Line integrals and vector fields | Multivariable Calculus | Khan Academy

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Using line integrals to find the work done on a particle moving through a vector field
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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.
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[@turbothrottletrouble4217]

You know why I like Sal, he assumes you know nothing, then he makes you feel like a genius because he makes you think about what's going on rather than having to rotely memorise everything that he says. Never quit Sal, this is God's work you are doing.

[@barracuda7361]

Yess i feel like a genius..

[@howmathematicianscreatemat9226]

"The only really valuable thing is intuition" [Albert Einstein]. Most professors don't seem to understand it but Mr. Khan Academy teacher does.

[@Monster-cv3pz]

Yeah exactly

[@mohithalder3169]

His name is Sal, the other person in this series is Mr. Sanderson

[@Monster-cv3pz]

@@mohithalder3169 cool!

[@cruiserkid1]

You sir are a legend....

[@avdeshkumar8500]

I Know that... NO NEED TO TELL ME.

[@Amanpreetkaur-cp4fs]

You sir are a legend ❌ Sir you are a legend ✔️

[@cruiserkid1]

@@Amanpreetkaur-cp4fs how can she slap??!!

[@Amanpreetkaur-cp4fs]

@@cruiserkid1 what I can't figure out what are you saying

[@Amanpreetkaur-cp4fs]

@@cruiserkid1 what

[@mariannejasmijn8656]

You explained this is a 20 minute video so much better than my 1,170 page textbook

[@積分定数C-o5n]

thank you so much. you are saving my life

[@BoutinMathieu]

Sal is by far my favorite teacher! :D

[@dekippiesip]

Good job, the khan academy needs to expand and reach a wider public btw! It has to get as big as wikipedia or facebook, that would truely revolutionize education. Btw, why use i,j,k notation for vectors? Representing vectors as rows or columns is much more convenient, at least for me. And it allows you to work in more than 3 dimensions!

[@passivlyassertive9696]

Thank you so much for these videos, it's absolutely incredible how intuitive this is, and to think you do this for almost every level of every subject....

[@jakeaudi297]

Sal i like the way u talk to us, u talk like a BUDDY or a friend not as a teacher or a tutor. I felt like i was listening to a friend of mine which makes it easy!

[@brace99]

you’re the best Sal, you’re the only reason I haven’t failed my physics degree so far

[@KuldeepSrivastava-w7k]

What area/volume can we think of calculating here? Like in the case of 2d->1d we saw the area of wall was what line integral eventually meant. In case of vector fields, or let's say precisely 2D->2D here, which is very similar to a transformation, which area will the line integral give??? Is it going to be the determinant related? Like let's sat A mapped to A' and B mapped to B', then line integral gives us AA'BB'?? I'm looking to build an intuition over this.

[@bschuske]

Sal the world is better off with you in it man. Thank you for all your videos.

[@Rajbhandari88]

Because of you I'm loving maths more! Thank you very much!

[@JaredBGoode1]

This was much more concise than my professor's explanations. I appreciate it a lot. I was really worried about my quiz tomorrow.

[@silverchair28]

whooaaa omg everything just clicked for me! thanks so much!

[@marcoardanese6013]

the best explanation on youtube.

[@lorettali3842]

Can’t believe this is much more clear than my university’s lecture…

[@AntonioMudge]

you are so funny dude. you're great! your videos touches a little of everything watching you is funny, entertaining and educational!

[@benedictejelonu]

I love when sal attempts to say "parameterize" at the first go ... Always cracks me up... This is 2021 , 10 + years later and your videos are still relevant sal. Thank you ❤

[@donthularuchitha7552]

This one...prepared fr me.. 11 years ago

[@316Darius]

capital N for Newtons

[@blackwolfkills]

Great video ....simple enough to understand detailed enough for indepth knowledge.

[@S1CKDRIFT3R]

you are the greatest human being alive!

