23: Scalar and Vector Field Surface Integrals - Valuable Vector Calculus

[@Caleepo]

This is the only clear explanation, I found on yt. I dont know why some profs dont give the visual intuition behind this, while its actually so easy to understand.

[@moosaawectison6008]

Because they themselves don't know these geometrical meanings ig. Each time when I try to debate on these visual intuitions with my professor, either he would roast me 😁 or try to end the session instantly. btw this man is doing great job. I learn a lot from this channel.

[@paradox6647]

I watched the first part of this and due to the thing at 4:15, it was a bit hard to understand, but I eventually pieced it all together, it is a really complex topic to explain, you did a much better job then if I were to explain it, even if I were to write it down for my future self when I forget. This is the best I’ve seen on yotuube, by a large margin and trust me, I searched far and wide, excellent work!

[@sportsgig7537]

This video is still relevant even today (2024). Thank you for making the video. It has made me appreciate the concept of surface integral of a vector field

[@3manthing]

Maybe i'm not the originally targeted part of audince, as i have studied maths, so this things are fairly easy to me, as i'm only refreshing my memory, so i cannot give this channel a proper assessment, not content-wise anyway. When it comes to math, i'm quickly pleased. Channels such as this one, offers me a fun revising of theoretical stuff, with some examples. You might be thinking, why don't i just pick up some math text book. I would, but i'm very lazy. What i can say is your explanations are simply amazing. It is by how you are explaining this things show, how well you understand it on deeper, more intuitive level. And at your age... 😯 👏 bravo, just bravo

[@briandwi2504]

That was brilliant. So concise and clear. Many thanks for passing on your insight into this topic. I shall watch that again and take notes. Really great lesson.

[@Wiik415]

to be honest you're saving my life, thank you

[@KyaBroderick]

this video saved me before my final. Its so much easier than I thought! Amazing explanation thank you

[@alicebobson2868]

this was so useful, ive just started going over my notes and to understand multivriable calc and this was one of the best videos for surface integrals, way better than my lecterur. Youre saving my grades lol

[@MoguinYT]

holy shit, how can someone explain something so good and so fast, propss my man!!

[@fairouztiti90]

Thank you from Algeria , this is really helping me 💗

[@bentupper4614]

Excellent. Clear and to the point. No frills needed.

[@hikmatullahpakhtoon3694]

Amazingly and beautifully explained. Thanks professor.

[@rahulbhavsar1402]

This explanation is unique all you tube video

[@luismendez933]

Increíble!!! 💯 Muy bien explicado, súper recomendado.

[@thabanivshoko4275]

best explanation for surface integrals

[@txikitofandango]

Just like line integrals can be thought of as a chain of varying density, a surface integral can be a curved sheet with varying density.

[@txikitofandango]

but I like the idea of flattening the surface onto 2-D and getting its height as a function of 2-D location

[@abdofast5]

brilliant! I think I'm going to watch all of your videos just for fun.

[@kinzakanwal471]

Thank u sir ...your lecture is very helpfull ,....Everything is clear now ....

[@MuskaanMittal]

At 7:00 , shouldn't the parallelogram's endpoints be r(u, v), r(u+du, v) etc?

[@MuPrimeMath]

That's correct. As is implied at 4:09, I'm using the ordered pairs as shorthand for the corresponding points on the surface.

[@kancer9725]

Thank you for this videos,beacuse of you i am planning to study mathematics

[@strippins]

I spent four years doing a physics degree starting in 2003. KZbin existed since 2005 and this sort of content was certainly not available until after I finished. I always found the unengaging lectures difficult to follow, printed lecture notes missing insight and text books impossibly heavy. I wonder how much more I could have got out of that education had content like this been around to enhance conceptual understanding .

[@nuclearcatapult]

So the reason I was having trouble visualizing a surface integral is because I'm not a 4-dimensional being. That makes sense.

[@Kdd160]

Wow!! You explained this so nicely man!!!

[@kamvc72]

great video.. many things got cleared here.

[@sreajan]

Great Lecture Sir. Respect

[@prateekkumar.1325]

U rock brother! Thanks a lot for making such videos. It inspires me a lot. Thank u vei much.!

[@academicstuff548]

thanks for such clear explanation.

[@celkat]

Thank you for your excellent explanation videos! 🙏 One issue is confusing me: 4:15 when you start explaining the parallelogram in terms of u and v, do you actually mean r(u,v), r(u+du,v) etc, given that this parallelogram is on the surface S?

[@MuPrimeMath]

Yes; we can think of taking the parallelogram in terms of u,v and evaluating r(u,v) for each corner.

[@parniamotamedi2694]

perfect explanation

[@Wan-vp9tp]

thanks for this explanation video!

[@saiakash707]

Excellent Video, Thanks a lot🎉

[@alishaanjum1108]

Beyond excellent😍😍

[@anmolmishra4784]

Amazing Thank you so much I do appreciate it ❤

[@hikmatullahpakhtoon3694]

Fair explanation.

