Surface Integrals // Formulas & Applications // Vector Calculus

In this video we come up formulas for surface integrals, which are when we accumulate the values of a scalar function over a surface. This is analogous to a line integral over a curve. As surfaces can be described implicitly, explicitly or parametrically - we come up with a surface area formula for each type. One application is finding the total mass of a surface with variable mass density, and a second is computing the average temperature on the surface of the earth.

0:00 Surface Area Formulas
1:50 Surface Integral Formulas
3:55 Application #1: Mass
4:30 Application #2 Averages

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[rmbennet]

Calculus 3 is the highest grade I have gotten since 4th grade. These videos have been lifesaving. Thank you!

[shakennotstired]

You are a fantastic teacher. Your videos help me to put things in perspective, to better understand what is going on, to better appreciate the theorems, instead of just memorizing formulas. Also, your manner of presentation is very engaging. It grabs one's attention and rouses one's interest, tremendously helpful to students of mathematics. I wish I had a teacher like you when I was young and in college. I am also studying your linear algebra lectures and planning on your discrete math lectures too down the road. God bless you

[jmgpqcx]

He's single handedly saving all freshmen

[mohandoshi]

I wish every university had a teacher like you. Then math would not have been the burden it usually seems to be and sometimes actually is. Your videos light up the world of math.

[highkey]

Amazing, amazing video. I had a textbook reading for this and I barely understood it, but this video flipped that switch in my brain and now I feel like I've mastered these! Thank you so much for doing this for us.

P.S. that temperature function was shockingly good😳20th century average temperature is around 14.0 C!

[samb.]

This may be the single most helpful calculus video of all time. I was completely lost with all of these different formulas and when to apply which one. This video is so helpful🤯

[michaelwilliams]

Literally am reading about these topics as you are releasing videos on them. Pretty cool!

[naveennaidu]

Wow, especially at the applications, till that part of video, I've no idea with the creepy function in between them . Thank u so much, it's been 24 hrs I've been searching for that, ur presentation has answered it finally, it's a mass density function.

[Festus]

Nice video. Which video can I find the derivation of the a^2sin-phi---> seen @ about 5.55? Thanks a lot.

[AlgoJerViA]

There is a missing closing parenthesis at 6:45 in the second line. Guess it might be a bit late for a fix, however.

[j.o.]

This answers my theorizing from earlier, that a "thickness" would have to be calculated in the 4th dimension. Question to self: what would happen if this was the 5th dimension? What would that look like? Well, I'm assuming that since we have a range of temperatures for each position in space, perhaps we'd have another variable that changes with the temperature (like humidity)? Although that would feel like cheating a bit, since it's exactly the same as having another variable vary with the position on the surface. So how do I think about this: we've essentially taken a 2D object that exists in 3D space, given it another variable (such that it's a 3D object) that creates a function G(x, y, z) that lives in 4D. So if we had a 3D object that is bent in 4D, meaning as we traverse the æ variable in the function H(x, y, z, æ), the complete value of the other systems would change. A 4th dimension function G(x, y, z) could be viewed as having a constant æ. So say we have a planet earth, with æ = constant = 2021, the function we have is G(x, y, z) could become the fourth dimensional H(x, y, z, æ), where æ represents a point in time during earth's history (perhaps this would be a sin function, as temperature changes cyclically at earth). Meaning, the 5th dimensional value would take in 4 variables from the 4th-dimensional H(x, y, z, æ) and get the temperature (or any other parameter for that sake) at a given point in time. Which equates well to each temperature point changing depending on some variable. For each point f(x, y) in the plane, there's a height which gives position. For each position G(x, y, z), there's a temperature, which gives temperature at a given position. For each temperature at a given position H(x, y, z, æ), there is a time variable, which gives temperature at a given position at that certain time. Not that instead of time, we could use position in space as the 4th variable, and that using both position in space and time as variables wouldn't work in the 5th dimension. It would however, create a 6th dimension that takes in K(x, y, z, æ, ø), where ø is position in space, meaning that for each temperature at a given position on planet earth at a given time in history, since the variable ø could be different and earth's position in space could be different, we'd have to create multiple universes (where, although the time is the same, the position in space is different). This would in fact happen no matter what variable we add in here, it doesn't have to be position in space. If could be the position of a certain bumblebee. That 6th dimensional function creates another universe (if we use the time, æ, as a variable). So essentially, we have K(x, y, z, æ, ø), which gives us a specific universe in which these conditions are true, that's the 6th dimension. What would the 7th one look like? Before I look into that, it could be useful to evaluate what the fomrulas would look like oh no wait. Nvm, now I see why we created another dimension. So we already have three position variable x, y, z. The moment we create another one ø, we automatically get a new dimension, which again, makes it more trivial. Anyways, it's interesting that we can create a new dimension without creating another universe, by taking in the 3 position variables, as well as time. Actually, instead of time we could take in the variable temperature. use H(x, y, z, G(x, y, z)). So we would get a variable H that, for each time the temperature is a certain way at a position of earth, there is a corresponding H value (such as time). I don't see any other variable that could be used. Oh, but, no, I got an idea. We although we won't be able to accurately pinpoint it, we could use ODE solutions to estimate the chance that something happens. Alright, I'll stop there, I'm writing way too much. But it's just so exciting, can't help it.

[ylap]

The example of the average temperature of earth made me believe in calculus again.

[ratata]

I really loved the part where you ignored what G represents while teaching how to compute but after that you mentioned it, I think that should be the way its done, the way that you do gives the impression of being a universal tool, however had you said like "this G represents mass" some people like me would say "Cool but I really don't care about computing mass."

Thanks for this amazing video.

[anjalidabhade]

please make a video on vector potential function

[sergiolucas]

Great video indeed, the temperature example is just amazing :)

[ishaanmanhar]

can we think that normal vector ru cross rv as some kind of jacobian?

[loganm]

M.V. and vector calc test on Monday... still just making sense of some things 😅

[aashsyed]

what is r sub u is it the partial derivative of r?

[persianmeme]

I heard before before before before till I end up here

[wf.i.]

Just like the professors, no example solving in real time, going step by step.