Professor Leonard!!!!!! my fiancé and I really love you! we even named our foster puppy Leonard because of you. Instead of watching netflix at night we watch your videos together(we are classmates). He also says that he loves that you are super muscular and his dream is to train at the gym with you lol. Anyways, I just wanted to thank for all the effort you put to really clarify complicated things to your students and people who watch your videos. We are very thankful to have access to it. You just make it so easy to understand and you really can see where the students can possibly make mistakes and you point it to us even before we notice it. That is talent! I wish my teacher have half of the talent you have. Wish there was a way to do something nice in return. the least we can do is to let you know how awesome you are! God bless you and your family! Thank you so much!!!!
This is the most impactful description I've seen of TNB curvature ever. I remember covering this in Calc 3 section of James Stewart. All I really understood were how the derivations worked and definition of unit tangent, normal, and binormal, but you sir Professor Leonard gave best informal explanation on why we need them. To understand curvilinear motion in moving reference frames and how the arc-length can serve as a parameter that simplifies the calculations immensely. Its always good to see the big picture, cheers.
best part: The Leonard bursting into song at 57:00. no seriously, the technique and explanation throughout not only made me understand on both an intuitive and mathematical level, but saved me from mistake opportunities throughout the calculation. Thanks once again, Mr. The Leonard.
Watching this as a review and I have to say, I've never had a professor even half as good as you. That being said I've had some great professors. Amazing work Leonard!
@@CamGersh I know this is four years later but this shit happened to me, some dude ive known for a couple of years at my bjj place popped up in a random comment section on the most random video ever and I was just like wtf and its weirder that this guy is RUclips famous and ive known him for years
Professor Leonard, I just want you to know that not only you have help me understand math and the theory at a deeper level, but also have fun during the process. thank you for your wealth of knowledge!!
Great Teacher. at 22:30, wish he had mentioned that Torsion is merely existence of change in Bi-Normal, just like Curvature exists with change in Normal. The vector perpendicular to a plane of paper (biNormal), when it starts changing its perpendicularity, Torsion creeps in.
Thank you so much for what you are doing! Our calculus 3 teacher was out during this section and had us teach ourselves. I nearly cried doing the homework but you are helping me through it! Thank you!!!!
Professor Leonard ,thank you for another mammoth and lengthy video/lecture on TNB Frames, Curvatures,Torsion and Encapsulation in Multivariable Calculus. From watching the lecture and doing problems, I understand the material from start to finish.This material is useful in Statics, Dynamics,Physics and Applied Mathematics.
I appreciate the planet earth because there is professor Lionard in it. The way ur explaining and the energy u have I havent seen yet. You really helped me. Thanks 10^n times.
In Sri Lanka the way our "Lecturers" lecture us is very different ...If we had people like prof.Leonard .... we'd pass our exams real easily. They don't teach us like you do. We HAVE to depend on RUclips! People like you are GODS!
You are so Awesome +Professor Leonard If it weren't for your Statistic videos I would have never gotten through Intro to Stats and Prob with a 93% Thank you so much for posting videos. You have no idea how much they have helped me and many others!!! =0)
Sir You are phenomenal.. You are the definition of a great teacher...You turn the complicated things into easy...I was having trouble with multivariable calculus but then I found your videos and you are amazing...I wanna Thank you from the bottom of my heart for making maths interesting..I honestly can watch your lectures for the whole day ...If it's in my destiny then hopefully I'll meet you one day and touch your feet(Indian tradition for a way of showing respect to elders)..May you and your family stay in the pink of your health, may you find all the happiness and success in life and may u keep posting awesome math videos..I'll even watch your storytime videos 😛...I TRUST THE LEONARD!!
At 1:11:35 , isn't T x N = (sqrt(2) / 2) < sint(t) , cos(t) , 1 > ? Where did the -cos(t) come from? For the cross product, the k component is -( - cos(t)).
The formula for the unit binormal at the beginning of this video it is said that it equals both TxN and (r'xr'')/||r'xr"|| but this was never explained and I don't understand where it came from . . .
I would love to take precal - linear algebra with you professor! Im in calc 3 and it feels real "bare bones" compared to your in depth lectures. Love the brake downs and buildups.
Brandon Ramirez Well, I know that this reply is too late , but I got to explain to those who want to know . In the eq. of plane you must plug the value (-1) in the both side Which means (-y=-1 ) so it doesn't matter if it was +ve or -ve
So curvature is how fast the direction of the motion of the particle is changing per the speed of the particle. If the nose of the plane is changing direction slowly but the plane is moving quickly, then the plane is moving in a fairly straight line. If the plane is moving slowly, however, and the direction the nose is pointing in is changing quickly, then the plane is doing spirals.
