- #1
mr.tea
- 102
- 12
Homework Statement
prove, using the definition, that the mapping
## \mathbf{u}=\mathbf{u}(u_1(x_1,x_2),u_2(x_1,x_2))## where
##u_1=\tan(x_1)+x_2 ##
##u_2=x_2^3##
is a bijection from the strip ##-\frac{pi}{2}<x_1<\frac{pi}{2}## in the ## x_1x_2##-plane onto the entire ##u_1u_2##-plane, and find the inverse.
Homework Equations
The Attempt at a Solution
Finding the inverse is not the problem. My problem is that when I am trying to apply the inverse function theorem, I get that at ##x_2=0## the Jacobian is 0. The Jacobian that I have found is:
[itex]\frac{\partial (u_1,u_2)}{\partial (x_1,x_2)}=
\begin{vmatrix}
\frac{1}{cos^2(x_1)} & 1\\
0& 3x_2^2\\
\end{vmatrix}
[/itex]
What am I missing here?
Thank you.