# 6.5 Divergence and curl  (Page 7/9)

 Page 7 / 9

## Key equations

• Curl
$\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{F}=\left({R}_{y}-{Q}_{z}\right)\text{i}+\left({P}_{z}-{R}_{x}\right)\text{j}+\left({Q}_{x}-{P}_{y}\right)\text{k}$
• Divergence
$\nabla ·\text{F}={P}_{x}+{Q}_{y}+{R}_{z}$
• Divergence of curl is zero
$\nabla ·\left(\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{F}\right)=0$
• Curl of a gradient is the zero vector
$\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{∇}f\right)=0$

For the following exercises, determine whether the statement is true or false .

If the coordinate functions of $\text{F}:{ℝ}^{3}\to {ℝ}^{3}$ have continuous second partial derivatives, then $\text{curl}\phantom{\rule{0.2em}{0ex}}\left(\text{div}\left(\text{F}\right)\right)$ equals zero.

$\nabla ·\left(x\text{i}+y\text{j}+z\text{k}\right)=1.$

False

All vector fields of the form $\text{F}\left(x,y,z\right)=f\left(x\right)\text{i}+g\left(y\right)\text{j}+h\left(z\right)\text{k}$ are conservative.

If $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0,$ then F is conservative.

True

If F is a constant vector field then $\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=0.$

If F is a constant vector field then $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0.$

True

For the following exercises, find the curl of F .

$\text{F}\left(x,y,z\right)=x{y}^{2}{z}^{4}\text{i}+\left(2{x}^{2}y+z\right)\text{j}+{y}^{3}{z}^{2}\text{k}$

$\text{F}\left(x,y,z\right)={x}^{2}z\text{i}+{y}^{2}x\text{j}+\left(y+2z\right)\text{k}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\text{i}+{x}^{2}\text{j}+{y}^{2}\text{k}$

$\text{F}\left(x,y,z\right)=3xy{z}^{2}\text{i}+{y}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}z\text{j}+x{e}^{2z}\text{k}$

$\text{F}\left(x,y,z\right)={x}^{2}yz\text{i}+x{y}^{2}z\text{j}+xy{z}^{2}\text{k}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\left(x{z}^{2}-x{y}^{2}\right)\text{i}+\left({x}^{2}y-y{z}^{2}\right)\text{j}+\left({y}^{2}z-{x}^{2}z\right)\text{k}$

$\text{F}\left(x,y,z\right)=\left(x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}y\right)\text{i}+x{y}^{2}\text{j}$

$\text{F}\left(x,y,z\right)=\left(x-y\right)\text{i}+\left(y-z\right)\text{j}+\left(z-x\right)\text{k}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\text{i}+\text{j}+\text{k}$

$\text{F}\left(x,y,z\right)=xyz\text{i}+{x}^{2}{y}^{2}{z}^{2}\text{j}+{y}^{2}{z}^{3}\text{k}$

$\text{F}\left(x,y,z\right)=xy\text{i}+yz\text{j}+xz\text{k}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\text{−}y\text{i}-z\text{j}-x\text{k}$

$\text{F}\left(x,y,z\right)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k}$

$\text{F}\left(x,y,z\right)=ax\text{i}+by\text{j}+c\text{k}$ for constants a , b , c

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0$

For the following exercises, find the divergence of F .

$\text{F}\left(x,y,z\right)={x}^{2}z\text{i}+{y}^{2}x\text{j}+\left(y+2z\right)\text{k}$

$\text{F}\left(x,y,z\right)=3xy{z}^{2}\text{i}+{y}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}z\text{j}+x{e}^{2z}\text{k}$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=3y{z}^{2}+2y\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}z+2x{e}^{2z}$

$\text{F}\left(x,y\right)=\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)\text{i}+\left(\text{cos}\phantom{\rule{0.1em}{0ex}}y\right)\text{j}$

$\text{F}\left(x,y,z\right)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k}$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=2\left(x+y+z\right)$

$\text{F}\left(x,y,z\right)=\left(x-y\right)\text{i}+\left(y-z\right)\text{j}+\left(z-x\right)\text{k}$

$\text{F}\left(x,y\right)=\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\text{i}+\frac{y}{\sqrt{{x}^{2}+{y}^{2}}}\text{j}$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}$

$\text{F}\left(x,y\right)=x\text{i}-y\text{j}$

$\text{F}\left(x,y,z\right)=ax\text{i}+by\text{j}+c\text{k}$ for constants a , b , c

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=a+b$

$\text{F}\left(x,y,z\right)=xyz\text{i}+{x}^{2}{y}^{2}{z}^{2}\text{j}+{y}^{2}{z}^{3}\text{k}$

$\text{F}\left(x,y,z\right)=xy\text{i}+yz\text{j}+xz\text{k}$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=x+y+z$

For the following exercises, determine whether each of the given scalar functions is harmonic.