For torsion: t = (r'(t) x r''(t))*r'''(t)/(...) is the triple prime supposed to be a double prime? because later on you write, torsion=-dB/ds*N, and the N has a double prime (r''(t)) or T'(t)
I am so scared to fail in this course. Our PI passed everything so fast and now he waits from us to be able to solve everything until double integrals. I have 1 exam tomorrow but I am eating from its time just to be prepared for this course's exam. Well we will see I have 4 days. If I do well on that exam I'll rewrite all of this courses notes in LaTeX and will share it with you guys that you can just relax and watch, follow with a good document of Leonard's Notes.
Professor Leonard - Do we ever add the magnitude of the velocity to be derived. I am working on this topic now and my book is adding the magnitude to each variable and then derives it. Thank You!
I'm thinking there may be some engineering students out there- as helpful as Prof Leonard is for Calc, are there any channels that are as helpful for Physics?
EE here. I really liked SBCC Physics when I was in Physics II. I don't know how the newtonian mechanics portion is, but I thought the E&M videos were great lectures. Here's the link: ruclips.net/video/0_ITe2aSGMU/видео.html Hope that helps!
Technically, yes, however sometimes you are only interested in the line that contains that unit vector. For example, if you want to define a plane, then you only need a vector that is normal to that plane, and that vector can point in either direction.
Wouldn't the curvature of the motion of something also depend on the metric? Example Curvature of the earth To us it appears flat But further away, it is definitely spherical... Or to even go to another extraction Say another spacial dimension? What if the curvature is so big that we don't notice it? But if we shrink to that level we can?
Hi Leonard. Im learning a lot from your videos. In some sources i found that curvature is defined as T'(s) . N(s) (similar to torsion) to give the curvature a sign. is that any different? i noticed that if you take the abs value like you do torsion will always be positive.
I know I’m a little bit late to the party, but I can’t find a proof of why the radius of the osculating circle is exactly 1/K. Could someone point me in the right direction?
iam just wondering shouldn't ||T'(t)|| be equal to ||r'(t)Xr"(t)||? for some reason when i calculate for the former i get sqrt(2)/2 and just sqrt(2) for the latter. am i understanding this correctly?
Derivative of tangent is perpendicular to the normal vector, so because we want a vector that is perpendicular to the tangent and have length of 1 we get unit normal=T’/|T’|
And yes “one” of the normal vector is the derivative of the tangent vector, but there is infinitely many normal vectors (due to scaling). In this case we want a normal vector that have a lenght of one so there is just two of them (T’/|T|) and (-T’/|T|)
@@rafiihsanalfathin9479 thanks rafi , doubt not fully cleared i think i need to workout little extra, i missed the logic in normal vector side. let me prepare my question one's more!
My logic is, 1.derivative of vector function is r(t)(position vector)= r'(t)(velocity/tangent vector) 2. derivative of r'(t) is = normal vector ....right? . unit normal vector=T’/|T’|.....right?
0:00:00 Curvature
0:21:28 Torsion
0:26:00 Osculating Circle
0:43:00 Example 1
0:56:50 Singing
1:42:22 Example 2
1:56:32 Example 3
2:28:52 Example 4
Doing god's work here. Appreciate you.
Thank you sir.
god bless
Thanks mate
lol, that singing part got me, I was going over and over again, thought I missed some concept named singing"
Professor Leonard!!!!!! my fiancé and I really love you! we even named our foster puppy Leonard because of you. Instead of watching netflix at night we watch your videos together(we are classmates). He also says that he loves that you are super muscular and his dream is to train at the gym with you lol. Anyways, I just wanted to thank for all the effort you put to really clarify complicated things to your students and people who watch your videos. We are very thankful to have access to it. You just make it so easy to understand and you really can see where the students can possibly make mistakes and you point it to us even before we notice it. That is talent! I wish my teacher have half of the talent you have. Wish there was a way to do something nice in return. the least we can do is to let you know how awesome you are! God bless you and your family! Thank you so much!!!!
+Giovana Fraenkel Thanks very much! Give little Leonard a puppy biscuit for me :)
Professor Leonard's muscles are awesome! He has so much for us to look up to!!!
