Outstanding lecture. Correction: OUTSTANDING lecture. I'm self-studying calc. 3 using Thomas' text. This lecture was invaluable in explaining some of the finer details. just wanted to say thanks.
I go to an Ivy League and something I've found is that while professors here are undoubtedly brilliant in their fields, this doesn't always translate to them being equally great lecturers and educators. You sir, are BOTH! Thanks for getting me through Calc 2 last semester and Multi this semester! You are incredible! :)
The most frustrating thing about these lectures is how I find myself thinking of topics and insights Prof makes as obvious whereas before they were totally oblivious to me! Love these lectures. They are helping me get through my Calc 3 course while all universities have gone online.
Professor Leonard has been saving mine and so many of my friends' asses and I just wonder if he even knows. He makes calc easy and I cannot express how great his lectures are. Forever grateful and happy to have these lectures available to me.
2:32 - You just decreased my trepidation seeing radical squares a ton. You seem to answer all these questions i have at the back of my mind. 8:48 49:35 50:25 53:58
The TNB frames is about to come up in my Elementary Differential Geometry course, but I was able to watch this and make a link between the TNB frames and the Galilean and Lorentz Transformation matrices. I was astonished to come to the realization of how much math impacts the physical world, especially when it slaps me in the face like the TNB frames did. I love these math lectures! I hope to see more throughout the rest of my career as a math major and beyond :)
I majored in physics and working as a software developer, and have decided to try and write a physics engine from scratch just for fun. Turns out TNB frames are extremely important when modeling particles, and its been ages so this and the next video have been a great refresher!
When you explain , it seems like everything is in the front of the eye . Thanks for improving my imagination about math,i really enjoy math, specially calculus now :)
Wow this is my first time watching your videos for calc 3. I’ve always passed by them because of the length, but wow these are well worth it. I truly feel like I’m a part of the class.
Finished the whole playlist (some of them watched twice). Thank you professor Leonard for helping out everyone who has professors who are great in MATH, but bad at teaching.
Weirdly enough, I really enjoy that I can see this video at 60FPS. Keep up the good work Leonard. I'm a 17 year old living in Spain, and I plan to start learning calculus through watching your videos (Here in Spain we don't exactly learn calculus, but some parts of it I think (So far I've never had a math class called ''Calculus''). Hope I reach this video someday (Starting from your calculus 1 :) )
+Maksym Hoyda Start early man. It's definitely worth it. About 8 months ago, Leonard introduced me to calc 1 and 2. His approach was so intuitive that it led me to take online classes on multi-variable calculus, vector calculus and even differential equations. I'm currently studying Linear Algebra and I hope to do upper division mathematics courses soon. This is all thanks to Leonard's awesome introduction. I swear. I went from a recent high school graduate who thought math was useless to just a math lover in 8 months. If I had the money, I'd double major in mathematics.
I'll second this. Leonard is a gift. I'm glad his videos are available for my son's as they come up through math (including his videos on precalc topics).
The lectures are just amazing. I rarely comment on someone's video, but as I am watching more of your video's I am truly appreciating the beauty of Mathematics. Thank you, Professor Leonard.
Raise your hand if you think Leonard is GODLIKE! Seriously, Professor. My current Calc 3 instructor admits to lecturing without preparation, constantly stumps himself, and has ABSOLUTELY no structure in his whiteboard notes. The ship was sinking and SuperLeonard came to carry it off sea and fly it to safety, That is why your profile picture is fitting! Thank you!
Thank you so much for these lectures, I will forever be in debt. The whole point of school to me is to learn and I can definitely say I have learned from your lectures
This dude is so good that he is making my robotics thesis easy!! Thanks Professor Leonard. You are helping me learn math since my first day of college. Still going after four years!!!!
"Walk the curve" thats a nice description of parametrization by arc length. Another description is , a curve r(t) is parametrized by arclength when t = s. So instead of using time to point to the curve via the tips of arrows, we use arclength. Walk the curve also captures this idea too.
