matrix, given a function, but I don't know how to start here or what I can use to solve it. I just need a tip to start.
Best Answer
Rewriting some equations of your problem for convenience
$$h(x) = h_1(x), \, h(y) = h_2(y)$$
$$g(x,y) = \left[\begin{matrix}g_1(x,y)\\g_2(x,y)\end{matrix}\right]$$
$$f(h(x),g(x,y),h(y)) = f(h_1(x),g_1(x,y),g_2(x,y),h_2(y)) = \left[\begin{matrix}f_1(h_1(x),g_1(x,y),g_2(x,y),h_2(y))\\f_2(h_1(x),g_1(x,y),g_2(x,y),h_2(y))\end{matrix}\right] = \left[\begin{matrix}0\\0\end{matrix}\right]$$
$$J_f(h_1(x),g_1(x,y),g_2(x,y),h_2(y)) = \left[\matrix{\frac{\partial f_1}{\partial h_1}\\\frac{\partial f_2}{\partial h_1}}\matrix{\frac{\partial f_1}{\partial g_1}\\\frac{\partial f_2}{\partial g_1}}\matrix{\frac{\partial f_1}{\partial g_2}\\\frac{\partial f_2}{\partial g_2}}\matrix{\frac{\partial f_1}{\partial h_2}\\\frac{\partial f_2}{\partial h_2}}\right]$$
$$J_g(x,y) = \left[\matrix{\frac{\partial g_1}{\partial x}\\\frac{\partial g_2}{\partial x}}\matrix{\frac{\partial g_1}{\partial y}\\\frac{\partial g_2}{\partial y}}\right]$$
$$J_{h_1}(x) = \frac{\partial h_1}{\partial x}$$
$$J_{h_2}(y) = \frac{\partial h_2}{\partial y}$$
From the chain rule, we know that for $i={1,2}$
$$ \frac{\partial f_i}{\partial x} = 0 = \frac{\partial f_i}{\partial h_1} \frac{\partial h_1}{\partial x} + \frac{\partial f_i}{\partial g_1} \frac{\partial g_1}{\partial x} + \frac{\partial f_i}{\partial g_2} \frac{\partial g_2}{\partial x} $$
$$ \frac{\partial f_i}{\partial y} = 0 = \frac{\partial f_i}{\partial h_2} \frac{\partial h_2}{\partial y} + \frac{\partial f_i}{\partial g_1} \frac{\partial g_1}{\partial y} + \frac{\partial f_i}{\partial g_2} \frac{\partial g_2}{\partial y} $$
It is given that
$J_f(h(x_0),g_1(x_0,y_0),g_2(x_0,y_0),h(y_0)) = \bigg[\matrix{1\\-2}\matrix{7\\0}\matrix{0\\-5}\matrix{2\\5}\bigg], J_{h_1}(x_0) = \big[\matrix{14}\big], J_{h_2}(y_0) = \big[\matrix{10}\big]$
So it is possible to write the system
$$ 14 + 7 \frac{\partial g_1}{\partial x} (x_0,y_0) = 0 $$
$$ 20 + 7 \frac{\partial g_1}{\partial y} (x_0,y_0) = 0 $$
$$ -28 -5 \frac{\partial g_2}{\partial x} (x_0,y_0) = 0 $$
$$ 50 - 5 \frac{\partial g_2}{\partial y} (x_0,y_0) = 0 $$