I hope your fiance isnt flirting with professor leonard, LOL. Please check that
This is the most impactful description I've seen of TNB curvature ever. I remember covering this in Calc 3 section of James Stewart. All I really understood were how the derivations worked and definition of unit tangent, normal, and binormal, but you sir Professor Leonard gave best informal explanation on why we need them. To understand curvilinear motion in moving reference frames and how the arc-length can serve as a parameter that simplifies the calculations immensely. Its always good to see the big picture, cheers.
I'm now at my last year of university in Italy and you're actually one of best professor I've ever had. Thanks professor Leonard.
best part: The Leonard bursting into song at 57:00. no seriously, the technique and explanation throughout not only made me understand on both an intuitive and mathematical level, but saved me from mistake opportunities throughout the calculation. Thanks once again, Mr. The Leonard.
8 years and this lecture is still relevant, wow
Watching this as a review and I have to say, I've never had a professor even half as good as you. That being said I've had some great professors. Amazing work Leonard!
Some people are Christian, some are Muslim and others are Jewish. Me personally, I'm a Leonardian.
😄😍😍👍💜 I'm a Leonardian too
+thebigVLOG OMG !! this is hilarious... Btw, I am leonardian , too !!!
Sou Ling please,do you know what is the horoscope sign of professor leonard? I'm waiting for your answer....
hhhhhhhhh you're stupid
At least , Prof Leonhard doesn't talk gibberish
Ok, Prof. Leonard, you're my professor this summer, since my remote class is not going as well as I hoped. Thank you!
I know this is late, but isn't at 15:00, ||dT/dt|| = ||T'(t)|| instead of just T'(t).
Also, thank for so much for these video lectures!
yes
YEAP I noticed that too
I'm praying that you're teaching linear algebra next semester.
funny seeing you here Adam, lmao I recognized your name
Can someone tell what uni does he teach at? 4 years on amd still saving GPAs😭
@@CamGersh I know this is four years later but this shit happened to me,
some dude ive known for a couple of years at my bjj place popped up in a random comment section on the most random video ever and I was just like wtf
and its weirder that this guy is RUclips famous and ive known him for years
Professor Leonard, I just want you to know that not only you have help me understand math and the theory at a deeper level, but also have fun during the process. thank you for your wealth of knowledge!!
Great Teacher. at 22:30, wish he had mentioned that Torsion is merely existence of change in Bi-Normal, just like Curvature exists with change in Normal. The vector perpendicular to a plane of paper (biNormal), when it starts changing its perpendicularity, Torsion creeps in.
I got an A+ on my Calc 3 exam. Thank you Professor Leonard!
at around minute 15, you forgot to put the magnitude for the derivative of T in the curvature function.
You helped me get a B in calc 2. I wish I found this for calc 1 as I would've done better but now I am ready for the next level. Thank you sir.
I love your lecture!!!!! Man, my prof does not give this type of intuition with regard to the formula. He just dumps formula on us. YOU ARE AWESOME!!!
Thank you so much for what you are doing! Our calculus 3 teacher was out during this section and had us teach ourselves. I nearly cried doing the homework but you are helping me through it! Thank you!!!!
Professor Leonard ,thank you for another mammoth and lengthy video/lecture on TNB Frames, Curvatures,Torsion and Encapsulation in Multivariable Calculus. From watching the lecture and doing problems, I understand the material from start to finish.This material is useful in Statics, Dynamics,Physics and Applied Mathematics.
I appreciate the planet earth because there is professor Lionard in it. The way ur explaining and the energy u have I havent seen yet.
You really helped me. Thanks 10^n times.
56:52 Come on professor, you know it is against the law to drink and derive. (:
No worries, he doesn't drink and derive, he derives while he drinks.
so glad you are doing calc 3 just intime for my summer course
I missed lecture this day at my college. Thank you so much for this!
This man is a legend
These videos are like finding gold on YT. Thank you Prof.
people will remember this men for the good or the bad for saving a whole generation of engeneers.
Professor Leonard!!!!! You are really really so great teacher. May God bless you. thanks a lot.
2:23:56 "and that's a big but" 😂😂😂😂 that made my day.
Hocam(Sir) whenever i need sth related with calculus you are the one help me a lot .Thax for everthing
you are really amazing i am watching your lectures from Pakistan and never seen a teacher that teaches like you
The only thing I'll miss about taking Calc III is Leonard. Literally my savior
In Sri Lanka the way our "Lecturers" lecture us is very different ...If we had people like prof.Leonard .... we'd pass our exams real easily. They don't teach us like you do. We HAVE to depend on RUclips! People like you are GODS!
