定义
The divergence of a vector field a ( x , y , z ) a(x, y, z) a(x,y,z) is defined by
d i v a = ∇ ⋅ a = ∂ a x ∂ x + ∂ a y ∂ y + ∂ a z ∂ z , div \boldsymbol{a}=\nabla \cdot \boldsymbol{a}=\frac{\partial a_x}{\partial x}+\frac{\partial a_y}{\partial y}+\frac{\partial a_z}{\partial z}, diva=∇⋅a=∂x∂ax+∂y∂ay+∂z∂az,
where a x a_x ax, a y a_y ay and a z a_z az are the x x x-, y y y- and z z z- components of a \boldsymbol{a} a. Clearly, ∇ ⋅ a \nabla \cdot \boldsymbol{a} ∇⋅a is a scalar field. Any vector field a \boldsymbol{a} a for which ∇ ⋅ a = 0 \nabla \cdot \boldsymbol{a}=0 ∇⋅a=0 is said to be solenoidal.
Examples
Example 1
Find the divergence of the vector field a = x 2 y 2 i + y 2 z 2 j + x 2 z 2 k \boldsymbol{a}=x^2y^2\boldsymbol{i}+y^2z^2\boldsymbol{j}+x^2z^2\boldsymbol{k} a=x2y2i+y2z2j+x2z2k.
Solution
From the definition, the divergence of a vector field a ( x , y , z ) a(x, y, z) a(x,y,z) is given by
∇ ⋅ a = 2 x y 2 + 2 y z 2 + z x 2 z = 2 ( x y 2 + y z 2 + x 2 z ) . \nabla \cdot \boldsymbol{a}=2xy^2+2yz^2+zx^2z=2(xy^2+yz^2+x^2z). ∇⋅a=2xy2+2yz2+zx2z=2(xy2+yz2+x2z).
Geometrical properties
The divergence can be considered as a quantitative measure of how much a vector field diverges
(spreads out) or converges at any given point.
For example, if we consider the vector field v ( x , y , z ) v(x, y, z) v(x,y,z) describing the local velocity at any point in a fluid then ∇ ⋅ v \nabla \cdot \boldsymbol{v} ∇⋅v is equal to the net rate of outflow of fluid per unit volume, evaluated at a point (by letting a small volume at that point tend to zero).