To take a step back, the idea of a Taylor expansion is that we can express any C∞ function as an (infinite) polynomial. If you’re close enough to the point you’re expanding around, then a finite polynomial can be an arbitrarily good fit.
The central challenge here is that K(w) is pretty much never a polynomial. So the idea is to find a mapping, g, that lets us re-express w in terms of a new coordinate system, w=g(u). If we do this right, then we can express K(g(u)) (locally) as a polynomial in terms of the new coordinates, u.
What we’re doing here is we’re “fixing” the non-differentiable singularities in K(w) so that we can do a kind of Taylor expansion over the new coordinates. That’s why we have to introduce this new manifold, U, and mapping g.
To take a step back, the idea of a Taylor expansion is that we can express any C∞ function as an (infinite) polynomial. If you’re close enough to the point you’re expanding around, then a finite polynomial can be an arbitrarily good fit.
The central challenge here is that K(w) is pretty much never a polynomial. So the idea is to find a mapping, g, that lets us re-express w in terms of a new coordinate system, w=g(u). If we do this right, then we can express K(g(u)) (locally) as a polynomial in terms of the new coordinates, u.
What we’re doing here is we’re “fixing” the non-differentiable singularities in K(w) so that we can do a kind of Taylor expansion over the new coordinates. That’s why we have to introduce this new manifold, U, and mapping g.