Gradient-like vector field

In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient [vector field].
The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.

Definition

Given a Morse function f on a manifold M, a gradient-like vector field X for the function f is, informally:

away from critical points, X points "in the same direction as" the gradient of f, and
near a critical point, it equals the gradient of f, when f is written in standard form given in the Morse lemmas.

Formally:

away from critical points,
around every critical point there is a neighborhood on which f is given as in the Morse lemmas:

and on which X equals the gradient of ''f.''

The associated dynamical system of a gradient-like vector field, a gradient-like dynamical system, is a special case of a Morse–Smale system.