Q: For which real values c is the subset $f(x) = x_{1}^{2} + x_{1}^{3} - x_{2}^{2} + x_{3}x_{4} = c$ a smooth submanifold of $\mathbb{R}^4$?
Try: for it to be a smooth submanifold, $c$ has to be a regular value, so we compute the derivate matrix and see where it is an isomorphism, so that the inversion function and submersion theorem will provide the submanifold.
Now, that $f$ goes from $\mathbb{R}^4$ to $\mathbb{R}$, thus we consider the splitting and the function on it: $\phi: \mathbb{R}^{3} \times \mathbb{R} \to \mathbb{R}^{3} \times \mathbb{R}$ given by $\phi(x,y) = (x, f(x,y)).$ Then, $D\phi(v) = $ \begin{array}{ccc} I_{3 \times 3} & 0 \\ J_{1}f(v) & J_{2}f(v) \\ \end{array} where $J_{1}f(v)$ is the derivative matrix with respect to the $\mathbb{R}^{3}$ part of $v$ and $J_{2}f(v)$ is the derivative matrix with respect to the $\mathbb{R}$ part of $v$ (from the splitting).
And now, we just look to see where this is an isomorphism and plug it in to $f$ to get the resulting $c$?
Do i have the correct idea? Thanks in advance.
But then, $J_{2}f(v) = x_{3}$ and as long as its not 0 we have isomorphism. But that implies all $\mathbb{R}^{4}$ vectors such that $x_{3}$ is not 0 work. But what does that say about $c$?