Examples
Since any manifold can be locally parametrised, we can consider some explicit maps from two-space into two-space.
- Let . We can calculate the Jacobian matrix:
The Jacobian matrix has zero determinant if, and only if xy = 0. We see that f is a diffeomorphism away from the x-axis and the y-axis.
- Let where the and are arbitrary real numbers, and the omitted terms are of degree at least two in x and y. We can calculate the Jacobian matrix at 0:
We see that g is a local diffeomorphism at 0 if, and only if, i.e. the linear terms in the components of g are linearly independent as polynomials.
- Now let . We can calculate the Jacobian matrix:
The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.
Read more about this topic: Diffeomorphism
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