I took a 3 semester Calculus series at a local community college, but the last semester did not go deeply into this. This is really exciting stuff. So much better taught then classes I attended
So at 1:50:00, I believe your binormal vector is pointing in the wrong direction. If B = T x N then the binormal vector should be pointing out of the board if you utilize right hand rule. I'm confused though, because it would make sense for the plane to twist in the direction that you drew it, but that doesn't follow the formula.
Professor thank you for your videos. I am sure everyone would love if in the end of the videos, You could upload the Home Work you give to your students.
My university professor flies through her lectures and I don't process at her speed and can hardly keep up with her explanations. I come to these video after lecture and thank the stars for Professor leonard. Can we get a rush on Linear so I am not dying next semester! Thanks so much.
They always do, "The result is left to an exercise for the reader" when they don't want to do the work to explain how to properly use the formulas for the functions.
@@justanotherguy46 Depends on how insightful the derivation is, though. For example, it isn't really necessary to prove that the Vector Product has the distributive property over addition, because if the definition of the Vector Product has been explained well enough then it is extremely easy to prove that property - it's 100% vector algebra at that point.
The lecture videos our professor assigns us to watch are narrated by a woman who sounds like she would rather be doing literally anything else. They bore me to tears every time. Your vids are a breath of fresh air. Thanks for doing this 👍👍
I have noticed that a lot of math teachers have a tendency to always be super-serious, so Professor Leonard's much more relaxed and welcoming style is definitely really pleasant.
for anybody wondering, these videos cover the exact same material in the textbook Calculus Early Transcendentals 9th Ed. but the chapter #'s are off so for example this is section 13.3 for me, not 12.3
In the solution you wrote to the example at 52:27, you should change it from r(s) to r1(s). Writing r(s) inplies that you're just plugging in s into the original function r(t), implying that r(s)=r(s/rad14), which is incorrect
Thank you very much pro for whole your effort in making then extremely brilliant maths tutorials, words can't describe how delighted & privileged i am to have learned maths through you vidoes " GOD bless you"
Professor Leonard ,thank you for a fantastic video/lecture on Arc Length/Parameterization and a solid introduction to TNB Frames in Calculus Three. I am familiar with Arc Length from Calculus One/Two, however , the TNB frames are problematic.
Thanks for the wonderful lectures Sir. I really love Mathematics when i know the concept behind what i do. If i haven't found your videos i would have definitely failed my tests.
@@nuzhatazim8194 Truly hahaha never thought of it that way. That tone is probably part of what makes him such a great teacher. The audience naturally wants to listen when it sounds like a secret. This prof is the GOAT of calculus lecture lol
@@cassied9327 If teachers were this enthusiastic and passionate about the stuff they are teaching, students will also want to learn. The energy is a hugeee factor on how you deliever the information..
Your videos are great, but it would really help if you had some sort of time indexing showing which part of the video you did which topics, so that we don't have to spend a ton of times finding it. Thanks!
I think wings are Normal vector and Tail will be B imagine the plane going in a straight line on a runway. then T,N,B will coincide with XYZ where B will be in Z which is the direction of tail Please correct me if this is not the case
+Caarve He said in one of the recent videos in this playlist that he had a Master's Degree in math - I think it was in section 12.2. He has also said at a slightly earlier point in this playlist that he had been teaching math for 12 years - it was the video where he showed a graph on a computer towards the end, something like "I have been teaching for 12 years and they finally give me a computer that works!".
Why arc length parameterization directly gives us the unit tangent vector. What's the intuition behind? Why do we want to walk along the curve if pointing by vectors gives us almost everything we want?
I do think there is a slight mistake in the last question for finding the arc length. If I apply the direct formula for the integration rather than what professor Leonard did, I get a slightly different answer. Can someone please verify this and confirm???
28:00 Shouldn't we have a +1 at the end? Kind of like this: x(5e+1) = 5ex + x. So when we take x out, we need to make sure x/x = 1. EDIT: 28:15 OK, nvm.. Someone's pointed that out.