You are so Awesome +Professor Leonard If it weren't for your Statistic videos I would have never gotten through Intro to Stats and Prob with a 93% Thank you so much for posting videos. You have no idea how much they have helped me and many others!!! =0)
you know those "kiss the cook" aprons, I just got a mental image of a "kiss the curve" apron that would say "osculating" on its back
XD lol
thanks prof. Leonard🦸♂️
Best teacher hands down helped me with calc 3 soooooooooooooo much!!!!!!!!
Sir You are phenomenal.. You are the definition of a great teacher...You turn the complicated things into easy...I was having trouble with multivariable calculus but then I found your videos and you are amazing...I wanna Thank you from the bottom of my heart for making maths interesting..I honestly can watch your lectures for the whole day ...If it's in my destiny then hopefully I'll meet you one day and touch your feet(Indian tradition for a way of showing respect to elders)..May you and your family stay in the pink of your health, may you find all the happiness and success in life and may u keep posting awesome math videos..I'll even watch your storytime videos 😛...I TRUST THE LEONARD!!
You are an absolute legend. Thank you so much.
The formula for curvature at 15:10 the unit tangent numerator forgot the magnitude sign? @1:11:54 kinda confirms it. Thanks again Leonard
This guy is so good
10:19
18:16 That's a cute proof, saves us so much work.
19:36 Polynomials Only
2:08:18 Evaluate for T when Unit Tangent Vector is found.
1:09:06 it might be really confusing but if you do the work, the J component has 3 negatives - -(-cost)J ==> -costJ
@0:15:54 the part where he sat down to wrote on the board my respect increased by infinity
love you sir i have no words to explain my feelings, stay blessed forever
"You're twisting the airplane. So twist it, turn it..." --- I kept waiting for you to say "pull it" so I could say "bop it" :)
Awesome, really enjoyed your teaching.
Thank you once again for these videos you are the best man
Why does the formula at 19:45 works? Proof?
The best teacher
At 1:11:35 , isn't T x N = (sqrt(2) / 2) < sint(t) , cos(t) , 1 > ? Where did the -cos(t) come from? For the cross product, the k component is -( - cos(t)).
+Nathan Tonning B_y = T_z * N_x - T_x * N_z = 1 * (-cost) - (-sint) * 0 = -cost - 0 = -cost
Nathan you are correct. Luckily, cosine of pi/2 equals 0, so it had no effect on the equations of the planes. Good catch!
What does he mean at ~12:10 "we can make this little chain rule substitution... we can do this as just... a chain rule"?
Great video!
I think I miss something on the last example before he switched to the next/last session, @1:55:39, answer for K?
The formula for the unit binormal at the beginning of this video it is said that it equals both TxN and (r'xr'')/||r'xr"|| but this was never explained and I don't understand where it came from . . .
At 25:20 does he mix up N and B with his fingers. Shouldn't N be middle finger and B be thumb in the right handed TNB frame
very good lecture.....thanks
GOAT
I would love to take precal - linear algebra with you professor! Im in calc 3 and it feels real "bare bones" compared to your in depth lectures. Love the brake downs and buildups.
Hey, Professor Leonard! In this video at 2:21:28 isn't the cross product of i x k=-j? If so then wouldn't the answer by y=-1?
I don't see why j doesn't matter if positive or negative, like he says in the video. To me, it does matter.
Brandon Ramirez
Well, I know that this reply is too late , but I got to explain to those who want to know .
In the eq. of plane you must plug the value (-1) in the both side
Which means (-y=-1 ) so it doesn't matter if it was +ve or -ve
So curvature is how fast the direction of the motion of the particle is changing per the speed of the particle. If the nose of the plane is changing direction slowly but the plane is moving quickly, then the plane is moving in a fairly straight line. If the plane is moving slowly, however, and the direction the nose is pointing in is changing quickly, then the plane is doing spirals.
I gotta be honest, TNB frames made a whole lot more sense to me without the airplane analogy.
it did help though. think about tnb as an aerospace engineer or a pilot
52:33 Why did he change the magnitude from 1/sqrt(2) -> sqrt(2)/2? Can someone explain? Or even @Professor Leonard?
it's called rationalizing
Thank for this today in lecture my professor was crazy enough to find b and use vector product.
For torsion: t = (r'(t) x r''(t))*r'''(t)/(...) is the triple prime supposed to be a double prime? because later on you write, torsion=-dB/ds*N, and the N has a double prime (r''(t)) or T'(t)
You're the man
Another great class!
I am so scared to fail in this course. Our PI passed everything so fast and now he waits from us to be able to solve everything until double integrals. I have 1 exam tomorrow but I am eating from its time just to be prepared for this course's exam. Well we will see I have 4 days.
