differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, a mathematical discipline, and more specifically in

Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...

, 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 function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...

s, 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

''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 In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of ''f,'' and * near a critical point (in the neighborhood of a critical point), it equals the gradient of ''f,'' when ''f'' is written in standard form given in the

Morse lemma In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...

s. Formally:p. 63
/ref> * away from critical points,

X \cdot f > 0,

* around every critical point there is a neighborhood on which ''f'' is given as in the Morse lemmas: :

f(x) = f(b) - x_1^2 - \cdots - x_^2 + x_^2 + \cdots + x_n^2

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

Dynamical system

The associated

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

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

References

* An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: Gradient-like vector fields
p. 56–69

Gradient-Like Vector Fields Exist
September 25, 2009 Morse theory Differential topology {{topology-stub