How long is a curve?? The Arclength Formula in 3D
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- Опубліковано 24 сер 2024
- What is the arclength of a vector-valued function or curve in 3D? In this video we break the length into a sum of little straight lines, we add up the lengths of the straight lines, take a limit and voila, we get an integral formula to compute arclength.
We've previously studied arclength for single variable functions back in Calculus II, click here for a refresher: • Arclength Formula | De... Now that we are in multivariable calculus the formula is more or less the same, just has more terms under the square root.
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I've never commented on a youtube video before but I had to share my love for how beautiful you explained this concept. THANK YOU!
You are so welcome!
This is one of the most amazing ‘moves’ I’ve ever seen in math. You are, ever so cleverly, finding the length of a curve in space, by converting the problem to finding the volume under a surface. This is actually done quite naturally when you realize that the approach of an infinite series is identical to an integral.
I learned this about a year ago with a curve in the plane, and used Desmos to plot both that curve and the curve whose ‘integration area’ was equal to the length of the main subject. It just makes you smile to see them together.
I’m actually not sure about the length of a curve in 3-space being the equivalent of volume under a surface. That’s just my intuition based on the length of a curve in the plane being equal to the area under some other curve. That is definitely a thing (that makes me smile, as stated).
Seriously, folks, it doesn’t get any clearer than this. If you still don’t get it, watch again (and again). Do not try to find a better explanation elsewhere. You won’t.
Great choice of colours - curves in blue, line segments in red and purple for points on the blue curves really brings clarity to the graph. Wonderfully done!
I am feeling so blessed to have a chance to watch your videos. It feels like it is getting into the bones. This is great, the way you explained this concept. I really feel like having " 2 - minutes silence" because of the joy and excitement it brought to me.
Wow, thank you!
best mathematics professor i have ever seen in my life, makes mathematics so easier to understand :)
It suffices to say, I have a math crush on you, Sir. My whole degree is on your channel! Much love from South Africa!
Professor you're really amazing , I've saved all your playlist and watching one by one. Thank you I'm getting intuition of concepts ❤❤
I'm learning multi-variable calculus by watching all the videos in your Calc3 playlist, one by one. Can't thank you enough for this. Because of you, calculus started making a lot of sense. Love from India. 💖💖💖
Glad they are all helping, good luck! And don't forget to give a celebratory comment when you make it to the end:D
This video series is underrated. Teaching techniques are perfect. ❤️
Thank you!!
Great explanation, this playlist is underrated - I will recommend it to all my friends who study in IT!
Thank You, Dr!!! You never failed to make me understand mathematical concepts!
This is how maths (and all subjects) must be explained. U are an example... no, an example no, a cannon of how teaching is!
This is an great explanation. Thank you!
Thank you professor,for teaching the concepts which my college lecturer's cannot.
nice! i liked how you talked about the derivation and not just the formula - not only that but the derivation was elegantly explained
Thank you!
man this is the best calc III course on youtube. Thank you!
Glad you think so!
These are wonderful summaries for refreshing my calc III knowledge. Thanks dude!
Thank you for making this series. It has helped me so much. The animation is awesome, it gives an intuition of the things that we studying.. Just woow😍..
Lots of thanks🌈
Thank you so much! I was looking for some extra stuff on top of our awkward online lessons and these videos have been super helpful for learning these concepts better.
This is an amazing explanation thank you
Preparing for Analysis 2. Dunno where I am heading. This playlist feels right. Thank you so much
You are a great teacher. I love the multivariable calculus course. Your UA-cam site has become a major hangout for me. Thank you. 👍🇨🇦
Your interpretation make me sense clearly about the algorithm. Thank you!
Eureka⭐⭐⭐,Thank U sir, you made me to understand truly the integral concept...
In the part of Africa where I write from when someone excel themselves at what they do.You ululate.I can't do that so I'll just say bravo Herr Doktor.
Prof Trefor; very nice and clear - thank you
So great to be taking the same contents in my calculus lessons as the ones you are presenting here lately. Well ilustrated exemples, and your explanation is also amazing. Thanks!
thank you, sir. I love your teaching.
Thanks sir nice explantion also 1st comment
Amazing explanation!
thanks
That's so sweet!!💝💝💝
Calculus of variation!!!
These videos are excellent.
Love it!!!
Thank you sir 🔥🔥🔥
U r a legend sir👍👍👍
Awesome video! Thank you!
I like your videos, thank you.
Thanku bhai
Great video! Where is the Video on eigenvalues and eigenvectors
how come you can move the bottom delta t into the square roots and squares?
Thankuu very much sir..it is very helpful ❤️❤️.
GREAT JOB
sir ,which sourse i use for problems solving ? suggest any book related this
What are a and b?
Like for integration in only x and y
a and b are like the lines x=a and x=b, what are a and b in this case?
just lovely!
We need more pause montages like the one in this weeks video for math 100
Sir,how do you generate these visuals?
Plus I am jealous of your beard.
