Evaluating Surface Integrals

Surface integrals are kind of like higher-dimensional line integrals, it's just that instead of integrating over a curve C, we are integrating over a surface S. This can be tricky, but it has lots of applications, so let's learn how to do these things!
Script by Howard Whittle
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[@theblackstoneproject9917]

Bravo! You have done what no one else on the internet (or my calc 3 professor) had been able to do: explain the many different cases you encounter when you do surface integrals.

[@cameronsantiago3155]

In my opinion, this topic is the most difficult to grasp out of all the topics in Calculus. However, difficulty is subjective

[@MBKfreestyle]

you've surely not studied partial differential equations yet

[@williambaird6386]

@@MBKfreestyle 🤓

[@swagf3a438]

​@@MBKfreestylehe said that difficulty is subjective, oH yOu ArE sMaRt

[@secretaryfig5364]

@@MBKfreestyle PROVING THIS GUY'S POINT, I'D ARGUE THIS IS MORE DIFFICULT THAN PARTIAL DIFFERENTIAL EQUATIONS

[@VALUE-INVESTOR]

@@MBKfreestylewe had this after PDE's

[@feeneyko]

That works for me! I really appreciate how concise it is! Many awesome videos drill deep into these concepts, but what I really need is ready-to-use skills!

[@infinitasalo472]

Just in time for my Calc III final, thank you so much

[@ritishgupta5201]

same story a year later lmao. Hope your final went good lol cuz I am gonna get fked in 2 hours

[@infinitasalo472]

@@ritishgupta5201 It went alright, best of luck on your final 🙏

[@devan6012]

@@ritishgupta5201 I have mine today lol, how did yours go?

[@deletioninducedin7days919]

@@ritishgupta5201 lol update cmon

[@deletioninducedin7days919]

@@devan6012 hope youll do well, if its done, how did it go?

[@rajashruthim5451]

I love all ur lectures, they were cool and clear 🔥♥️💯👍

[@chokonma3420]

This is very well explained, thank you for this!

[@codatheseus5060]

Ooo I'm here to help with my understanding of exterior algebra, linear algebra, and geometric algebra. Thanks Dave, I've been loving your back and forth with narcissist James Tour and you're helping me out with my other passions too. I love it.

[@sotirisbakas194]

You just explained it better than my Calculus II did in 6 months. Thank you!

[@MrCEO-jw1vm]

Thank you so much, sir! I thought I won't catch up in my vector calc class, but I'm getting ahead now!

[@j.k.sharma3669]

Nicely explained . Thanks

[@n8likesmath]

I've been looking for a derivation of dS. seems all the videos online like to skip this detail. thank you

[@swarnalakshminarayanan9691]

Thank you verymuch for the informative presentation

[@stepheneugene]

whoa! thanks professor finally understood surface integrals.Took some time but finally its in , thanks again

[@justpaulo]

Thank you for the video! I could actually calculate the comprehension problem in my head b/c it's quite intuitive. Basically the surface S is an inclined plane (z=1+x) that given the x limits goes from z=1 up to z=2. Since we want to integrate f=z on that plane, we can take the area of the plane (i.e. 2*√2), and multiply it by the average value of z (i.e. z=1.5), leading to the final answer of 3*√2. NOTE: The area of the plane is 2*√2 b/c the length of one side of the plane is the y range (equal to 2) and the other is √2 b/c that other side of the plane goes across 1 unit (x=0 -> x=1) as it goes up 1 unit too (z=1 -> z=2).

[@ivoryas1696]

justpaulo &Look at Nikola Tesla over here 🙄- Good on you M8!

[@finnholden636]

Thanks so much for making this a lot clearer than it was in my head :)

[@parvizhasanzade271]

excellent explanation

[@wstryc]

So easy, thanks!

[@ArhamKhan05]

Thank You So Much Sir.

[@saurabhsingh-ow7ue]

thank you sir.....

[@45powerj]

This might be the best video I’ve ever seen on math, god bless you Dr Dave

[@mage9219]

This helps me with the book introduction to electrodynamics, by Griffiths. Also we can compute the integral for any kind of surface.

[@unrelentingawesomeness7501]

same, the special integrals werent explained very thoroughly in griffiths e&m

[@pawelo2698]

Thank you master!

[@jam8077]

THANK YOU SO MUCH! ❤

[@vijaysinghchauhan7079]

Beautifully explained.

