2 The divergence of a vector field

Consider the vector field $\underset{̲}{F}=F_1\underset{̲}{i}+F_2\underset{̲}{j}+F_3\underset{̲}{k}.$



In 3D cartesian coordinates the divergence of $\underset{̲}{F}$ is defined to be

$\text{div}\underset{̲}{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$

Note that $\underset{̲}{F}$ is a vector field but div $\underset{̲}{F}$ is a scalar.

In terms of the differential operator $\underset{̲}{\nabla }$ , div $\underset{̲}{F}=\underset{̲}{\nabla }\cdot \underset{̲}{F}$ since

$\underset{̲}{\nabla }\cdot \underset{̲}{F}=\left(\underset{̲}{i}\frac{\partial }{\partial x}+\underset{̲}{j}\frac{\partial }{\partial y}+\underset{̲}{k}\frac{\partial }{\partial z}\right)\cdot \left(F_1\underset{̲}{i}+F_2\underset{̲}{j}+F_3\underset{̲}{k}\right)=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$

Physical Significance of the Divergence



The meaning of the divergence is most easily understood by considering the behaviour of a fluid and hence is relevant to engineering topics such as thermodynamics. The divergence (of the vector field representing velocity) at a point in a fluid (liquid or gas) is a measure of the rate per unit volume at which the fluid is flowing away from the point. A negative divergence is a convergence indicating a flow towards the point. Physically positive divergence means that either the fluid is expanding or that fluid is being supplied by a source external to the field. Conversely convergence means a contraction or the presence of a sink through which fluid is removed from the field. The lines of flow diverge from a source and converge to a sink.



If there is no gain or loss of fluid anywhere then div $\underset{̲}{v}=0$ which is the equation of continuity for an incompressible fluid.



The divergence also enters engineering topics such as electromagnetism. A magnetic field $\left(\underset{̲}{B}\right)$ has the property $\underset{̲}{\nabla }\cdot \underset{̲}{B}=0,$ that is there are no isolated sources or sinks of magnetic field (no magnetic monopoles).

Example 11

Find the divergence of the following vector fields.

1. $\underset{̲}{F}=x^2\underset{̲}{i}+y^2\underset{̲}{j}+z^2\underset{̲}{k}$
2. $\underset{̲}{r}=x\underset{̲}{i}+y\underset{̲}{j}+z\underset{̲}{k}$
3. $\underset{̲}{v}=-x\underset{̲}{i}+y\underset{̲}{j}+2\underset{̲}{k}$

Solution

1. div $\underset{̲}{F}=\frac{\partial }{\partial x}\left(x^2\right)+\frac{\partial }{\partial y}\left(y^2\right)+\frac{\partial }{\partial z}\left(z^2\right)=2x+2y+2z$
2. div $\underset{̲}{r}=\frac{\partial }{\partial x}\left(x\right)+\frac{\partial }{\partial y}\left(y\right)+\frac{\partial }{\partial z}\left(z\right)=1+1+1=3$
3. div $\underset{̲}{v}=\frac{\partial }{\partial x}\left(-x\right)+\frac{\partial }{\partial y}\left(y\right)+\frac{\partial }{\partial z}\left(2\right)=-1+1+0=0$

Example 12

Find the value of $a$ for which $\underset{̲}{F}=\left(2x^{2}y+z^2\right)\underset{̲}{i}+\left(x{y}^2-x^{2}z\right)\underset{̲}{j}+\left(axyz-2x^2{y}^2\right)\underset{̲}{k}$ is incompressible.

Solution

$\underset{̲}{F}$ is incompressible if div $\underset{̲}{F}=0$ .



div $\underset{̲}{F}=\frac{\partial }{\partial x}\left(2x^{2}y+z^2\right)+\frac{\partial }{\partial y}\left(x{y}^2-x^{2}z\right)+\frac{\partial }{\partial z}\left(axyz-2x^2{y}^2\right)=4xy+2xy+axy$

which is zero if $a=-6.$

Task!

1. Find the divergence of the following vector field, in general terms and at the point $\left(1,0,3\right)$ .

   ${\underset{̲}{F}}_1=x^3\underset{̲}{i}+y^3\underset{̲}{j}+z^3\underset{̲}{k}$

Answer

Task!

1. Find the divergence of ${\underset{̲}{F}}_2=x^{2}y\underset{̲}{i}-2x{y}^2\underset{̲}{j}$ , in general terms and at $\left(1,0,3\right)$ .

Answer

Task!

1. Find the divergence of ${\underset{̲}{F}}_3=x^{2}z\underset{̲}{i}-2y^3{z}^3\underset{̲}{j}+xy{z}^2\underset{̲}{k}$ , in general terms and at the point $\left(1,0,3\right)$ .

Answer