I really hope one of his student's got professor Leonard a light saber. If not I'll order him one after I get an A+ in calc 3. Also, thanks for helping with the A+ in calc 2 ;)
To all the people who responded on my horrible spelling of various words throughout my comment: It is a riff off of the misspelling of "particle" as "particale" in the video. Not being a spelling nazi here; Professor Leonard pokes fun at is as well. TLDR: Learn to appreciate sarcasm, your life will be less filled with rage.
How could we know that it would be a good idea to distribute [cos(t) - sin(t)]^2 and [cos(t)+sin(t)]^2 at 29:30 Because before distributing it, it seemed like it would just make me take longer to answer since I'd be doing more calculations.
hello professor Leonard, why is that when we integrate an arc length of a helix vector function, the initial and final point final point for integration always comes from the z-coordinates?
At time 1:01:20 you're missing the prime on the definition of s(t) = the integral of the magnitude of the DERIVATIVE of the magnitude of the vector function.
The lecture was really awesome sir, but I've a small doubt. Since both Normal and Binormal are perpendicular to tangent vector. How could we confirm that T' ( Tangent vector to the Tangent Vector) actually gives Normal (Turning) and not Binormal (Twisting)? Please help me out with this. Thanks a lot BTW for the lecture series.
if i may suggest a thought process, try thinking of Plane curves, ie curve completely resting in 2-D plane, say a Paper sheet. You can easily visualize the tangent, the normal lying in the plane (can be obtained by rotating the tangent by 90 deg, in the plane), and a 3rd vector going out of the paper sheet. Now think of slowly pulling this curve out of the paper, and imagine the corresponding vectors. Mathematically, you can get a curve-Normal, by arc-length reparametrization followed by second differential. Then move on to biNormal by crossing the Tangent and curve-Normal. Hope it helps !!
professor leonard is a role model for future math teachers and professors
shut up
@@wakeupthisisntreal8168 nice one man
Outstanding lecture. Correction: OUTSTANDING lecture. I'm self-studying calc. 3 using Thomas' text. This lecture was invaluable in explaining some of the finer details. just wanted to say thanks.
@Kim Timothy EnghThomas Calculus textbook
this lecture is way too long its so much easier learning from the textbook
@@zakusa9891 you know you can speed it up right. i watch it on 2.25x speed and it works for me, plus cuts it to just about an hour of lecture
@@JasbrinaD I'm done with calculus so goodbye I'm unsubscribe to professor Leonard because I don't want to see any more
I am studying calculus 5 lol
I go to an Ivy League and something I've found is that while professors here are undoubtedly brilliant in their fields, this doesn't always translate to them being equally great lecturers and educators. You sir, are BOTH! Thanks for getting me through Calc 2 last semester and Multi this semester! You are incredible! :)
The most frustrating thing about these lectures is how I find myself thinking of topics and insights Prof makes as obvious whereas before they were totally oblivious to me! Love these lectures. They are helping me get through my Calc 3 course while all universities have gone online.
shut up
Reparameterization starts around 40:00 for those interested. At 1:35:10 TNB intro starts.
Literally was looking for a comment about reparameterization
shut up
godsend, thank you
Professor Leonard has been saving mine and so many of my friends' asses and I just wonder if he even knows. He makes calc easy and I cannot express how great his lectures are. Forever grateful and happy to have these lectures available to me.
shut up
2:32 - You just decreased my trepidation seeing radical squares a ton. You seem to answer all these questions i have at the back of my mind.
8:48
49:35
50:25
53:58
The TNB frames is about to come up in my Elementary Differential Geometry course, but I was able to watch this and make a link between the TNB frames and the Galilean and Lorentz Transformation matrices. I was astonished to come to the realization of how much math impacts the physical world, especially when it slaps me in the face like the TNB frames did.
I love these math lectures! I hope to see more throughout the rest of my career as a math major and beyond :)
I majored in physics and working as a software developer, and have decided to try and write a physics engine from scratch just for fun. Turns out TNB frames are extremely important when modeling particles, and its been ages so this and the next video have been a great refresher!
Its hard to find a professor that understands right not rote. I appreciate the videos.
When you explain , it seems like everything is in the front of the eye .
