Introduction to the surface integral | Multivariable Calculus | Khan Academy

[@broudwauy]

Awesome explanation. Remembering that the cross product was the area of the parallelogram defined by the two vectors was an absolute revelation. I had absolutely no idea why the fundamental vector product showed up in surface integrals.

[@raydencreed1524]

Inconspicuous Bear Wrestler If you want something “fundamental”, then the cross product is not a good choice. It’s defined only in three dimensions. The wedge product, on the other hand, is a much more natural form of the cross product and can generalize to any number of dimensions you want!

[@whoisbobby3]

Khan, you are literally the reason I went from cal 1- to linear and diff. You are the best!

[@DiomedesStrosMkai]

That's why Khan is awesome.. A lot of the time my frustration comes when the book just skips a step and leaves me thinking '"what the hell?". Usually it's explained somewhere, but it's always off the cuff or several chapters back. This way it's *thoroughly* explained.. I'm not missing *anything*. Every piece of the puzzle is shown

[@1Man2Go]

dude..i do not know how much you're getting paid for this.. but it's not enough.. how you explained this is outright brilliant. something my teachers always fall short of.

[@malefetsanekoalane4549]

I have finally figured out what makes your explanations so great. You answer questions before they are asked. You replace computational ability with deep understanding,which is education for life. May you be spared for other generations.

[@DavideSLiuni]

This man is the only reason of why I'm passing calculus in university

[@rheejoan]

"now lets go do this for t!!!...i am running out of colors..." Love this man

[@gianlucacastro5281]

That was the only thorough explanation I could find as to way the surface integral takes this cross product the way it does, thank you!!

[@HotPepperLala]

God seriously needs to create more people like you

[@vukstojiljkovic7181]

Thank you for this. They never did something like this, they just introduced it and started calculating.....

[@wontpower]

The cross product in the integral reminds me of the Jacobian. The area of dsdt doesn't quite match up with the surface area of the differential surface area in x y and z, so you need to multiply by abs(dr/ds x dr/dt) in order to account for that difference.

[@vaggs75]

Okay short explaination. If we take the double integral of a function that is dependent on three variables, then we get the volume of a shape. This shape is a 3d shape. the bottom of the shape is the projection of the surface onto the zy plane, just like if we had a torch and lit it from exactly above. the top of the shape are the "hills that are formed" from the values of the fucntions. Now what the surface integral calculates is the bottom of the same shape, but with a twist. the top stays the same but the bottom becomes the projection onto another surface that is not the zy plane but rather maybe a diagonal surface or something of that sort.

[@hakuchaku27]

this must be longest video of khan academy

[@oyster4545]

🥰🥰🥰Sal you deserve noble prize ❤❤

[@legendarysannin65]

Please connect your videos with a link on the screen so we can jump from the video before. Very good explanation!

[@sandracordoba6090]

You are indeed a brilliant person when it comes to explain this concepts. All my respect

[@FlippyBrown]

Sir, am I right to say that every single point on the region is associated to a parallelogram. And we are adding all the parallelograms(its area) associated to every single point on the region, in order to find the surface area?

[@raydencreed1524]

Nicholas That sounds right to me :)

[@fernandolk4536]

This can be used to explain the animal transmutation and the underlying rationale of why the Nazi killed specifically.

[@pedrogaleano6722]

Man how come 39 people not like this video? Thank you very much Sal!

[@oneinabillion654]

Sal, I remember in linear algebra u said that. U will be remembered.

[@mchegrmchegr290]

Very informative. Simply excellent.

[@TennisGvy]

@Liaomiao That's if you are integrating a function in two directions. Integrating a function adds another dimension so to speak.

[@gotnerdy]

Thanks for this and also making the video downloadable. Also what software/hardware are you using for this?

[@jitulborah_10]

the 3d reference khan made is beautiful . 😊

[@moszczynskieng]

Another fantastic vector calculus video from Sal Khan! Many thanks!

[@haroldh6678]

Man... just amazing... just amazing

[@ryanmonte]

Yup. It's in a 2nd year calculus class at my university.

[@mrbrohere]

What are doing after university?😀

[@VinothKumar-qo7ry]

U are one f the excellent Proff.. I seen

[@Barak314159]

You sir are amazing! You teach so well and in so many topics and I can just thank you infinite amount of times!

[@lucidmath5481]

Sal can you please do some vids on intuitively explaining the volume integral?

[@zh1412]

wow thanks for helping me with the fundamentals surface integrals:)

[@Liaomiao]

why are we doing a double integral? all the vid saids was that we're going in 2 directions. But all the other times we've used double integrals is to find a volume under a surface...

