Describing Surfaces Explicitly, Implicitly & Parametrically // Vector Calculus
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- čas přidán 6. 08. 2024
- How can we describe two-dimensional surfaces, even if they are embedded in 3D space? Similar to the three ways to describe curves in 2D, we can do this explicitly, implicitly, or parametrically. In the case of parametrically we get TWO parameters and choose them to try and naturally represent symmetries in the space. We specifically focus on the example of the cone and see how we can use cylindrical coordinates as a base to build out a parameterization of this space.
0:00 Intro to Surfaces
1:23 Descriptions of Curves
3:24 Descriptions of Surfaces
4:24 Cone Example
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You know what? I'm stunned how much effort you've put into these playlists just for people like me. You did this on your own and you did this without any other reason than to teach people. Thank you so much. Without people like you this world would be a much darker place.
I've always thought of surfaces as being 3D. But in actual fact they are still 2D, living in 3D world. Learn something every day. Brilliant.
This isn't really a good way of thinking about it. These surfaces are just a map from 2D to 3D.
It baffles that you don't have more subscribers. Your work is very helpful
Haha glad it helps:)
I’ve taken Vector calc ages ago. I did well and finished fine, but felt our instructor always deprived his explanations of the intuition or logic. This is simply amazing
Thank you!!
The animations really help in visualizing surfaces. Thank you so much!
First 4 minutes cleared my so many doubts. Thank you Trefor
simply the best calc prof out there! thank u dr. trefor!!
This is simply the best explanation I have encountered. Bravo!
Wow, thanks!
Oh My God! How Brilliant is this explanation!
Very nice image of the meaning of parameterization at the end of the video. Such elegance in thought.
Highly insightful, keep up the good work. Cheers!
THOSE VIDEOS ARE INCREDIBLES! Gave me a lot of great insights I''ve never had!
brilliant visualization and demonstration!
Wonderful!!! You have exposed the explanations which remain hidden: 2D surface to 3D surface parameterisation.
This really helped a lot! Our professor didnt teach much and it was really confusing before this video!!
Just what I needed for this week’s assignment! Thanks!
perfect timing!
I really want to thanks you for this videos, you increase my knwoledge in vector calculus and I love it! you are the best
Happy to hear that!
i really loved that last point of view! It would have been nice if you put some points in the r-theta axes and see where is it on the x-y-z space
nice video!!
thank you so much!
Great work sir, thanks
binging the entire playlist calculus exam in 3 hours 😅💜
Beautifully explained.
Thank you 🙂
Great video :)
Best explanation ever. I am very appreciated your work.
Thank you so much!!
Thank you very much sir 🔥🔥🔥
Thank you for this topic dear Professor !
You’re most welcome!
I really really appreciate you thanks man 🌹❤️
It's my pleasure!
Really cool video 🔥🔥🔥🔥
Glad you liked it!
If you read this, you've helped me a lot. Thank you!!!
Glad I could help!
Thanks for the explanation, it is really good and your videos are really helping! Also, one question: if you were given a parametric surface, i.e. the equation only r(r, theta), what would be your approach in trying to find the shape of this surface? Let's say that you are asked to draw the curve and it is not an impossible one hahaha
Dr Trefor, could you make a video about extended Green´s Theorem, please? Thanks in advance
Wow I get how cones are the fundamental shape of implicit quadratic functions it’s just a circle that has a height that stretches the radius by that height I guess that’s why space time is a cone
sir how do you make such things in a simpler way. Your method and techniques of teaching are really wonderful. Always feel amazing at your vector calculus video notification.
So nice of you to say!
In the cone example, I think the phi corner (top corner with the z-axis) should also be taken into consideration, as r runs from 0 to 3. For me the a cone consist of piled circles (disks) arranged from small to large. By taking the top corner into consideration the latter is fixed. So why not using three parameters (r, theta, phi) instead of two (r, theta)?
