exponential map in a sentence

Examples

1. The exponential map of a nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connected, simply-connected Lie group.
2. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:
3. In the tangent space at any point on a quaternion Slerp curve, the inverse of the exponential map transforms the curve into a line segment.
4. Over the real numbers ( or more generally any field of characteristic 0 ) the exponential map takes any nilpotent square matrix to a unipotent matrix.
5. Only one quarter of this plane is the image of the exponential map, so the logarithm is only defined on that quarter ( quadrant ).
6. That is to say, the starting point p \ in H goes to the tangent plane from H through the inverse of the exponential map.
7. :I'd guess the OP wants something for binomials like the exponential map formula for Lie groups, but that's fairly hairy.
8. The exponential map is onto in each case, since the polynomial function lies in a given Lie subalgebra if and do and are sufficiently small.
9. For an alternative derivation based on this exponential relationship, see Axis angle representation # Exponential map from so ( 3 ) to SO ( 3 ).
10. Then travels from one tangent plane to the other through the isometry \ psi, and then down to the surface S'with another exponential map.