Great video! (I am watching a bunch of your videos to help my high school-er in multi variable calc, and I am a bit rusty.)
owww my god!! i think i am in love. this is beautifull :-)
well explained sir,thanks..you are doing great job keep this up
I had some problem with vector calculus on this topic which is arc length but I always knows there is one solution of my problems on higher mathematics that is Dr Trefor - Bazett . Thank you sir ☺😊
Question: When calc textbooks say to do numerical integration for these kinds of integrals, do they mean finding a corresponding MacLaurin power series, or do they mean other numerical techniques (Simpsons's Rule, Trapezoid Rule, etc?), or do they mean something else altogether?
These videos are really good. However it would be nice if there was a link to many problems and their solutions to help someone really program these concepts into their brains, you know?
Ah yes. After years of calculus I can finally calculate the length of a spring
So let's finally clearly add that s'(t) = | v(t) | where v(t) is a vector
So the differential arc length ds is the thing inside the integral sign ?
very nice!
Nice video, but this integral also represents the integral of the tangent vectors magnitudes along the curve for the original function. Doesn't it seem odd that adding up the tangent vector magnitudes would equal the curve length between 2 points?
I did not get the part whre you used the Pythagorean theorem to calculate Delta L.
One thing I am not getting is we have xyz space could we find tangent by differentiating with time.
I mean we don't have time axis so how could i look at it
Dude. Thank you
I have a question. What do u think about giving this deeping demostrations on a normal class, with a schedule defined and the puplis drafted to stay in, may being bored at that momment?
In my own classes I might give this presentaion, but then the students are actively involved for the rest of the class trying to actually do math!
Very interesting!
weird question which i feel must obviously be "that won't work" but... why couldn't you use the Pythagorean theorem? if you were to flatten the helix into a circle, or a piece of a circle, or a circle that overlaps itself many times (depending on how 'long' the helix was)... and then find the arc length of that (piece of) circle, and 'pull' that length into a straight line (The line is simply a line of the length of the arc length of the circle)... then you take one end point and "drag" it "down" in 2d space. this describes a right triangle with the original arc length as the top edge, the vertical edge is the amount that we drug it down (you would probably drag it down as far as the helix was tall) and the new diagonal is the 'lengthened' arc length which is the hypotonus (and hopefully the length of the helix). then it's just simple the square root of a^2 (the arc length) + b^2 (the height of the helix). would that work?!?! it seems to be too simple but in my head there is no reason that shouldn't also supply the answer... or at the very least a VERY close approximation. anyone?
Ok let me challenge you this way. I agree for some curves it would be a close approximation. But can you think of an example of a curve where it ISN"T close?
@@DrTrefor thanks so much for responding! the only thing i can think of would be a helix with an ever decreasing radius... but then the issue wouldn't be the Pythagorean part, it would be that finding the arc length of that spiral would be different than finding of it of a proper circle. so... no i can't really think of a good example (that starts with a regular circle) that isn't close. i'm honestly still stuck on "why isn't this just exactly the answer". i'm not a mathematician and it's been a while since school.
@@DrTrefor hmm and certainly a helix that 'stretches' at a non constant rate. But again, if it's a proper circle and stretches at a constant rate, I'm not sure why this wouldn't work.
This only gives you the arc length of the image though, not the set {(x, T(x)) | for all x in ...}
nice!!
Damn you're good. Made me smile in the end haha.
Keep up the good work sir.
Would be a huge improvement if you got a better microphone! Get's you 10 times the views as well I think.
I’ve actually recently upgraded, my old room was SO ad for echo
@@DrTrefor That's great!!! Cheers :)
Physics help to understand.
is also possible the proof by means of the mean value theorem?.
please add some examples
Three people tripped and fell onto the dislike button.
How long is a piece of string?
Subscribed!
I get dt/dt is one and multiplying it changes nothing.
But how u get denom dt under each term of the sqrt and under sqrt itself
What property or algebra manipulation did u use?
Nevermind I got
Sir , in the bracket of derivative ( at last moment ) it's partial or full ?
great ❤
Great
This formula is great but it spits out integrals that are very difficult to compute sometimes...
That's very true! Square roots are the bane of my existence:D
greaaat
I am just playing with youtube algorithms, here.
that's all
anyway, I love the way you teach
Haha thank you!
Sir is that same as taking derivative of position vector, find magnitude of it and integrate?
Not entirely. If you only had a y function of x, it wouldn't simply be the integral of the magnitude. Instead, we'd first add it with the square of dx/dx which of course is simply equal to 1. So arc length with just one function is L=integral sqrt(1 + (d/dx)^2) dx
In the multidimensional case with multiple functions, it is the sum of the derivative squares, all under a square root sign.
I dont get why ΔL= [ (Δx)^2 + (Δy)^2 + (Δz)^2 ] ^1/2 . Is it because of the ortogonality of the axis and pythagoras?
No idea but it seems some of these concepts are rooted in analysis or require fairly complex proofs as my professor also presented it in this way and we kind of have to take it at face value. It's a weird 3D extension of pythagoras' theorem I guess.
It’s not complicated, it’s simply the length of a line in R^3. You can find the formula by applying the Pythagorean theorem twice for a general line.
i love u
goattttttttt
If you attached a mic to your collar, your videos would be perfect. It can be difficult to hear some words at times.
∆L_i is just ∆r, except it's not a vector?
No! ∆L = ||v||(dr/dt)dt
Can those dt's "cancel"?
Noice
It sounds recorded in the bathroom. Hope there's nobody using it.