5.7 The Divergence of a Vector Field

The Divergence of a Vector Field

Introduction

Definition of the Divergence of a Vector Field

Measuring the Change in Strength of a Vector Field

In this preview activity, we will look at several two-dimensional vector fields and try to assess when the vector field has increased or decreased in strength over a given region. We begin with graphs of the three vector fields, $\vF$, $\vG$, and $\vH$. Parts , , and ask you to answer the same three questions about the vector field and square illustrated in each of the figures. Part asks you to think further about the third vector field.

For each of the vector fields $\vF$, $\vG$, and $\vH$ and the square centered on $P_1$, $P_2$, and $P_3$ (respectively), which statement do you think best applies? - More of the vector field is going into the square than going out. - Less of the vector field is going into the square than going out. - The same amount of the vector field is going into the square as is going out.

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Look at the plot of the vector field $\vG$ in and state whether you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in each of the four quadrants. You can make your argument in terms of the change in magnitude along the flow of the vector field or in terms of the net flow into or out of a small region on the plane. You may need to make separate arguments for each of the four quadrants.

Look at the plot of the vector field $\vH$ in below and state whether you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in each of the four quadrants. You can make your argument in terms of the change in magnitude along the flow of the vector field or in terms of the net flow into or out of a small region on the plane. You may need to make separate arguments for each of the four quadrants.

Calculate the divergence of the vector fields given below. -$\vF(x,y)=\langle -x,-y\rangle$-$\vG(x,y)=\langle y,x \rangle$-$\vH(x,y)=\langle xy,1-x\rangle$

Explain how your answers to the questions in Activity 5.7.1 can be explained by using your results from part of this activity.

• Let $\vF=\langle{F_1,F_2,F_3}\rangle$ and let

Show that $\divg(\vG)=\vec{0}$. - Vector fields with a zero divergence everywhere in their domain are called divergence-free or incompressible vector fields. Which of the following vector fields are divergence-free? -$\vF=\langle{-y,z,x}\rangle$-$\vF=\langle{\cos(yz),3xe^{z-x},6(x+y+z)^3}\rangle$-$\vF=\langle{4xyz,y^2z,yz^2}\rangle$-$\vF=\nabla f$ where $f$ is a scalar function of $x$, $y$, and $z$- Let $\vF_1=\langle{3(x-z)^2,2\cos(x)+3yz+y,-(z-1)^2+e^{xy}}\rangle$. Calculate the divergence of $\vF_1$ and give a point where $\divg(\vF_1)=0$. - Is $\vF_1$ a divergence free vector field?