# Prove that near the orign $0 \in \mathbb R^{4}$, its solutions from the graph of a continuously differentiable function $G: R^{2} \rightarrow R^{2}$.

Consider the system of equations in the variables $$u,v,s,t$$:

$$(uv)^{4}+ (u+s)^{3}+t=0$$

$$sin(uv)+e^{v}+t^{2}-1=0$$

Prove that near the origin $$0 \in \mathbb R^{4}$$, its solutions from the graph of a continuously differentiable function $$G: R^{2} \rightarrow R^{2}$$. Clearly indicate the dependent and independent variables.

My idea: I used implicit function theorem to solve this question. Firstly, for independent and dependent variable, I find the matrix of transformation

$$[A]= \begin{bmatrix}F_{1s}~~F_{1t}~~F_{1u}~~F_{1v}\\F_{2s}~~F_{2t}~~F_{2u}~~F_{2v} \end{bmatrix}$$

At origin, $$[A]= \begin{bmatrix}0~~1~~0~~0\\0~~0~~0~~1 \end{bmatrix}.$$

From this matrix, I observe that $$t$$ and $$v$$ variables are independent, then I apply Implicit function theorem for solution.

Anyone can suggest me whether this is the correct way or not to calculate independent and dependent variables.