Thanks for improving my imagination about math,i really enjoy math, specially calculus now :)
Wow this is my first time watching your videos for calc 3. I’ve always passed by them because of the length, but wow these are well worth it. I truly feel like I’m a part of the class.
shut up
@@wakeupthisisntreal8168 no u
ratio
@@boppe2235 uno reverse
Finished the whole playlist (some of them watched twice). Thank you professor Leonard for helping out everyone who has professors who are great in MATH, but bad at teaching.
Weirdly enough, I really enjoy that I can see this video at 60FPS. Keep up the good work Leonard.
I'm a 17 year old living in Spain, and I plan to start learning calculus through watching your videos (Here in Spain we don't exactly learn calculus, but some parts of it I think (So far I've never had a math class called ''Calculus''). Hope I reach this video someday (Starting from your calculus 1 :) )
+Maksym Hoyda Start early man. It's definitely worth it. About 8 months ago, Leonard introduced me to calc 1 and 2. His approach was so intuitive that it led me to take online classes on multi-variable calculus, vector calculus and even differential equations.
I'm currently studying Linear Algebra and I hope to do upper division mathematics courses soon.
This is all thanks to Leonard's awesome introduction. I swear. I went from a recent high school graduate who thought math was useless to just a math lover in 8 months. If I had the money, I'd double major in mathematics.
I'll second this. Leonard is a gift. I'm glad his videos are available for my son's as they come up through math (including his videos on precalc topics).
did you reach it?
y wud someone WANT to go thru the torture of calc
I wonder what you are doing now.
this guy teaches the concept so clear ! Thank you so much Leonard
The lectures are just amazing. I rarely comment on someone's video, but as I am watching more of your video's I am truly appreciating the beauty of Mathematics. Thank you, Professor Leonard.
As a prior mechanic on airplanes in the USAF, the use of an aircraft to describe the TNB was so helpful!
Raise your hand if you think Leonard is GODLIKE!
Seriously, Professor. My current Calc 3 instructor admits to lecturing without preparation, constantly stumps himself, and has ABSOLUTELY no structure in his whiteboard notes. The ship was sinking and SuperLeonard came to carry it off sea and fly it to safety, That is why your profile picture is fitting! Thank you!
Thank you so much for these lectures, I will forever be in debt. The whole point of school to me is to learn and I can definitely say I have learned from your lectures
shut up
This dude is so good that he is making my robotics thesis easy!! Thanks Professor Leonard. You are helping me learn math since my first day of college. Still going after four years!!!!
I just wanna say thank you so much for all of these videos, your style of lecture breaks everything down so efficiently!
shut up
"Walk the curve" thats a nice description of parametrization by arc length. Another description is , a curve r(t) is parametrized by arclength when t = s. So instead of using time to point to the curve via the tips of arrows, we use arclength. Walk the curve also captures this idea too.
1:40:00 Tangent, Normal and Binormal vectors
I took a 3 semester Calculus series at a local community college, but the last semester did not go deeply into this. This is really exciting stuff. So much better taught then classes I attended
wonderful lecture with an engaging professor -- watched the whole thing in one sitting! thank you for the amazing lectures!!
You're one of the best professors I've ever seen!!!
Thank you professor , for the first time in my life , I'm not memorizing maths , and everything makes sense to me.
I really feel more than in class with your videos. thanks a lot professor may GOD bless you more and more with knowledge
So at 1:50:00, I believe your binormal vector is pointing in the wrong direction. If B = T x N then the binormal vector should be pointing out of the board if you utilize right hand rule. I'm confused though, because it would make sense for the plane to twist in the direction that you drew it, but that doesn't follow the formula.
Yeah, I think he swapped N and B accidentally.
Professor thank you for your videos. I am sure everyone would love if in the end of the videos, You could upload the Home Work you give to your students.
shut up
with exams approaching , you are a real hope . Many thanks !
My university professor flies through her lectures and I don't process at her speed and can hardly keep up with her explanations. I come to these video after lecture and thank the stars for Professor leonard. Can we get a rush on Linear so I am not dying next semester! Thanks so much.
