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.
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.
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!
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
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
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.
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
I spent four years doing a physics degree starting in 2003. UA-cam 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 .
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?
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!
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?
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.
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!
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?
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.
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?
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.
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.
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!
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
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
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.
So the reason I was having trouble visualizing a surface integral is because I'm not a 4-dimensional being. That makes sense.
this video saved me before my final. Its so much easier than I thought! Amazing explanation thank you
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
to be honest you're saving my life, thank you
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.
but I like the idea of flattening the surface onto 2-D and getting its height as a function of 2-D location
Excellent. Clear and to the point. No frills needed.
This explanation is unique all you tube video
Thank you from Algeria , this is really helping me 💗
holy shit, how can someone explain something so good and so fast, propss my man!!
Increíble!!! 💯
Muy bien explicado, súper recomendado.
Amazingly and beautifully explained. Thanks professor.
best explanation for surface integrals
I spent four years doing a physics degree starting in 2003. UA-cam 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 .
Thank u sir ...your lecture is very helpfull ,....Everything is clear now ....
brilliant! I think I'm going to watch all of your videos just for fun.
perfect explanation
Great Lecture Sir. Respect
Thank you for this videos,beacuse of you i am planning to study mathematics
At 7:00 , shouldn't the parallelogram's endpoints be r(u, v), r(u+du, v) etc?
That's correct. As is implied at 4:09, I'm using the ordered pairs as shorthand for the corresponding points on the surface.
thank you so very much
may god bless you
great video.. many things got cleared here.
thanks for such clear explanation.
Wow!! You explained this so nicely man!!!
Beyond excellent😍😍
U rock brother! Thanks a lot for making such videos. It inspires me a lot. Thank u vei much.!
Fair explanation.
Riemann hypothesis, please.
thanks for this explanation video!
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?
Yes; we can think of taking the parallelogram in terms of u,v and evaluating r(u,v) for each corner.
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!
See Fubini's Theorem
Excellent Video, Thanks a lot🎉
Amazing Thank you so much
I do appreciate it ❤
Beautiful 🎉
so what does the surface integral on scalar field and surface integral on vector field gives us ??
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?
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.
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!
Thanks for the video. However, I found that the first surface integral is equal to 48π for some reason. What did I do wrong?
Love u sir.
Given 2 likes from two id...
Does the magnitude of cross product in the surface integral have anything to do with the Jacobian..It seems to be similar ones
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!
@@MuPrimeMath
Thats Cool😍💕
Thanks a lot！You are the best！
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?
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.
@ ohhh, i see! thank you so much!
What is the difference between left hand side and right hand side
Very nice
thank you very much, ARE you s university professor? which university?
Great vid
is || ru x rv || = || rv x ru ||
thank you
The cross product is anticommutative, meaning that b × a = -(a × b). As a result, the magnitudes of the two are equal.
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?
This is also called shadow integral can you please explain that too
Thank you
Awesome
I have one problem.. can you solve it for me please?!
Great
I like this
yeo caltech class of 2024!!
The result of the last example is 12π not 9π.
Both of the integrals shown at 26:38 evaluate to 9pi
@@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.
...as y²
You are wow...!
Pretty sure not four dimensions… still two dimensions even though in 3D
Oops I found it; I forgot to square the 4 in (4sin(theta))²
❤️
i going to faillllll!
tsaL
Sorry, but I'm disappointed in your explanation of a surface integral over a vector field ... I've seen better ...