[@MohamadKasem-r9o]

This is amazing the best exlpination I have seen until now. I just wondered why do we assume that dS is a parallelogram and not a square as only u or v is changing between each point. Why isn't dS written as du x dv?

[@MuPrimeMath]

The reason it's a parallelogram is that we're looking at how changing the input parameters u,v affects the output point on the surface. Changing u and changing v will each move the output in a particular direction along the surface, and those two directions are not necessarily perpendicular, so the result will not necessarily be a square or rectangle.

[@MohamadKasem-r9o]

Ah alright. I thought u and v always moved in term of x, y, z, and not as their own vectors. Makes much more sense now. Thank you!

[@mossy60661]

thank you so very much may god bless you

[@geniusmathematics9123]

Love u sir. Given 2 likes from two id...

[@latifmuhammad8874]

Thanks for the video. However, I found that the first surface integral is equal to 48π for some reason. What did I do wrong?

[@samrachkem2801]

As far as I know, the order of double integral is not interchangeable. Maybe I could be missing some part of the video but which variable should I be integrate firstly when solving surface integral? Thank you very much!

[@MuPrimeMath]

See Fubini's Theorem

[@eyuelbegashaw8609]

so what does the surface integral on scalar field and surface integral on vector field gives us ??

[@jaydenc6472]

Hi, may I know how to solve this, if we do not parameterize it, instead we use the formula dS=sqrt(1 + (dz/dx)^2 + (dz/dy)^2 )dA? What should we substitute in order to eliminate z?

[@the.lemon.linguist]

I understand that in the case of some curve r(t) that traces a curve, dr is a tiny change on r, and it can be found with r'(t)dr either by thinking of it (not very rigorously) as taking the dt from dr/dt and moving it over to the other side, effectively finding the "infinitesimal rise" by multiplying the derivative by an "infinitesimal run" or alternatively by thinking of it as converting a 0-form to a 1-form if you think of it in the nature of differential forms. In this case, would the analogy apply similarly with this? Would the partial derivative w.r.t. u times the differential du give that tiny change by similarly multiplying the rate by a tiny "run" of sorts?

[@MuPrimeMath]

If the second variable v is held constant, then r(u,v) traces a curve in the variable u. Therefore the same reasoning applies as in the single-variable case as long as we assume that v doesn't change. So du times the partial derivative with respect to u gives a change in the curve along the u direction.

[@the.lemon.linguist]

@ ohhh, i see! thank you so much!

[@mingdonghe9169]

Thanks a lot！You are the best！

[@iyadindia862]

Does the magnitude of cross product in the surface integral have anything to do with the Jacobian..It seems to be similar ones

[@MuPrimeMath]

Yes, they are related! One way to think about a two-variable substitution (x,y) → (u,v) is to think of the original (x,y) region as a flat surface. Then the substitution is a parametrization that looks like r(u,v) = [ x(u,v), y(u,v), 0 ] If you compute the cross product rᵤ x rᵥ, it ends up being equal to the Jacobian in two dimensions!

[@iyadindia862]

@@MuPrimeMath Thats Cool😍💕

[@andrewgraybar4984]

Riemann hypothesis, please.

[@rivaille8867]

Beautiful 🎉

[@pushkarsinghkaushik300]

What is the difference between left hand side and right hand side

[@learnsimple108]

thank you very much, ARE you s university professor? which university?

[@ofbguppies2325]

Great vid

[@ranam]

This is also called shadow integral can you please explain that too

[@abaidanwer8962]

Very nice

[@ahmedelshiekh9536]

I have one problem.. can you solve it for me please?!

[@LinhTran-uh6lt]

is || ru x rv || = || rv x ru || thank you

[@MuPrimeMath]

The cross product is anticommutative, meaning that b × a = -(a × b). As a result, the magnitudes of the two are equal.

[@hmt001]

Thank you

[@danielvolinski8319]

The result of the last example is 12π not 9π.

[@MuPrimeMath]

Both of the integrals shown at 26:38 evaluate to 9pi

[@danielvolinski8319]

@@MuPrimeMath OK, I see my error: the z in the first component of the vector field looks like a 2 so instead of z/x, I wrote 2/x.

[@rohaniyer4672]

yeo caltech class of 2024!!

[@Satya1621]

Awesome

[@kelfinmunene5941]

I like this

[@latifmuhammad8874]

Oops I found it; I forgot to square the 4 in (4sin(theta))²

[@latifmuhammad8874]

...as y²

[@ehsanAnsar628]

Great

[@derrickbecker9856]

Pretty sure not four dimensions… still two dimensions even though in 3D

[@swaroopdewal4626]

You are wow...!

[@irwanahmed001]

i going to faillllll!

[@trigon7015]

tsaL

[@sarkarsubhadipofficial]

❤️

[@bulldawg4498]

Sorry, but I'm disappointed in your explanation of a surface integral over a vector field ... I've seen better ...