Hi, what's your experience with the linear class? In case you need help, reach me out
Thanks for the lecture, my professor just wrote down all the equations without any examples and expect us to know how to use them.
They always do, "The result is left to an exercise for the reader" when they don't want to do the work to explain how to properly use the formulas for the functions.
@@justanotherguy46
Depends on how insightful the derivation is, though.
For example, it isn't really necessary to prove that the Vector Product has the distributive property over addition, because if the definition of the Vector Product has been explained well enough then it is extremely easy to prove that property - it's 100% vector algebra at that point.
The lecture videos our professor assigns us to watch are narrated by a woman who sounds like she would rather be doing literally anything else. They bore me to tears every time. Your vids are a breath of fresh air. Thanks for doing this 👍👍
I have noticed that a lot of math teachers have a tendency to always be super-serious, so Professor Leonard's much more relaxed and welcoming style is definitely really pleasant.
I have been watching his lectures since calc 1. Finished calc 1 2 3 with As. Hoping to ace vector calc now!
Amazing!
Greatest calculus professor ever.
for anybody wondering, these videos cover the exact same material in the textbook Calculus Early Transcendentals 9th Ed. but the chapter #'s are off so for example this is section 13.3 for me, not 12.3
the mind blowing part of this lecture to me was @ 2:07:07 , i love when proofs uses prior knowledege and you feel rewarded for paying attention
Nice to see a fellow Nigerian.
This guy motivates me to study calculus and work on biceps at the same time
In the solution you wrote to the example at 52:27, you should change it from r(s) to r1(s). Writing r(s) inplies that you're just plugging in s into the original function r(t), implying that r(s)=r(s/rad14), which is incorrect
Thank you very much pro for whole your effort in making then extremely brilliant maths tutorials, words can't describe how delighted & privileged i am to have learned maths through you vidoes " GOD bless you"
Can you also add worksheets or the list of examples you covered in the video for us to try on our own. Thank you! Love the work that you are doing 🤝
Professor Leonard ,thank you for a fantastic video/lecture on Arc Length/Parameterization and a solid introduction to TNB Frames in Calculus Three. I am familiar with Arc Length from Calculus One/Two, however , the TNB frames are problematic.
Not gonna lie I was not a fan at first. but you have helped me understand and remember how things work, and for that I thank you and keep it up.
I cannot thank enough to you sir. You are an angel.
Thanks for the wonderful lectures Sir. I really love Mathematics when i know the concept behind what i do.
If i haven't found your videos i would have definitely failed my tests.
shut up
Why does it feel like everything he's telling is a secret LOL
Fr😂
I can never look at him the same way tbh lol
@@nuzhatazim8194 Truly hahaha never thought of it that way.
That tone is probably part of what makes him such a great teacher. The audience naturally wants to listen when it sounds like a secret.
This prof is the GOAT of calculus lecture lol
@@cassied9327 If teachers were this enthusiastic and passionate about the stuff they are teaching, students will also want to learn. The energy is a hugeee factor on how you deliever the information..
@@nuzhatazim8194 Love this! Couldn't agree more :)
I am Indian still i love to understand these lectures. These are just awesome
Thank you for all your efforts Professor!
Professor Leonard do you have lectures for Differential Equations?
You are so helpful you have no idea. I find it very difficult to find stuff online about this.
OMG! This is amazing. I'm so glad I found this channel. Thanks for your help!
i have to say, your videos helps me a lot to understand math, and thank you so much for putting this up!!!
For those just starting the video, he gets to the hard part when the Professor changes outfit.
Your videos are great, but it would really help if you had some sort of time indexing showing which part of the video you did which topics, so that we don't have to spend a ton of times finding it. Thanks!
If you didn't spend 1 min writing a stupid comment you would have found the section that is relevant to you already. It takes 20 sec.
this is a feature that youtube needs to implement
Just look where adds pop up...thats where it gets broken down
I just finished my electromagnetism course. Thank you!!!
Ap? Cuz I’m doing physics C.
