Be it F (x, y, z ) = (x + y + z , x cos(y^2 + z^2 ) + z ) and consider the system of equations F (x, y, z ) = 0.
a) What pair of variables should you choose in order to describe the solutions of the system, in a neighborhood of the origin, as functions of the third variable?
I did DF(x,y,z) and got:
\(\displaystyle \begin{bmatrix}1 & 1 & 1 \\ cos(y^2 + z^2) & -2xy*sin(y^2 + z^2) & -2xz*sin(y^2 + z^2) +1 \end{bmatrix}\) replacing with the origin (0,0,0):
\(\displaystyle \begin{bmatrix}1 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}\)
And in my opinion in the origin, (0,0,0) I can have (x,y) = f(z) or (y,z) = f(x) right? Because the det[dF/d(x,y)] != 0 and det[dF/d(y,z)] != 0
Am I wrong or am I right?? I hope it's my last post on the subject... I'm sorry I'm being such a pain but I'm studying this without classes...
a) What pair of variables should you choose in order to describe the solutions of the system, in a neighborhood of the origin, as functions of the third variable?
I did DF(x,y,z) and got:
\(\displaystyle \begin{bmatrix}1 & 1 & 1 \\ cos(y^2 + z^2) & -2xy*sin(y^2 + z^2) & -2xz*sin(y^2 + z^2) +1 \end{bmatrix}\) replacing with the origin (0,0,0):
\(\displaystyle \begin{bmatrix}1 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}\)
And in my opinion in the origin, (0,0,0) I can have (x,y) = f(z) or (y,z) = f(x) right? Because the det[dF/d(x,y)] != 0 and det[dF/d(y,z)] != 0
Am I wrong or am I right?? I hope it's my last post on the subject... I'm sorry I'm being such a pain but I'm studying this without classes...
