Mappings and Vector Fields

Abstract

If we associate each point z of some domain 𝔇 of the z-plane with a certain complex value w according to a given law then w is called a function of z. Two geometrical interpretations of the functional relation are particularly useful. One uses one plane, the other two planes. The value w belonging to the point z (or, if more expedient, $\bar w$ ) can be thought of as a vector acting on the point z ; in this way a vector field is defined in the domain 𝔇. In the other interpretation, the value w associated with the point z in the z-plane is conceived as a point in another complex plane (w-plane). In this way the domain 𝔇 is mapped onto a certain point set of the w-plane.