[@YassinMohamedTV]

finally i understand .. thank you sal

[@joycewang5859]

Hi, thank you ! love your courses ! I have a small question , hope you can help me . At 8:00 in the video , the graph is a x-y plan . but why not a i - j plan ? I see the x and y is only parameter while the axel sould be i and j ? Here the x and y is like parameter 'a' and 'b' in a f (a, b) funktion?

[@jaelee9406]

Thank you so much Sal :)!

[@jasonferguson921]

Will you be producing videos on abstract algebra on subjects like rings, groups and fields :)

[@1992Dadi]

Thanks for the clear explaination .....

[@georgeanons301]

Calculus 3 exam tomorrow. Sal is my only hope.

[@jessechen4971]

Great vid! Tho it would be best if there was audio on both L and R channels =)

[@jessechen4971]

***** funny, it's stereo for me too now

[@sajidkhan-th4mr]

Excellent Mr. English man

[@joangonzalvez9865]

this video is just awesome.

[@joshuasegal4161]

Amazing Video!!! Really understand this now!

[@NehadHirmiz]

Thank you for these wonderful tutorials

[@kavoos1000]

you are great, i love your vids, simple and very nice for review and to learn, thank you very very much for posting them ive learned a lot .. god bless you

[@s.fofandi]

10:34 ENHANCE!!!!

[@ScilexGuitar]

Thank you so much, now I actually understand what Im integrating!

[@hobihobi1563]

i can't thank you enough for this

[@weiyangli9619]

This video is simply awesome man Thank you very much

[@cruzcastillo6984]

I'm going to think of line integrales in terms of work

[@levisim997]

Thanks ! Great video

[@ManBearPigMike]

GREAT VIDEO T.T you're a life saver

[@cristubek2416]

Warning: Line integral playlist is totally messed up (updated @ 10/16/2019)

[@Topclasscricket625]

Right

[@comprehensiveboy]

I'm just off to the gym. As a vector field gravity is a biatch! So much work moving around in it. Even sideways, or standing still. :)

[@xTheLuckySe7en]

comprehensiveboy Well, no work is done if force and displacement are perpendicular to each other "moving sideways" or if there is no displacement at all "standing still". ;)

[@Waranle]

Sweet, sweet... Thank you Sal

[@cdkilo77]

I HATE WORK! Great video as always!

[@adosar7261]

when we represent a vector field F(x,y)=(x,y) is the domain of F the plane x-y or it can be also time-temperature ?

[@powernap1234]

So we can use the line integral to find how much a curve and a vectorfield align and to find the arae under a function in R3 along a curve? Thaanks

[@Lun4812]

So if *dr* is a small change in *r* that is tangent to *r* then what is *dr/dt* ? I mean, isn't the derivative the tangent already? Why separate the *dr*?

[@daomowon7784]

dr/dt is the sloape of the tangent line of the vector curve r. dr is the infinitesimal change in r and dt is the infinitesimal change in t. Because they are two different variables, in order to integrate they need to separated. (integral a dr = integral b dt)

[@ben_swain]

daomo won r is the vector at each instant, and t is just some independent variable like time. I don't see how dr/dt would give the slope of the tangent line to the vector curve r. I'm confusing myself trying to think about it.

[@kunal_chand]

As far a i can understand from the video , dr is the tangent to the curve ( a infinitesimally small vector along the direction of tangent ) and by dividing with dt ( a very small value ) , we made the magnitude of the vector larger . so the just like Sal said, the velocity increases. That is what he gave a explanation to visualise what dr/dt is , but dr/dt is slope of the tangent line of vector curve r ( said by other people ) represented in form of green color long vector . This is the thing what i interpreted , i can be wrong .

[@satarneeraj9044]

excellent video collection

[@kiley1672]

Thank you so much for these videos they are SOOOOO helpful and insightful. They help me understand the foundation of the math I'm using!! Amazing work

[@infernoj2]

2-D space

[@GuyAffricot]

You rocks so hard Sal.