Hi dr. Trefor first of all thanks for the video, i got a guestion is there a specific way of who we find the parametric equation of a function?
In which video have you explained cylindrical coordinates as you said?
How would you take the gradient of a vector function with r(u,v) when it has three components of only two variables?
Sir if we are given some vector function f(x,y,z) and a surface in xy plane which is bounded by x^2+y^2=z^2 and the plane z=4 ....then what will be the double integral of f.n dS?
Is there a good practice problem manuel that you can recommend for vector calculus (with full solutions)? Stewart only has the final answers and that doesn’t help much… 😞
So...Here we gotta choose the parameterisation such a way ,so that we can see how the surface is sektched by r(t) in xy plane and z axis individually. Is there anything more to it ,why we did that parametrisation ?
Edit: Also why we need two parameters in 3D in the first place ? So that if we hide one parameter it gives the curve in XY pane and hiding the other gives the Z axis component ,right ?
This seems like, and I haven't watched forward yet, like we're gonna smash that line integral into 2D or 3D and use Jacobians.
r is not z-height right? r is the projection length of the point. did I miss? Thanks for the video. Finally I understand the intuition behind parameterization. never too late to learn it properly
You are awesome sir I get very help from this video ❤❤❤ I am also physics undergrad student 1st year
Thank you so much!
Mention not sir
You're called blessed, brother. Trust me, by Prof's videos, you'll love Physics, especially Electromagnetism. I'm a physics grad.
Super
Thanks!
Could you pls do class on Laplace and Fourier transforms
Have a whole playlist on laplace transforms:)
Question to myself: so it seems that we can parameterize anything to turn it into what's essentially a square (or perhaps a cube if you're looking for volume). That would be a natural parameterization. I can see how we find that for this concrete example. But is there a general formula for this? And what would that look like? I know there is cylindrical coordinates and spherical coordinates, but these seem hard to find. Looking forwards to finding out if there's an answer to this, how to select a natural parameterization.
Broadly speaking, a useful parametrization is one that captures naturally symmetries in the constraints of a system.
@@DrTrefor Interesting, I'll think a bit about that, thank you
3:55 can the bound of u be dependant on the value of v?
Yup that is fine. It gets messier when it is nonrectanular in the uv space, but certainly not imposible. It's just like integrating over a nonrectangular region in the xy plane.
Why did we need to introduce two variables u and v, instead of just leaving the variable t?
Dear Dr. Trefor, your presentation is nice and clear. May I ask you, how do you make the animations? I am a Ph. D. student of Mathematics, want to learn this animation. I would be happy if you please reply to my comments. Can I contact you via email? Thanks
Why do we need two parameters exactly to describe a surface in 3D? Why not 1 or 3 parameters?
It’s sort of a definition. We call a 2D surface an object described by two parameters
Why were you not my professor in uni :(
haha if only!
😂
Shouldn't the limits of theta be exclusive of 2 Pi, so that you don't double count it with zero? In the video it says less than or equal to 2 Pi.
This is going to be more relevant when we actually use these descriptions, but we often want to be able to describe the boundaries of a region. Take the simple unit circle which we paameterize with theta between 0 and 2 pi inclusive. Now it has two defined endpoints, one starting at 0 and one ending at 2pi and can do an integral from 0 to 2pi. If we used strictly less than 2pi, then it no longer has two endpoints.
jordan leyva here
x^(2/3)+y^(3/2)=1
why do you say you're describing a 2 dimensional surface, when a cone is a 3 dimensional structure? Not sure what you mean by a 2D cone embedded in 3D.
It’s similar to the surface of the earth. We can only walk two directions, north/south or East/west. Yes there is a third direction up into the air but if constrained to the earth we can’t access it, so “2 degrees of freedom embedded in 3D”
@@DrTrefor thank you! I like your videos
No chalk and blackboard with all its distracting handwriting and obscurity of text and diagrams by lecturers body and arms. Excellent use of 3d effectsand color in clear diagrams.!.