Divergence of a Vector Field

Topics: Vector Field - Calculus

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(definition)

The divergence of a vector field $F(x,y,z)=⟨F_1(x,y,z),F_2(x,y,z),F_3(x,y,z)⟩$ is the scalar function:

$$\text{div}F(x,y,z)=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$$

…defined at all points at which all the indicated partial derivatives exist.

Alternative Notation

More simply, with a bit of notation abuse, we can say that the divergence of a vector field $F$ is:

$$\text{div}F=\nabla \cdot F$$

…where $\nabla =⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩$; see inner product.

Divergence tells us how much the vectors in the vector field “grow” or “shrink”.

When the divergence of a vector field is $0$, we say that the vector field is incompressible.