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

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


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.


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