[@kunal_chand]

For Calculating the Work done , why do we take the line Integral ? Wont the definite integral do the work ? And how a line Integral can be equal to a definite integral ?

[@purplefire5]

1. so what's the difference between an equation modelling a vector field f(x,y) and a vector position function (r) they both give you vector why is there distinction between the two are there any differences? 2. I was also wondering why wer'e suddenly using dr to represent the small interval of path length along the curve, why can't we use ds isn't that what we've always used to represent small intervals along a curve, what's the use of bringing a vector dr into the equation, especially since it's linear, won't it be less accurate? Any help would be appreciated guys! (srry if the questions are silly i'm lacking in my intuition of all of this)

[@michael_zhou]

I'm a bit late. A vector field has a vector, (visualised as an arrow) for every point on the xy plane (thus vector *field*) where as a vector position function traces out a path, or a curve (thus *position*). You can also visualise a vector field as wind or water currents, at each point on the 'field', there is an amount of water (magnetude) moving in a direction. But there is always a vector to represent that movement for *every* point on that field (even if there's no movement ). ds is scalar and dr is a vector. Because these curves are vectors it makes more sense to use vectors, they indicate *direction*. ds and dr are both linear because an infinitesimal part of a curve, scalar or vector, is a line segment. Please correct me if I am wrong. Thanks.

[@yjay7789]

idk why this feels like a physics lesson

[@kananabbasov744]

I couldn't understand why don't we multiply the F to cos from previous video I remember the formula W=dr*F cos(eta)

[@joluju2375]

It is explained in the previous video that the dot product does exactly the same job, and it's cool because you don't even need to know theta.

[@markmitchell6098]

Did you just forget the unit vectors in that last integral you wrote down or is there some reason you can leave them out?

[@yichizhang795]

super good video!

[@vladibudnitski3790]

Thank you!

[@ee4life623]

Difference between vector valued functions and vector fields, please? Are vector valued functions only defined through a parameter "t" and vector fields multivariable functions, but both have a vector result? Please help and thanks.

[@andrewolesen8773]

I hope I'm not too late, but a vector valued function is any function that has a vector input and output. A vector field is one way of interpreting the results of a function with a vector output. That being said a vector valued input is not required for a vector valued output.

[@harishli2020]

thank u.............

[@杨峰-z7m]

Could you tell me what kind of software do you use to record the vedio? It's instersting. Thank you for your vedio.

[@prateeksharaikar7273]

good one

[@thespectre4914]

Man, line intergrals were giving area, but now there is no surface so how do we calculate the area?

[@NSBeverything]

which series is this...i want to watch it full...is it multivariable calculus?

[@sgut1947]

Yes

[@orzHaiz]

thank you so much~

[@pianoman47]

What happened to i and j at the end? They weren't included in the final integral expression.

[@boblacolle]

i*i = 1 and j*j =1. Multiplicating a unit vector with itself = 1

[@rahibrehman4245]

What if the vector field was i + j + k what would the graph look like ?

[@265HITMAN265]

Absolutely impressive :P

[@wongtsh]

all multical professor can just go home, this video is enough

[@MultiLomp]

Which program do you use for writing these slides? :))

[@lucasm4299]

Why do you mention line integrals? I keep thinking of a "curtain", but I don't see it.

[@andrewolesen8773]

A curtain wouldn't make sense in this case. There is a two dimensional input and a two dimensional output, the curtain worked with a two dimensional input and one dimensional output. When he says line integral I think he is referring to the total arc length, which is the single variable analog of line integrals.

[@amk5431]

may the force be with you!

[@alriashi7]

it means 2 Dimensions

[@shr28]

awesome one again :D

[@AhmedMohamed-gh1cx]

.please add the arabic subtitle to benefit more from the explaining :)

[@andrerossa8553]

tks

[@djgamer457]

what software and hardware do you use? want to use this in my classroom :)

[@zuesr3277]

It's Artrage.

