I'm finding it a little harder to visualize rotation numbers, though. My best attempt at a description is to imagine continuously tracing the curve '\gamma(t)', going through every point that it passes through, while looking top-down on it. At every point on the curve, the vector field 'v' produces a vector 'v(\gamma(t))' that begins at '\gamma(t)', lies flat on the sphere (i.e. is tangent to the sphere), and is of nonzero length. (The last assumption is the assumption we are making for contradiction).

The idea is that, as we trace the curve '\gamma(t)', we are constantly measuring the angle - with a positive-negative sign - between (a) the tangent vector 'v(\gamma(t))', and (b) the current velocity vector of '\gamma(t)'. As we trace the curve, if this angle rotates counterclockwise 0...90...180...270...0, we add "1" to our rotation number, and we subtract one for a clockwise rotation 0...-90...-180...-270...0.