If I do well on that exam I'll rewrite all of this courses notes in LaTeX and will share it with you guys that you can just relax and watch, follow with a good document of Leonard's Notes.
1.56
This is late, but at 49:40, shouldn't the thing he writes be T(t) instead of T'(t)?
You're right, he did correct it though @ 52:08
Professor Leonard - Do we ever add the magnitude of the velocity to be derived. I am working on this topic now and my book is adding the magnitude to each variable and then derives it. Thank You!
you should posts your old exams :)
I'm thinking there may be some engineering students out there- as helpful as Prof Leonard is for Calc, are there any channels that are as helpful for Physics?
EE here.
I really liked SBCC Physics when I was in Physics II. I don't know how the newtonian mechanics portion is, but I thought the E&M videos were great lectures. Here's the link:
ruclips.net/video/0_ITe2aSGMU/видео.html
Hope that helps!
MICHEL VAN BIEZEN IS THE GOD for Physics!!
the most viewed channel of mine
towards the end, shouldn't î x k = -j instead of j?
2021 Fall Math1014 w/o Leonard: B
2022 Spring Math2011 w/ Leonard: (A+?)
2:22:30 isnt i cross k equal to -J ?
Technically, yes, however sometimes you are only interested in the line that contains that unit vector.
For example, if you want to define a plane, then you only need a vector that is normal to that plane, and that vector can point in either direction.
Wouldn't the curvature of the motion of something also depend on the metric?
Example
Curvature of the earth
To us it appears flat
But further away, it is definitely spherical...
Or to even go to another extraction
Say another spacial dimension?
What if the curvature is so big that we don't notice it? But if we shrink to that level we can?
You are AWESOME!!!
2:21:30
I x k = -j not +j
He explained why the sign doesn't matter
He explained why you can either use positive or negative j for that case.
this shit popped in my recommendations the day after i learned about its content at school 😱
Hi Leonard. Im learning a lot from your videos. In some sources i found that curvature is defined as T'(s) . N(s) (similar to torsion) to give the curvature a sign. is that any different? i noticed that if you take the abs value like you do torsion will always be positive.
56:50 -- what an amazing lecture xD
reminder 1:43:22
I know I’m a little bit late to the party, but I can’t find a proof of why the radius of the osculating circle is exactly 1/K. Could someone point me in the right direction?
unit normal is derivtive of tangent by magnitude of derivative of tangent
one doubt, whats the derivative of tangent, i mean just tangent "is it normal vector
first time i saw professor leopard
Can you do Differential Equations!!! Please
What book is this? It's definitely not Stewart's 8th. Thank you-
iam just wondering shouldn't ||T'(t)|| be equal to ||r'(t)Xr"(t)||? for some reason when i calculate for the former i get sqrt(2)/2 and just sqrt(2) for the latter. am i understanding this correctly?
+Hussein Elsherbini oops major mistake in my logic, i equated a part of a ratio to the other which isn't necessarily true.
Prof, I love u
15:55 Little Leonard
56:11-57:11. Made my day haha.
Is it gonna take long for function of two variables to be in the channel? Please hurry 😢
+Raghad Abdulaziz Starting today. Check back later.
I feel so lucky I hadn't started my calc3 2 years ago haha. you're amazing professor.
also your puns are amazing and always have a point! :)
then normal is derivative of tangent right, i mean unit normal. right? am so confused.
Derivative of tangent is perpendicular to the normal vector, so because we want a vector that is perpendicular to the tangent and have length of 1 we get unit normal=T’/|T’|
And yes “one” of the normal vector is the derivative of the tangent vector, but there is infinitely many normal vectors (due to scaling). In this case we want a normal vector that have a lenght of one so there is just two of them (T’/|T|) and (-T’/|T|)
@@rafiihsanalfathin9479 thanks rafi , doubt not fully cleared i think i need to workout little extra, i missed the logic in normal vector side. let me prepare my question one's more!
My logic is, 1.derivative of vector function is r(t)(position vector)= r'(t)(velocity/tangent vector) 2. derivative of r'(t) is = normal vector ....right? . unit normal vector=T’/|T’|.....right?
then what is the derivative of normal vector "is there any derivative for normal vector or is it possible to take derivative of normal vector"
you seem like a good teacher. i think you should split your vids into smaller sections
That was messed up that they didn’t finish that frozen bit and left us hanging🥺
1:14:03
This video made me POOP!
1:56:31
Than