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how does one explain the following:

"The sphere can be covered by 2 open sets using stereographic projection in such a way that the intersection of these 2 sets is a connected set $W$.Let $p \in W$, if the jacobian of transitions maps is negative then intechange the parameters, so the jacobian will be positive.Since $W$ is connected, the jacobian is positive for every $p \in W$"


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up vote 3 down vote accepted

Since the Jacobian matrix of partial derivatives is nonsingular, the determinant will be nonzero throughout the intersection of the two coordinate charts. So it is either positive or negative since the intersection is connected. Therefore, you can change one of the coordinate charts by an orientation-reversing map (which has negative determinant) such as $(u,v)\mapsto (v,u)$ to make the determinant of the Jacobian matrix positive.

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