[@anmolabhayjain9721]

Can anyone provide me with an explanation as to why during bijective transformation the boundary surface of one closed curve forms the boundary surface of the other closed curve or closed volume? That is why the boundary in u-v plane is also the boundary in x-y(or x,y,z for closed volume) plane of the transformed curve. Also assume that the transformation has a continuous first partial derivative over u and v.

[@ShashankShekhar-de4ld]

thanks for giving this great intuition

[@山卂丂丂山卂]

Khan academy is the best

[@MatthewFricke]

@abhi99ps people use different notations for magnitude

[@xandersafrunek2151]

This series of videos is highly underrated.

[@bangaloremathematicalinsti5351]

One of the fantastic Explanation, Kudos to the Professor :)

[@jingxie3126]

which one is "the last video" referred at the beginning?

[@ausaramun]

Parial Derivatives of vector-valued functions. (I know...a little late...)

[@anonymous_4276]

@@ausaramun yeah. It's a "little" late.

[@sakki3378]

@@ausaramun just a little lol

[@yvonneho876]

@@sakki3378 a tiny little bit

[@agrajyadav2951]

@@yvonneho876 miniscule (btw this reply might be a teeny tiny bit late as well)

[@haidynagi7314]

Solve integration of. X^2/(x^4 +x^3 -1)

[@bladox971]

good luck guys

[@Sam-dc9bg]

Thank you! Wow this was a good introduction.

[@intellagent7622]

makes sense now thanks

[@GlorifiedTruth]

Thanks again, Sal.

[@y0n1n1x]

The dislikes are from color blinds

[@Cyclingdino]

I have one question, when we establish the domain of the parameters (s,t) , do we have to make sure the set of values is convex or not at all? I'd appreciate it if anyone could answer. Thank you

[@smellypotatoes2292]

Hi thanks to your videos I learnt calculus at 6th grade

[@bheemeshbommireddy4807]

LETS STOP ARTICLE 13 sure bud

[@Think0Like0Cheese]

Thank you so much !!

[@coolinglifee]

you teach really good!

[@dhimanroy1671]

The third function may be charge density.

[@jingxie3126]

Thanks!

[@thegoonist]

wow is this university level? ive never encountered this concept before

[@DecaSpace]

I feel that my scholarships, grants and money go to college or classroom that doesn't even explain this content as clear and well as a free KZbin video. I'm starting to really accept the idea that college education is overrated. You are just paying thousands of dollars to "certify" you - sometimes not even teach you well.

[@lizjakubowski]

Can someone explain why the area of each parallelogram is given by the magnitude cross product TIMES ds dt? i understand the magnitude of the cross product, wouldn't that itself give the area of each parallelogram, and therefore the area of the surface when integrated? are ds dt representative of the tiny depth of that parallelogram?

[@liquidstl]

@rachmaniralf you dont need differential equations or linear algebra?

[@SotraEngine4]

is the partial cross product the Jacobian determinant?

[@parthasarathym1882]

I wonder why d(sigma) is not just equal to (dr/dt)X(dr/dt)

[@hastyz7325]

the magnitude of the cross product of two parellel vectors is 0. that wont span the surface

[@norwayte]

Very good. Keep on going.

[@mrbrohere]

Hey! How's life going after 11 years?

[@jea1080]

absolutely amazing!!!

[@mandeep1244]

truly helpful!

[@TheAllboutwin]

@karkrashful someone's missing something inside.

[@vidhyalakshmim7616]

Which is the last video that he is referring to?

[@binthui]

what's the video before this one :((?

[@TheSunshineRequiem]

partial derivatives of vector valued functions

[@zuesr3277]

It's about partials derivative of the vector valued functions.

[@mrfrankincense]

What playlist is this in?

[@zuesr3277]

Calculus playlist

[@abhi99ps]

Shouldn't you put a double absolute value symbol for magnitude

[@raydencreed1524]

abhi99ps It doesn’t matter. If you know what is meant by the symbols, then being anal about formalities like that isn’t going to help anyone

[@nickvenanzi1607]

what program do u use

[@eljapi9346]

AMAZING

[@Andrew6James]

Given we have a square in the 2d space, why when we look at the volume in the 3d space do the lines now become curved?

[@Harshhaze]

The 2D graph is looking from the top down, and you can't see the curves on tube 3D graph from above

[@That_One_Guy...]

That's because they lies on 2 different planes (actually there's no special reason that the path in x,y,z is curved. It could also be a square too) and the fact that t and s variables are parametrizing ("writing") the path that lies on x,y,z plane but those 2 variable doesn't lies on that plane. The t,s variable graph form a square because you let one variable constant, another varies, and you also limit the value that the varying variable can take, meanwhile t,s variables form a curved path on x,y,z because they form 3 different 2 variables function (what i meant mathematically : x = f(t,s), y = g(t,s), z = h(t,s) ). Because this is a parametrized path, then x,y,z variables could be independent or dependent on each other, if z dependent on y and x then it forms a surface (z = h(t,s) = a(x,y) = a(f(t,s), g(t,s))

[@That_One_Guy...]