[@hakimbrahimi1015]

Thank you so much Prof Dave!

[@muhammadhusni3097]

thanks prof

[@astridsaldana9302]

what book did you use? or what book do you recommend for study?

[@sabirfly]

thank dude

[@flaviocamacho986]

thanks prof.

[@nehadas8103]

THANK YOU

[@tamizhtamizh412]

I'm so happy... Love you professor 👍❤

[@Maxwell_Integral]

This is simple I just can't remember anything on how to parameterize surfaces. WHICH I'm going to you for help lol. Amazing as always

[@EzekielChirchir-yc5hy]

Requires patient and time ,positive mindset to grasp well this concept.

[@portreemathstutor]

Thank you.

[@nomthethomaseko1833]

thank you

[@taylorcagle742]

If you start out with a 2-D vector, will it still be a double integral? Or single?

[@LucidTheLifestrider]

i have no idea what I’m watching and I love it

[@daniellabinjo6046]

Watch it before going to bed and try and ask yoir subconscious mind to hepp you understand it

[@henryhung2795]

the room vibe

[@nnass262]

@@daniellabinjo6046 Even the devil couldn't teach me this crap

[@schifoso]

Thus was almost as hard as your last couple of Italian language videos...

[@rajjolal23]

correction: @ 5:32 its mag of Rx X Ry. not Ru X Rv.

[@maxproietti8211]

Does the parameters stay constant for each example. If not how do you choose them?

[@gianlucacastro5281]

Helped me a lot, thank you!

[@itsazizkazemi]

We did it guys. We've successfully removed the numbers from math.

[@kosiscatharsis7943]

In the problem the f will come out as (0,0,1+x), rx=(1,0,1), ry=(0,1,0), rx*ry=(-1,0,1). And the answer after solving the double integral is coming as 3 not 3(2)^1/2.

[@sushimeng]

Thanks for your video, it helps me comprehend the main concepts of surface integrals! And one thing that confused me is that at 9:00, why there is a vector r instead of the unit normal vector n?

[@bowenying2391]

i have the same question

[@marelowehecheveria7455]

any update about the comprehension? help

[@sroydetroy6404]

But I didn't understand why is that that when the function is described with the parameters x and y- you decide to give it a certain formula -pdz/dx-qdz/dy+R basing it on a particular excercise that you came up with and than applying it to a completley different question at 9:06? can someone please explain because it's unclear to me why should you use specifically that formula if it came from a completely different question?

[@STKeTcH]

All the problems in my calculus book are 10x harder than this. Especially the first steps, which are just given in this video. I wish you would raise the challenge a bit in your video, because this way I'm missing the vital part.

[@ManojKumar-cj7oj]

Great

[@broodlingg]

hmmmm. it doesn't change the outcome of the answer but when you were taking the determinant for finding Ru x Rv you got -cosv as the j component but i do not understand why it is negative. I am fairly certain that it should be positive. however when taking the magnitude of it the negative goes away so it doesn't actually change the answer. idk correct me if I am wrong.

[@kyleparsotan6799]

11:48 @Professor Dave Explains I got 3 as the answer. I think I did something wrong. Used double integral (-Pdelz/delx -Qdelz/dely + R)dxdy

[@milesonyoutube8222]

I got 3 too after i parameterized z=1+x and did all that stuff so idk what’s up

[@xyblade4625]

yeah isnt it 3?

[@eggxecution]

yep got 3 too I got stuck at this maybe we did something quite wrong i don't know where is it

[@eggxecution]

So I noticed that the sqrt(2) is missing it seems he used the formula 6:05 it seem we are given a scalar field or a function or f(x,y,z) so this is the formula for scalar field / function when the surface "z" is in terms of x and y parametric equation which is at 6:10, the one he used at 11:48 is the formula for the vector field F. And if we plug it in, we'll get 3*sqrt(2)

[@eggxecution]

Ight update I got it guys

[@ImASDFx2]

I had to laugh out loud. The "double integral of the magnitude of the cross product of the derivative of r with respect to u and the derivative of r with respect to v". It's nearly the entire semester in one problem...

[@eggxecution]

made it, it seems the problem uses the formula of 6:10 because we have a function or a scalar field f(x,y,z) an not a vector field F or so that's how I comprehend it

[@EricPham-ui6bt]

We could simplify so that the derivative of z is tan of angle of normal vector z and cross product with line integral of z normal vector?

