# 6.7 Stokes’ theorem

 Page 7 / 10

Calculate the curl of electric field E if the corresponding magnetic field is $\text{B}\left(t\right)=⟨tx,ty,-2tz⟩,0\le t<\infty .$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{E}=⟨x,y,-2z⟩$

Notice that the curl of the electric field does not change over time, although the magnetic field does change over time.

## Key concepts

• Stokes’ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes’ theorem is a higher dimensional version of Green’s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.
• Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.
• Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary C .
• Faraday’s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes’ theorem can be used to derive Faraday’s law.

## Key equations

• Stokes’ theorem
${\int }_{C}\text{F}·d\text{r}={\iint }_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S}$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}$ over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

$\text{F}\left(x,y,z\right)={y}^{2}\text{i}+{z}^{2}\text{j}+{x}^{2}\text{k}\text{;}$ S is the first-octant portion of plane $x+y+z=1.$

$\text{F}\left(x,y,z\right)=z\text{i}+x\text{j}+y\text{k}\text{;}$ S is hemisphere $z={\left({a}^{2}-{x}^{2}-{y}^{2}\right)}^{1\text{/}2}.$

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=\pi {a}^{2}$

$\text{F}\left(x,y,z\right)={y}^{2}\text{i}+2x\text{j}+5\text{k}\text{;}$ S is hemisphere $z={\left(4-{x}^{2}-{y}^{2}\right)}^{1\text{/}2}.$

$\text{F}\left(x,y,z\right)=z\text{i}+2x\text{j}+3y\text{k}\text{;}$ S is upper hemisphere $z=\sqrt{9-{x}^{2}-{y}^{2}}.$

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=18\pi$

$\text{F}\left(x,y,z\right)=\left(x+2z\right)\text{i}+\left(y-x\right)\text{j}+\left(z-y\right)\text{k}\text{;}$ S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

$\text{F}\left(x,y,z\right)=2y\text{i}-6z\text{j}+3x\text{k}\text{;}$ S is a portion of paraboloid $z=4-{x}^{2}-{y}^{2}$ and is above the xy -plane.

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=-8\pi$

For the following exercises, use Stokes’ theorem to evaluate ${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS$ for the vector fields and surface.

$\text{F}\left(x,y,z\right)=xy\text{i}-z\text{j}$ and S is the surface of the cube $0\le x\le 1,0\le y\le 1,0\le z\le 1,$ except for the face where $z=0,$ and using the outward unit normal vector.

$\text{F}\left(x,y,z\right)=xy\text{i}+{x}^{2}\text{j}+{z}^{2}\text{k}\text{;}$ and C is the intersection of paraboloid $z={x}^{2}+{y}^{2}$ and plane $z=y,$ and using the outward normal vector.

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=0$

$\text{F}\left(x,y,z\right)=4y\text{i}+z\text{j}+2y\text{k}$ and C is the intersection of sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ with plane $z=0,$ and using the outward normal vector

Use Stokes’ theorem to evaluate $\underset{C}{\int }\left[2x{y}^{2}zdx+2{x}^{2}yzdy+\left({x}^{2}{y}^{2}-2z\right)dz\right],$ where C is the curve given by $x=\text{cos}\phantom{\rule{0.2em}{0ex}}t,y=\text{sin}\phantom{\rule{0.2em}{0ex}}t,z=\text{sin}\phantom{\rule{0.2em}{0ex}}t,0\le t\le 2\pi ,$ traversed in the direction of increasing t .

${\int }_{C}\text{F}·d\text{S}=0$

[T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral $\underset{C}{\int }\left(ydx+zdy+xdz\right),$ where C is the intersection of plane $x+y=2$ and surface ${x}^{2}+{y}^{2}+{z}^{2}=2\left(x+y\right),$ traversed counterclockwise viewed from the origin.

[T] Use a CAS and Stokes’ theorem to approximate line integral $\underset{C}{\int }\left(3ydx+2zdy-5xdz\right),$ where C is the intersection of the xy -plane and hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}},$ traversed counterclockwise viewed from the top—that is, from the positive z -axis toward the xy -plane.

${\int }_{C}\text{F}·d\text{S}=-9.4248$

[T] Use a CAS and Stokes’ theorem to approximate line integral $\underset{C}{\int }\left[\left(1+y\right)zdx+\left(1+z\right)xdy+\left(1+x\right)ydz\right],$ where C is a triangle with vertices $\left(1,0,0\right),$ $\left(0,1,0\right),$ and $\left(0,0,1\right)$ oriented counterclockwise.

Use Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S},$ where $\text{F}\left(x,y,z\right)={e}^{xy}\text{cos}\phantom{\rule{0.2em}{0ex}}z\text{i}+{x}^{2}z\text{j}+xy\text{k},$ and S is half of sphere $x=\sqrt{1-{y}^{2}-{z}^{2}},$ oriented out toward the positive x -axis.