Example for 2D parametrization : A circle can be formed by : x = cos(t), y = sin(t) On xt plane, the equation form a *cosine graph*, on yt plane it form *sine graph*, but on xy plane it form a *circle*. Proof : t = arccos(x) (or arcsin(t)) => y = sin(arccos(x)) We know that cos(arccos(x)) = x = x/1, using pytaghorean and sine definition : sin(arccos(x)) = sqrt(1-x^2)/1 = sqrt(1-x^2) y = sqrt(1-x^2) => x^2 + y^2 = 1

[@That_One_Guy...]

Unfortunately we can't always express parameter function as function in term of x/y/z or some of them , so we just leave them as function of some variable t,s, whatever variable (as input) where x/y/z is the output variables

[@adoado16]

@thegoonist: It's multivariable calc, so most likely. :)

[@akshatjindal6851]

legend

[@christofferbouwer8057]

He sounds like James Spader at times

[@speedsterfilms2992]

songs to sleep to XD

[@吳柏慶-t9l]

make sense nice

[@vtron9832]

The only thing I know which I regret saying is that Sal sucks at making nice looking graphs

[@jakiasultanajui4588]

can someone explain why this double integral .

[@PopularityTroll]

there are two variables

[@wontpower]

When you integrate over the area of the region, you're really integrating over dsdt

[@romantorres1093]

im watching this in my class while i have a test wish me luck ( my test is 2 hours long)

[@yanxu6907]

can u add subtitle

[@sabiansmasher2000]

Why would I need to know this for civil engineering...such b.s. Well explained though!

[@emlmm88]

Flux and surface integrals are imperative to understanding continuum mechanics topics (like crack propagation) and heat transfer processes between materials.

[@xxxcoolboyxxx]

@adamgil91 why does it matter?

[@k1rv0lak]

I understand this stuff and I'm an infant... What is wrong with me???

[@anonymoustraveller2254]

DIv haha , the comment you trolled was just above your comment😂

[@Abhijitsings07]

Ur no more an infant now.

[@MrLeifArnesen]

Would you like to take my calc 3 final?

[@victorian4j]

2 ppl dislike this video cause they failed their exams.. :(

[@JustAnotherPers0n]

looks like smooching lips 4:40

[@pebble2529]

JustAnotherPers0n ikr

[@beckhamroyjenkins4223]

4:22 the transformed surface looks like lips

[@FlowFusionMusic]

@rinwhr people make people make people.

[@zuesr3277]

I now know calculus and linear algebra with finance statistics economics and probability and I am at 9th grade is something wrong with me. Is it OK.

[@kendallcarter3491]

That's an astounding thing in 9th grade I struggle in algebra alone.

[@beiberu]

Wow

[@HL-iw1du]

6:31 “diffrential”

[@allenoh1569]

I am not a huge fan of Anthony Tromba.

[@Nutterbutterz95]

Too damn hard!

[@DakaloMudau-j9c]

am i the only one who sees the joker?

[@dillonberger4036]

I don't mean to be pedantic, but the graphs you used at 2:50 don't quite match up. I understand and follow what you're doing completely, but shouldn't your axes on the right graph be labeled s, t, and z?

[@dillonberger4036]

Or perhaps there ought to be a condition such that x=s and y=t.

[@magicguy1988]

Dillon Berger He was exploring how the st-plane is mapped to 3D space using r(s, t) as a position vector function for some surface. It's conventional to label 3D axes as x, y and z

[@gus-padovany]

On the 3D space the x-axis is not the same as the s-axis and the y-axis is not the same as the t-axis. x y and z are functions of s and t, as he pointed out. f( x(s,t), y(s,t), z(s,t)) . It's a transformation, they're not the same.

[@raydencreed1524]

Dillon Berger No, apparently you mean to be wrong.

[@coolinglifee]

Women make sandwiches.

[@aidantdavis]

I understand this stuff in 7th grade... What is wrong with me???

[@ich123binsimmernoch]

+aidantdavis be happy about it

[@hl7297]

You have superpower, go find Prof. X

[@rizvanahmedrafsan]

at 7th grade most of us were attention whores xD

[@financewithsom485]

Great

[@abdallahableel4373]

You're gonna be getting straight A's in college . lol I'm 20 & i still youtube this

[@muhammadmustaqimbinsamsudi4784]

Thank you! Wow this was a good introduction.