Jacobian and its determinant in inverse and implicit Theorem

[xoninhas]

Well I've made some questions already about the inv. Theorem and the implicit Theo. and I've noticed that I have a problem knowing after all what is that Jacobian, and what is the determinant of the Jacobian and when do I need them in the implicit and inverse theorem applications.

imagine F(x,y,z) = (xy + yz, z*y^2), so D:R^3->R^2

well to know if this function can be as f:R->R^2 using the implicit theorem.

What do I do? \(\displaystyle \begin{bmatrix}dF1/dx \\ dF2/dx \end{bmatrix}\) = \(\displaystyle \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) means it can't be used? and if \(\displaystyle \begin{bmatrix}dF1/d(y,z) \\ dF2/d(y,z) \end{bmatrix}\) != \(\displaystyle \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) means that it can be written (y,z) = f(x)?

when do i use determinant and when do i use just the matrix??

Thanks guys, you've been real help in my study!

 

[tkhunny]

#1 - It's not the right matrix if it isn't square. Determinants don't mean much otherwise. Remember that x and y in terms of t problem we did before? That needed two equations and it produced a 2x2 matrix for the Jacobian. In the case of the problem earlier today, you were a little confused because it wasn't as complicated as you wanted it to be. This is because we had only two variables and only one equation. This leads to a 1x1 matrix and the determinant of that matrix is not very tricky. You must reach out and grab the easy ones. Don't let them get away.