$\underset{S}{\iint }\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S}=0$

suppose you are at a dinner party with your friend.The fork, spoon, knife and glasses all produce noisy sounds.What psychological factors influence the discussion between you and your friend?
It may give me an impression to the other person that the person who makes the noise is nervous or either ill-mannered.
Indefatigable
I would think that people at the table are not comfortable which could be because of the environment or the topic of discussion or the thoughts working in their minds.
Ritu
depending how loud and frequently the sound are made the discussion can change
David
why functinalism?
which question is important in this chapter
what is the difference between retention and application and what can be done to improve the connection between the two
Clues should be developed and associations be formed
philip
what is the study of personality?
psychology
Racheal
individual traits
Utkarsh
Personality psychology
Riya
personality psychology is a branch of psychology that studies personality and its variation among individuals
Swapnil
what are the basic statistical concepts in psychological research ?
mean ,median ,NPC ,t ,z test
Soumyashree
what is the NPC?
Maen median mood stander daviation
Janat
Percentile also
Janat
NPC stans for normal probability curve
Janat
Normal probability curve (NPC)
Janat
yez
Soumyashree
Yhn koi b questione kr skta h?
Janat
is there any scale for measuring post COVID Anxiety?
I'd like to know if there's any?
Zino
I've heard about Coronavirus Anxiety Scale (CAS). It works as a mental health screener but I'm not sure anyone can do the test except psychologist
divani
Thanks !
Zino
no
Hausa
Thanks Divani zahra
Mohd
explain the function how intelligence and learning function together
Learning comes before intelligence Learning could be the process of acquiring knowledge. Or the ability to gain knowledge Whereas intelligence is the display of acquired knowledge.
Felix
I love it
Teena
Jade Baccus, this is kind of hard "to me" it is the root of living altogether, without both how can we live in a world like this? how can we make new discoveries?
intelligence: can be nourisht and remain indipendent from how much knowledge we aquire. But can be displayed thru use and application of knowledge itself
fabio
Hey, is there any classes by psychology grp
Smitha
Hlw Every0ne
Khalilur
learning can be work together with intelligence like a partner with the thinking of life
James
schools of psychology
hi
giri
hi
Swapnil
hi
Munir
hello
Imsekar
hai
Tu
Hi
piya
hello
kajal
Hello
Mohamed
hii
grace
hi
george
hi
Anuja
hi
kajal
gud night
grace
Good night everyone
Anuja
good night 😴🌃
Swapnil
hello
Ananya
What's the point of this chat section?
Mohamed
to say hello and to feel solidarity.
Tu
the early childhood can reference the emotions?
Our early childhood experiences are important in the development of our personalities. It kind of shapes our behaviour and mannerism.
Debarpita
Especially in choosing career
Felix
😅Well that's complicated. It differs from person to person.
Debarpita
The early childhood or let's say your upbringing plays a vital role in driving your emotions. If one did not have a pleasant childhood it's highly likely they are more vulnerable to depression or some other psychological condition.
Indefatigable
how to change the emotion?
change the thought at hand .
Shawnte
Then I'm assuming the mind will begin to let go of the prior emotion.
Shawnte
Yes right. Think about something else, then emotion will follow.
Annie
then join life
Munir
Munir
Emotions run hand in hand with the mind and environment. A person can be sad or angry on a certain things going on in their mind then something can happen in the environment around them that m adds keys them happy.
Douglas
how has the new legalization of marijuana effected the anti drug programs
how to change the emotional ?
Sammi
I am wrilly interisting about big problems psihycil nature between famillys/like parents bully some childe physically , even hall life and show some kinde of heath and \$inaid
same
Gaea
Never give minor issues space. Time creates space, more time more space therefore if u give a small issue more time it gets more space in your mind hence breeding stress
yes yes yes so true
Gaea
hi..my brother gets angry Everytime I say no to him for anything he ask for he don't talk to me after that.. Everytime I only have to say sorry so this time I decided not to say sorry he haven't talked to me for two weeks I am so stressed what should I do
Nush
give him so time. I think you should let him figure out his own feelings. plus, you did the right thing. he needs to know that not everyone will give him what he wants.
Gaea
Don't be stressed out . You did right thing. but remember one thing don't show angry gestures to him or don't try to make him realize by teasing or scolding. If you try to be humble and patient, he may understand his mistake on his own.
PARUL
yeah
Gaea
Hi everyone! So i'm gonna be a college freshman soon and i'm gonna major in Psychology😬 but lately i've been worrying about what job i'm gonna have after getting my Bachelor's degree. Any tips or advice?
I think minimum of a masters degree is needed in order to get a job once you do mphil and phd u gonna get licence to work as a clinical psychologist and a lot more options 😊best wishes
kitty
hi...my brother gets angry with me anytime I say no to him for anything he ask for and stop talking to me every time I only have to say sorry..so I decided this time not to say sorry he haven't talked to me at all what should I do please help me
Nush
I think it's just a matter of ego and pride.they may have a kind of feeling like if they apologize it hurts their integrity and pride. Some people never admit they are wrong and it creates stress in the other person. .some people are not empathetic and they always want to feel cared n lovedbyother
kitty
so what can we do..if you are a person who always says sorry it's a good indicator that you have such a broad mind where you are ready to solve problems n you love them even though they are bad at you..basically everyone in this whole world is selfish..humans are selfish but there are some people
kitty
like you whom God created just to solve problems and lead other people in a better way..what you should do is create a beautiful bonding bw your bro ask him softly about his problems..love him more..and eventually he may become softer and lovable
kitty
if your bro is an introvert there could be chances where he has got several problems he haven't discussed with anybody which brings out their egoistic behr and blunt nature...ask him to say about his problems..make him understand the importance of apologies and empathetic behr..and try not to end up
kitty
in a fight discussing these things...u should softly deal with it..even if other people let you down or didn't say sorry to you doesn't mean that you r wrong..u are saying sorry everytime because u have a good heart..
kitty
hope this helps🐒have a good day..good life!!😊
kitty
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?