[@TungNguyen-vb4on]

can anyone explain how to derive the formula for surface integral of vector field

[@eriklam33]

11:40 - I got 2 times square root 2 instead. The length of Z is square root 2 between X=0 and 1, length of Y is 2 between Y=0 and 2. Area = length of Z times length of Y. Not sure if I missed something?

[@sexy13ITCH0000]

no same I keep getting 2 sqrt(2) as well. Might be a mistake on the answer.

[@mesumabbas9481]

I thought: let's explore some higher mathematics. After 2 mins: I think I should close it now.

[@banpridev]

Think u need to color code ur graphs

[@alishaanjum1108]

Are da of equal lengths

[@alishaanjum1108]

Are da of equal lenths

[@manavparekh2833]

Hi I keep getting 3/2 as my answer for the question at the end. Could you please explain it to me thanks Manav

[@samarthtaneja3534]

In the problem starting at around 9:30 was the r supposed to be an n (normal vector)?

[@jimhuang9536]

yes. U R right!

[@Shannxy]

I need to find the check comprehension song so I can jam it on replay while I solve the task

[@ProfessorDaveExplains]

i have it on a five hour loop in a video in my "just for fun" playlst

[@Shannxy]

@@ProfessorDaveExplains Hahah thanks, it's the perfect vibe!

[@ChauNguyen-wd6fm]

I just hate how this didn't exist when I took this class at university

[@poro167]

NICE VIDEO, LETS GOO

[@shibah9345]

Can anybody explain to me how (sinV)^2 + (-cosV)^2 becomes 1? I know (sinV)^2 + (cosV)^2 equals 1 but is (sinV)^2 + (-cosV)^2 equal to (sinV)^2 + (cosV)^2 ? Please help me out.

[@Gamesanty334Blogspot]

When you square any negative number, the answer is always positive, since a negative number times a negative number equals a positive one. That said, if we evaluate, for example, (-2)², the answer will be 4, since (-2)(-2) = 4. Note how this is the same as evaluating 2², which also equals 4. Note how (-2)² equals (-2)(-2). This is not the same as -2², which equals -(2)(2), which is the same as -4. The same happens with cos(v) and (-cos(v)). Since we are evaluating (-cos(v))(-cos(v)), we get a positive answer. If we were evaluating -(cos(v))(cos(v)), the answer would be negative (be aware of the negative sign outside the parenthesis). Since we know (-cos(v))² is positive and has the same magnitude as (cos(v))², which is also positive, then they must evaluate to exactly the same. Since (-cos(v))² = (cos(v))², (sin(v))² + (-cos(v))² also equals one.

[@STKeTcH]

I just don't get what function to integrate. In my calculus book, they just take y and integrate it over the surface, but it's absolutely not clear to me what I should double integrate :((

[@ronnygranados]

if in a vector field you must, first of all, parameterize the surface you are interested in, i mean, the surface in which the vector field is flowing through, remember surface integral calculate the flux throughout a surface in R3. once you parameterize your surface [ r(u,v)=(u,v,g(u,v)) being u,v the two parametrization variables ] and then you calculate the cross product and there you got the normal vector. after that, the force field you are given must be changed in terms of the u and v variables. after that you are left with two vectors and calculate teh dot product of them and there it is what you need to double integrate

[@priyanshubhargav3953]

@Professor @4.45 it should be not

[@hassankipngetich8838]

Its just wonderful, clearly explained just try to eliminate those words appearing on screen that hinders clear view of the writings you're presenting, otherwise its so lit,,

[@ProfessorDaveExplains]

Just hit the CC button, my friend. It's on your end.

[@gauravladha5465]

@@ProfessorDaveExplains I got the answer as 3 for comprehension. Could you check?

[@nicolasbastien-besner167]

Thank you ! It helped me a lot

[@nicflorant4633]

Hey salut Nic, bonne chance demain là

[@nicolasbastien-besner167]

Nic Florant Ahahahah merci toi aussi big 😂😂😂 fuck Celse right

[@yuhao8430]

Hi any proof on why dS=|ru x rv|dudv? 1:29

[@JensenPlaysMC]

yes. this isnt rigorous but here it is. assume surface can be paramaterised into r(u,v) then a small change to r caused by one lf the variables is (dr) as dr/du is ratio of change in r caused by u, to u itself so dr/du *(du) is the actual change to r caused by u. this is a vector qauntaty. imagine limit sum of a difference in r's value being a tangential vector to the surface. now do the same for achange jn v an you have two vectors relating to a small change along the surface. by taking a cross product between these you then get a vector with a magnitude of the area of a small ds

[@pawelo2698]

Can parametrization of x,y,z be whatever I want? ( it doesnt have to be "x=ucosv, y=usinv, z=v" yes?)

