Question

Is it possible for a C¹ vector field in R³ to have only one source? Why or why not?

Answer 1

One geometric interpretation of the divergence at a point R ∈ D is as a measure of how much the point (R) ∈ D is a source or sink for the vector field. By source, we mean that the vector field has a positive net flow away from the point locally, whereas by the sink, we mean it has a negative net flow. To better quantify this and properly define these ideas, we need the notion of flux, which is defined in terms of integration. Intuitively, flux is the limiting (infinitesimal) measure of the amount of flow out of a small volume, minus the amount of flow into that volume. Though we will not yet discuss the formal definition with integrals, we nevertheless can consider a few examples to illustrate this intuition.

In other words,  a vector field is an assignment of a vector to each point in a subset of space. In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain.

Given a subset S in Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates (x1, ..., xn). If each component of V is continuous, then V is a continuous vector field, and more generally V is a Ck vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. Given two Ck-vector fields V, W defined on S and a real-valued Ck-function f defined on S, the two operations scalar multiplication and vector addition