I am having some trouble with a question recently regarding partial derivatives/multivariable calculus. Here is the question:
Suppose that $(x, y, u, v)$ $∈$ $R^4, v $$\ne$ −1, satisfy equations
$e^x + e^y + cos u + sin(2v) = 3$
$x + (e^2)^y + u^2 + log((v + 1)^2) = 1$
I have to find which two variables out of $x,y,u,v$ can be expressed uniquely in terms of the other two near the origin such that $(x,y,u,v) = (0,0,0,0).$
I have taken$z^T = (x,y,u,v)$ and $f(z) = $$\begin{pmatrix}e^x + e^y + cos(u) + sin(2v)-3\\x + (e^2)^y + u^2 + log((v+1)^2)-1\end{pmatrix}$$ , $$f(z)=0$
I then took the partial derivatives of each element, giving the Jacobian matrix to be:
$Jf(x) = $$\begin{pmatrix}e^x &e^y &sin(u)&cos(2v)\\1&2e^{2y} & 2u & {2\over v+1}\end{pmatrix}$
Giving $Jf(0,0,0,0) = $$\begin{pmatrix}1&1&0&2\\1&2&0&2\end{pmatrix}$
However I am unsure on how to proceed from here in finding which two variables can be expressed in terms of the other two. Any help would be greatly appreciated.
Implicit function theorem states that if the submatrix $$\begin{pmatrix} \frac{\partial f_1}{\partial \zeta} & \frac{\partial f_1}{\partial \xi} \\ \\ \frac{\partial f_{2}}{\partial \zeta} & \frac{\partial f_2}{\partial \xi} \end{pmatrix}$$ of $Jf$ is invertible, than you can solve the equation for $\xi$ and $\zeta$. In your case the submatrix $$ \begin{pmatrix} 1 & 1 \\ 1 & 2\end{pmatrix} $$ is invertible, so, the first two variables could be expressed in terms of second two.