[@someoneeternal7575]

Great!!!

[@peanutbuttersmooth4849]

never better

[@KLKCan]

Lets use 'N' instead of 'n'........ Unit is in the honor of Newton

[@intellectelite]

your awesome!

[@NASIMBINJASIM]

great !!!!!!! @khanacademy @salman

[@oceanview3165]

f(x,y) doesn't mean z axis? then why he was saying that the "vector field" in the x y plane ?

[@shabadabba]

fahim arko no because your output is a 2D vector

[@boblacolle]

x y plane where z stays constant. Function doesnt change in z.

[@tag_of_frank]

So these are not position vectors

[@ben_swain]

I'm confused about this: dr = x'dt + y'dt If x' = dx/dt, and y' = dy/dt, couldn't you say dr = dx + dy? and I thought r = < x(t) , y(t) >. I must have missed something

[@sebastiankarlsson4355]

+Ben Swain: You're reasoning are correct but why would you wanna see dr as "dx + dy" in this case? You want to integrate over dt :)

[@uttamcp5016]

I think I can help you with that. When you say that dr=x'dt+y'dt you are forgetting to consider dr as a vector. You can definitely say dr=dx+dy but with a small correction vector dr = dx(i)+dy(j). This way, its magnitude would be Magnitude(dr)=square root (dx^2+dy^2) In this case, he has considered it to be vector dr which essentially means vector dx in x direction + vector dy in y direction. But you can't just add them up as you have mentioend because they are vectors and not scalars. However, since the components dx and dy are perpendicular to each other, you can view it as a Pythogoras theorem problem and find its magnitude. Hope it helps

[@June28July]

*Puts noose hanging from my roof away*

[@nirajkumarmishra296]

*awesome*

[@kirkgardner2879]

Sal... parameterized... ;)

[@yli5531]

I wish he could have stopped using the ijs for a vector. Not every audience is that into physics notation.

[@zuhaibrashidbangroo5413]

Hello sal how to u?

[@RAVEN-rw7mo]

I'm the 100th Comment. After 9 yrs that is👀💯💯😅😂

[@ahuja56]

Hmmmmm

[@GreatWonderMoose]

I seem to be the only one who really doesn't like Sal's videos. He rambles so much, and it's tough to keep track of what could otherwise be a simple explanation of a concept. We could have been given the problem: "A block of ice is pulled with 10 Newtons of force and displaced 5 meters across a surface at a 60-degree angle. What is the work in this case?" But instead we got: "The classic example: maybe you have an ice cube (or some type of block; I just say ice so that there's not a lot of friction), maybe it's standing on a bigger sheet or lake or ice or something, and maybe you're pulling on that ice cube at an angle - let's say you're pulling at an angle like that. That is my force right there; let's say my force is equal to, well, tha- that's my force vector, let's say the magnitude of my force vector, so I'll say the magnitude (so let me put two brackets around there) the magnitude of my force vector, we could say, is, let's say it's 10 Newtons, and the direction of my force vector - right, any vector has to have a magnitude and a direction - and the direction, lets say it's 30- it has a 30-degree angle - let's say it's 60-degree angle above horizontal. So that's the direction I'm pulling in, and let's say I displace it - let's say I displa- this is all a bit of review, hopefully. If you're displacing it, let's say you displace it 5 Newtons. So let's say the displacement - that's the displacement vector right there and the magnitude of it is equal to 5 - sorry, not 5 Newtons - 5 meters."

[@anonymoustraveller2254]

WondrousMoose you are the only oneeee.cause khan saved my life.

[@manoor6391]

I am agree by somehow , it becomes biring sometimes and diffucult to concentrate .

[@murphyec]

LMAO

[@alial-musawi9898]

No one is forcing you to come here.

[@kruthikpimpre5921]

I partially agree

[@vivekchilakala2586]

THANK YOU SO MUCH