Vector Fields and Dynamical Systems

Abstract

This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.