$u\left(x,y,z\right)={e}^{\text{−}x}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}y-\text{sin}\phantom{\rule{0.1em}{0ex}}y\right)$

$w\left(x,y,z\right)={\left({x}^{2}+{y}^{2}+{z}^{2}\right)}^{\text{−}1\text{/}2}$

Harmonic

If $\text{F}\left(x,y,z\right)=2\text{i}+2x\text{j}+3y\text{k}$ and $\text{G}\left(x,y,z\right)=x\text{i}-y\text{j}+z\text{k},$ find $\text{curl}\phantom{\rule{0.2em}{0ex}}\left(\text{F}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{G}\right).$

If $\text{F}\left(x,y,z\right)=2\text{i}+2x\text{j}+3y\text{k}$ and $\text{G}\left(x,y,z\right)=x\text{i}-y\text{j}+z\text{k},$ find $\text{div}\phantom{\rule{0.2em}{0ex}}\left(\text{F}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{G}\right).$

$\text{div}\phantom{\rule{0.2em}{0ex}}\left(\text{F}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{G}\right)=2z+3x$

Find $\text{div}\phantom{\rule{0.2em}{0ex}}\text{F},$ given that $\text{F}=\text{∇}f,$ where $f\left(x,y,z\right)=x{y}^{3}{z}^{2}.$

Find the divergence of F for vector field $\text{F}\left(x,y,z\right)=\left({y}^{2}+{z}^{2}\right)\left(x+y\right)\text{i}+\left({z}^{2}+{x}^{2}\right)\left(y+z\right)\text{j}+\left({x}^{2}+{y}^{2}\right)\left(z+x\right)\text{k}.$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=2{r}^{2}$

Find the divergence of F for vector field $\text{F}\left(x,y,z\right)={f}_{1}\left(y,z\right)\text{i}+{f}_{2}\left(x,z\right)\text{j}+{f}_{3}\left(x,y\right)\text{k}.$

For the following exercises, use $r=|\text{r}|$ and $\text{r}=\left(x,y,z\right).$

Find the $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{r}.$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{r}=0$

Find the $\text{curl}\phantom{\rule{0.2em}{0ex}}\frac{\text{r}}{r}.$

Find the $\text{curl}\phantom{\rule{0.2em}{0ex}}\frac{\text{r}}{{r}^{3}}.$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\frac{\text{r}}{{r}^{3}}=0$

Let $\text{F}\left(x,y\right)=\frac{\text{−}y\text{i}+x\text{j}}{{x}^{2}+{y}^{2}},$ where F is defined on $\left\{\left(x,y\right)\in ℝ|\left(x,y\right)\ne \left(0,0\right)\right\}.$ Find $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}.$

For the following exercises, use a computer algebra system to find the curl of the given vector fields.

[T] $\text{F}\left(x,y,z\right)=\text{arctan}\left(\frac{x}{y}\right)\text{i}+\text{ln}\sqrt{{x}^{2}+{y}^{2}}\text{j}+\text{k}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\frac{2x}{{x}^{2}+{y}^{2}}\text{k}$

[T] $\text{F}\left(x,y,z\right)=\text{sin}\left(x-y\right)\text{i}+\text{sin}\left(y-z\right)\text{j}+\text{sin}\left(z-x\right)\text{k}$

For the following exercises, find the divergence of F at the given point.

$\text{F}\left(x,y,z\right)=\text{i}+\text{j}+\text{k}$ at $\left(2,-1,3\right)$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=0$

$\text{F}\left(x,y,z\right)=xyz\text{i}+y\text{j}+z\text{k}$ at $\left(1,2,3\right)$

$\text{F}\left(x,y,z\right)={e}^{\text{−}xy}\text{i}+{e}^{xz}\text{j}+{e}^{yz}\text{k}$ at $\left(3,2,0\right)$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=2-2{e}^{-6}$

$\text{F}\left(x,y,z\right)=xyz\text{i}+y\text{j}+z\text{k}$ at (1, 2, 1)

$\text{F}\left(x,y,z\right)={e}^{x}\text{sin}\phantom{\rule{0.1em}{0ex}}y\text{i}-{e}^{x}\text{cos}\phantom{\rule{0.1em}{0ex}}y\text{j}$ at (0, 0, 3)

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=0$

For the following exercises, find the curl of F at the given point.