[@supramayro434]

No,it actually depends on your function. He used those functions as an example. I know I'm a bit late,but for future generations it'll be useful:)

[@carultch]

You can parametrize it however you want, as long as you fully describe the surface in question, with your parametric equations. However, there often are ways you can parametrize it that will significantly simplify the remaining work. For instance, in your example, I could parametrize it as: x = u*cos(3*v) y = u*sin(3*v) z = 3*v But I wouldn't really accomplish anything by making a more complicated definition of my parameter v. I'll get the same results as if I kept it simple, and stuck with z=v, and defined everything else in terms of that.

[@toitozky155]

idk but you got the unit normal vector (n) in new way. what i mean is that i get the n in different way n= grad.S/|grad.S|

[@vanillapie8321]

nice

[@instantrhodes8068]

T shirt so cool

[@TraeHall]

What Playlist is this

[@ProfessorDaveExplains]

math

[@dragosstan2885]

8’59, flux F.ndS instead F.rdS...

[@AlexandreG]

I don't think you're right...

[@emmairiondo3360]

10/10

[@TheRojo387]

It sucks that you don't have a video explaining Bézier curves yet. (Those are constructed by iterating interpolations between straight paths to control points until just one control segment remains, if that makes even remote sense to you!) But the ultimate generalisation of geometric curves is also the industry standard: NURBS.

[@aaditiwar1]

dawg whaaat da haaaaaaaaleeeeeeee

[@fizika5562]

Dr I got 3 as the answer have i done an error??

[@ericwang557]

I also got 3, but I dont know what's wrong with my process....

[@dgirlsportable3196]

I got -3 also

[@georgigachev4175]

Me too

[@cmdrbobert9862]

😬I'm definitely more confused than when I started.

[@AjayPatel-te4kb]

👌👌👌

[@Gball708]

?

[@tylerkutschbach]

Are you gonna do Theodore Roosevelt tomorrow? You said you do the president videos once per month.

[@ProfessorDaveExplains]

I'm a little behind, it'll be next week.

[@bmohan5032]

Can please anyone help me with comprehension

[@fernandoportal5422]

this is late but z=x+1 is plane in 3d space that can be describe by = so you have something in the form of

[@sroydetroy6404]

@@fernandoportal5422 Can you elaborate?

[@oasishighersecondaryschool8804]

Thanku physics jesus

[@proxy8918]

T H A N K Y O U

[@ronaldmangasihutauruk3663]

Your texts subtitle make me confuse to see and to understand

[@altheoremoto3123]

💯💯💯

[@EmpyreanLightASMR]

Whoever came up with using u and v as the two variables would have chosen better variables if they'd had my handwriting. What on Earth were these people thinking. ςigma and τau would've worked just as nicely. μu or νu would've sufficed as well. u and v. picard facepalm.

[@HarpSeal]

Let’s not forget the people who chose to use ξ in sets

[@EmpyreanLightASMR]

@@HarpSeal I have a 1960s calc book that uses ξ to represent little Δx* when explaining Riemann sums, and also uses ξ elsewhere. it's kind of strange to see.

[@JohnyLangrin]

❤❤❤

[@dorothyjanepasaporte364]

Helppp

[@koolaidkid694]

final tmo, if i dont get an 80% i fail😿

[@sethguzman677]

i kid you not, thats exactly what I need too. 80% just to get a D. I’m mentally prepped to retake. Did you pass?

[@babarazamsucks]

First!

[@mesumabbas9481]

what happened to those hairs.

[@kammaganisricharan5850]

:-)

[@stefanusnanduwa4864]

i watched this and solved a question worth 15 marks in 2 minutes, my teacher gave me $200 for that

[@meharsamba]

See thats the thing...there was no practical application example here...like i am trying to sketch a Vane of impeller i am designing and...this is useless to me... Mathematics wouod be much more interesting if instead of just blatant numbers they would give us practical application of the formulaes...

[@heronimousbrapson863]

This makes absolutely no sense at all.

[@tempestandacomputer6951]

Get good scrub

[@josiahsoita708]

that intro is for kids