Divergence Operator

The divergence operator is a differentiation operator that operates on vector fields. It uses the dot product with partial derivatives.

Definition §

Divergence Operator ^definition

Let $\vec{F}$ be a vector field operating in Euclidean space, $\vec{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ where $\vec{F}(x_1,\dots ,x_n)=(f_1(x_1,\dots ,x_n),\dots ,f_n(x_1,\dots ,x_n))$. If $\vec{F}$ is $C^1$ (that is, all $f_i$ are $C_1$), then the divergence of $\vec{F}$ is:

$$\begin{array}{rl}\nabla \cdot \vec{F}=\text{div }\vec{F}& =\left(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n}\right)\cdot (f_1,\dots ,f_n)\\ & =\frac{\partial f_1}{\partial x_1}+\frac{\partial f_2}{\partial x_2}+\cdots +\frac{\partial f_n}{\partial x_n}\end{array}$$

• While $\vec{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$, we have $\nabla \cdot \vec{F}:D\to \mathbb{R}$
• In the case of a real vector field $\vec{F}:{\mathbb{R}}^3\to {\mathbb{R}}^3$, if we have $\vec{F}(x,y,z)=(u(x,y,z),v(x,y,z),w(x,y,z))$ where $u,v,w:{\mathbb{R}}^3\to \mathbb{R}$ are scalar functions, we have:

$$\nabla \cdot \vec{F}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$$

Compressibility Of A Vector Field §

The divergence operator can define the compressibility of a vector field, as well as define sources and sinks. Let $\vec{F}$ be a vector field:

• If $\nabla \cdot \vec{F}>0$ at a point $p$, then $p$ is a source. We can say the field is expanding.
• If $\nabla \cdot \vec{F}<0$ at a point $p$, then $p$ is a sink. We can say the field is compressing.
• If $\nabla \cdot \vec{F}=0$ at point $p$, then it is neither sink nor source. We can say the field is incompressible at point $p$.

Thus, the divergence operator defines an incompressible vector field if it is incompressible for all points $p$.

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Applications §

Fluids §

If $\vec{F}$ is the velocity vector field representing fluid flow, then the divergence gives the net transport of fluids in or out of a point:

• If $\nabla \cdot \vec{F}>0$ at a point $p$, then more fluid is flowing out then in and $p$ is a source. We can say the fluid is expanding.
• If $\nabla \cdot \vec{F}<0$ at a point $p$, then more fluid is flowing in then out and $p$ is a sink. We can say the fluid is compressing.
• If $\nabla \cdot \vec{F}=0$ at point $p$, then the fluid net transport is steady. If this holds for all $p$, the fluid has incompressible flow.

Examples §

1: Expanding Vector Field

Find the divergence of the given vector field:

$$\vec{F}(x,y)=(x,0)$$

Solution

$$\begin{array}{r}\nabla \cdot \vec{F}(x,y)=\frac{\partial }{\partial x}(x)+\frac{\partial }{\partial y}(0)=1+0=1\end{array}$$

Since the divergence is $>0$, we can say $\vec{F}$ is an expanding vector field.

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2: Compressing Vector Field

Find the divergence of the given vector field:

$$\vec{F}(x,y)=(0,-y)$$

Solution

$$\begin{array}{r}\nabla \cdot \vec{F}(x,y)=\frac{\partial }{\partial x}(0)+\frac{\partial }{\partial y}(-y)=0-1=-1\end{array}$$

Since the divergence is $<0$, we can say $\vec{F}$ is an compressing vector field.

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3: Incompressible Vector Field

Find the divergence of the given vector field:

$$\vec{F}(x,y)=(x,-y)$$

Solution

$$\begin{array}{r}\nabla \cdot \vec{F}(x,y)=\frac{\partial }{\partial x}(x)+\frac{\partial }{\partial y}(-y)=1-1=0\end{array}$$

Since the divergence is exactly zero, we can say $\vec{F}$ is an incompressible vector field.

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