Section 6.4 Divergence 

6.4 Divergence 

This section, we learn the divergence of a vector field. The motivation for this is flux of a vector field along a curve or a surface. First, we introduce the flux of a vector field along a curve. 

Definition: 

The flux of $\mathbf{F}$ across $C$ is line integral

$${\int }_C\overrightarrow{F}\cdot \overrightarrow{N(t)}ds$$

where $\overrightarrow{N(t)}$ is the unit normal vector of the curve.

Theorem:Calculating Flux across a Curve 

Let $\mathbf{F}=<P,Q>$ be a vector field over ${\mathbb{R}}^2$ and let $C$ be a smooth curve with parameterization $\overrightarrow{r(t)}=<x(t),y(t)>$, $a\le t\le b.$ Let $\overrightarrow{n(t)}=<y^{\prime }(t),-x^{\prime }(t)>$ . The flux of $\mathbf{F}$ across $C$ is 

$${\int }_C\overrightarrow{F(r(t))}\cdot \overrightarrow{n(t)}dt={\int }_a^{b}P(\overrightarrow{r(t)})y^{\prime }(t)dt-Q(\overrightarrow{r(t)})x^{\prime }(t)dt={\int }_{C}Pdy-Qdx$$

Example 1: Calculate the flux of $\overrightarrow{F}=<2x,2y>$ across a unit circle oriented counterclockwise.

Exercise 1: Calculate the flux of $\overrightarrow{F}=<y,-x>$ across a unit circle oriented counterclockwise.

Computing the integration over a curve may not be easy. Hence we want an alternative way for the computation. We introduce the divergence

of a vector field. 

Definition:

If $\mathbf{F}=<P,Q,R>$ or $\mathbf{F}=<P,Q>$ is a vector field in ${\mathbb{R}}^3$ or ${\mathbb{R}}^2$ and $P_x,Q_y$ and $R_z$ all exist, then the divergence of $\mathbf{F}$ is defined

by  $$\text{div}\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\text{ for }{\mathbb{R}}^3$$

or 

$$\text{div}\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\text{ for }{\mathbb{R}}^2.$$

Example 2: Find the divergence of $\mathbf{F}(x,y,z)=xy{e}^z\overrightarrow{i}+yz{e}^x\overrightarrow{k}.$

Exercise 2: Find the divergence of $\mathbf{F}(x,y,z)=x{e}^y\overrightarrow{j}+z{e}^y\overrightarrow{k}.$

Example 3: Find the divergence of $\mathbf{F}(x,y,z)=<e^x\text{sin}(y),e^y\text{sin}(z),e^z\text{sin}(x)>.$

Exercise 3: Find the divergence of $\mathbf{F}(x,y,z)=<e^y\text{cos}(z),e^x\text{cos}(y),e^z\text{cos}(x)>.$

Theorem:

If $\mathbf{F}=<P,Q,R>$ is a vector field on ${\mathbb{R}}^3$, and $P,Q$ and $R$ have continuous second order paritial derivatives then

$$\text{div}(\text{curl}(\mathbf{F}))=0.$$

Example 4: Is there a vector filed such that $\text{curl}(\mathbf{F}(x,y,z))=<x\text{sin}(y),\text{cos}(y),z-xy>$? Explain.

Exercise 4: Is there a vector filed such that $\text{curl}($$\mathbf{F}(x,y,z))=<e^y\text{cos}(z),e^x\text{cos}(y),e^z\text{cos}(x)>$? Explain.

Theorem: Calculating Flux across a Curve using Green’s Theorem 

Let $\mathbf{F}=<P,Q>$ be a vector field over ${\mathbb{R}}^2$, The flux of $\mathbf{F}$ across $C$ that enclose a region $D$ is

$${\int }_C\overrightarrow{F(r(t))}\cdot \overrightarrow{n(t)}dt=\int {\int }_D\text{div}\mathbf{F}dA$$

Example 5: Calculate the flux of $\overrightarrow{F}=<x^2+\text{cos}(y),\text{tan}(x)+3y>$ along a curve $C$ from $(0,0)$ to $(1,0)$ to $(1,1)$ to $(0,1)$ and back to $(0,0)$ with the positive orientation. 

Exercise 5: Calculate the flux of $\overrightarrow{F}=<e^x+{\text{sin}}^{-1}(y),e^y+\text{sec}(x)>$ along a curve $C$ from $(1,1)$ to $(2,1)$ to $(2,2)$ to $(1,2)$ and back to $(1,1)$ with the positive orientation. 

Example 6: Calculate the flux of $\overrightarrow{F}=<x^2+y^2,y^2-x^2>$ along a curve $C$ from $(1,0)$ to $(1,-1)$ to $(0,0)$ and back to $(1,0)$.

Exercise 6: Calculate the flux of $\overrightarrow{F}=<x^3-y^2,x^2-y^2>$ along a curve $C$ from $(0,0)$ to $(1,0)$ to $(1,1)$ and back to $(0,0)$.

If $\mathbf{F}(x,y,z)$ is the velocity of a fluid (or gas), then $\text{div}\mathbf{F}(x,y,z)$ represents the net rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point $(x,y,z)$ per unit volume. In other words, $\text{div}\mathbf{F}(x,y,z)$ measures the tendency of the fluid to diverge from the point $(x,y,z)$. If $\text{div}\mathbf{F}=0$, then $\mathbf{F}$ is said to be incompressible.

Group work:

1. Find the divergence of $\mathbf{F}$ at a given point $P$.

a. $\mathbf{F}=<x{y}^2{z}^4,2x^{2}y+z,y^3{z}^2>$ and $P=(1,1,-1)$.

b. $\mathbf{F}=e^x\text{sin}(y)\overrightarrow{i}-e^x\text{cos}(y)\overrightarrow{j}$ and $P=(0,0,2)$. 

c. $\mathbf{F}=<e^{-xy},e^{xz},e^{yz}>$ and $P=(3,2,0)$. 

2. Calculate the flux of $\overrightarrow{F}=<x^3-y^3,y^3+x^3>$ along an ellipse $C:x^2+9y^2=9$ with positive orientation.

3. Calculate the work done by $\overrightarrow{F}=<x^3-y^3,y^3+x^3>$ by moving an object along an ellipse $C:x^2+9y^2=9$ with positive orientation.