$\text{F}\left(x,y,z\right)=\text{i}+\text{j}+\text{k}$ at $\left(2,-1,3\right)$

$\text{F}\left(x,y,z\right)=xyz\text{i}+y\text{j}+x\text{k}$ at $\left(1,2,3\right)$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=\text{j}-3\text{k}$

$\text{F}\left(x,y,z\right)={e}^{\text{−}xy}\text{i}+{e}^{xz}\text{j}+{e}^{yz}\text{k}$ at (3, 2, 0)

$\text{F}\left(x,y,z\right)=xyz\text{i}+y\text{j}+z\text{k}$ at (1, 2, 1)

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=2\text{j}-\text{k}$

$\text{F}\left(x,y,z\right)={e}^{x}\text{sin}\phantom{\rule{0.1em}{0ex}}y\text{i}-{e}^{x}\text{cos}\phantom{\rule{0.1em}{0ex}}y\text{j}$ at (0, 0, 3)

Let $\text{F}\left(x,y,z\right)=\left(3{x}^{2}y+az\right)\text{i}+{x}^{3}\text{j}+\left(3x+3{z}^{2}\right)\text{k}.$ For what value of a is F conservative?

$a=3$

Given vector field $\text{F}\left(x,y\right)=\frac{1}{{x}^{2}+{y}^{2}}\left(\text{−}y,x\right)$ on domain $D=\frac{{ℝ}^{2}}{\left\{\left(0,0\right)\right\}}=\left\{\left(x,y\right)\in {ℝ}^{2}|\left(x,y\right)\ne \left(0,0\right)\right\},$ is F conservative?

Given vector field $\text{F}\left(x,y\right)=\frac{1}{{x}^{2}+{y}^{2}}\left(x,y\right)$ on domain $D=\frac{{ℝ}^{2}}{\left\{\left(0,0\right)\right\}},$ is F conservative?

F is conservative.

Find the work done by force field $\text{F}\left(x,y\right)={e}^{\text{−}y}\text{i}-x{e}^{\text{−}y}\text{j}$ in moving an object from P (0, 1) to Q (2, 0). Is the force field conservative?

Compute divergence $\text{F}=\left(\text{sinh}\phantom{\rule{0.2em}{0ex}}x\right)\text{i}+\left(\text{cosh}\phantom{\rule{0.2em}{0ex}}y\right)\text{j}-xyz\text{k}.$

$\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}=\text{cosh}\phantom{\rule{0.2em}{0ex}}x+\text{sinh}\phantom{\rule{0.2em}{0ex}}y-xy$

Compute curl $\text{F}=\left(\text{sinh}\phantom{\rule{0.2em}{0ex}}x\right)\text{i}+\left(\text{cosh}\phantom{\rule{0.2em}{0ex}}y\right)\text{j}-xyz\text{k}.$

For the following exercises, consider a rigid body that is rotating about the x -axis counterclockwise with constant angular velocity $\omega =⟨a,b,c⟩.$ If P is a point in the body located at $\text{r}=x\text{i}+y\text{j}+z\text{k}\text{,}$ the velocity at P is given by vector field $\text{F}=\omega \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{r}.$

Express F in terms of i , j , and k vectors.

$\left(bz-cy\right)\text{i}\left(cx-az\right)\text{j}+\left(ay-bx\right)\text{k}$

Find $\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}.$

Find $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=2\omega$

In the following exercises, suppose that $\nabla ·\text{F}=0$ and $\nabla ·\text{G}=0.$

Does $\text{F}+\text{G}$ necessarily have zero divergence?

Does $\text{F}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{G}$ necessarily have zero divergence?

$\text{F}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{G}$ does not have zero divergence.

In the following exercises, suppose a solid object in ${ℝ}^{3}$ has a temperature distribution given by $T\left(x,y,z\right).$ The heat flow vector field in the object is $\text{F}=\text{−}k\text{∇}T,$ where $k>0$ is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is $\nabla ·\text{F}=\text{−}k\nabla ·\text{∇}T=\text{−}k{\nabla }^{2}T.$

Compute the heat flow vector field.

Compute the divergence.

$\nabla ·\text{F}=-200k\left[1+2\left({x}^{2}+{y}^{2}+{z}^{2}\right)\right]{e}^{\text{−}{x}^{2}+{y}^{2}+{z}^{2}}$

[T] Consider rotational velocity field $\text{v}=⟨0,10z,-10y⟩.$ If a paddlewheel is placed in plane $x+y+z=1$ with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

#### Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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