@@mortiloniperpomontinson no, in junior year of undergrad, we have an electromagnetic theory course
Love ur teaching methods Leonard. Keep it up
I think wings are Normal vector and Tail will be B
imagine the plane going in a straight line on a runway. then T,N,B will coincide with XYZ where B will be in Z which is the direction of tail
Please correct me if this is not the case
I agree. The "turning in the direction of" would be the wings, not the "tail fin", I would think.
smart expansion and good explain .
Professor Leonard! what is your educational background? What degrees have you earned thus far in your life?
+Caarve
He said in one of the recent videos in this playlist that he had a Master's Degree in math - I think it was in section 12.2.
He has also said at a slightly earlier point in this playlist that he had been teaching math for 12 years - it was the video where he showed a graph on a computer towards the end, something like "I have been teaching for 12 years and they finally give me a computer that works!".
You must be Clark Kent, you change outfits so fast. Only Superman Can move that fast!
I just noticed that on the board's top left and right-hand corners, he wrote diagonal lines to demarcate where the video recording border is.
I have only one professor, Leonard.
Why arc length parameterization directly gives us the unit tangent vector. What's the intuition behind?
Why do we want to walk along the curve if pointing by vectors gives us almost everything we want?
Leonard does a real hack job explaining TNB here.
Loved this video it was great help!!!! Exactly what video were you talking about when you integrated sec^5?
I do think there is a slight mistake in the last question for finding the arc length. If I apply the direct formula for the integration rather than what professor Leonard did, I get a slightly different answer. Can someone please verify this and confirm???
Professor Leonard at 1.5x playback speed will get get me through Calculus 3.
I wish I could attend your lectures
Sir please make some videos on affine and euclidean geometry
28:00 Shouldn't we have a +1 at the end? Kind of like this:
x(5e+1) = 5ex + x.
So when we take x out, we need to make sure x/x = 1.
EDIT:
28:15 OK, nvm.. Someone's pointed that out.
Holy fuck man, you're just like my professor except clear, concise, and thorough.
sir you teach best
I think the orientation of B is wrong, if he's drawing it pointing "into the board" if B = T x N (1:48:05)
I really hope one of his student's got professor Leonard a light saber. If not I'll order him one after I get an A+ in calc 3.
Also, thanks for helping with the A+ in calc 2 ;)
TNB - 1:35:12
I am majoring in Particale physics, so thanks for bringing some light on my carrer of choise!
To all the people who responded on my horrible spelling of various words throughout my comment: It is a riff off of the misspelling of "particle" as "particale" in the video. Not being a spelling nazi here; Professor Leonard pokes fun at is as well.
TLDR: Learn to appreciate sarcasm, your life will be less filled with rage.
kinda hard to tell you were being sarcastic
The thing that is interesting is the fact that r'(t) is perpendicular to r"(t)... This is observed solely on vector functions?
Thank you sir, Godbless you
Thank you so much
How could we know that it would be a good idea to distribute [cos(t) - sin(t)]^2 and [cos(t)+sin(t)]^2 at 29:30
Because before distributing it, it seemed like it would just make me take longer to answer since I'd be doing more calculations.
Do you have something on finding osculating planes/circles? I can't seem to find it in this lecture. Big fan btw!
BIG FAN
>.< .You are the BEST. I like your way of teaching. Now I understand what I am studying *_*
hello professor Leonard, why is that when we integrate an arc length of a helix vector function, the initial and final point final point for integration always comes from the z-coordinates?
At time 1:01:20 you're missing the prime on the definition of s(t) = the integral of the magnitude of the DERIVATIVE of the magnitude of the vector function.
Where did you do your BA and MA?
wow, amazing video thank you for your work!
sorry sir, i want to ask at 1:32:57, it should be (sqrt 5) /20 right? not (sqrt 5)/10
Professor Leonardo can you please cover double integral section ; i have problem in finding the limits
Professor,,
Have u complete your all notes on bord,,
Becz we are making notes,,
Any thinking is mission ????
I want him as my teacher...he is so good in teaching😍😍😎
The lecture was really awesome sir, but I've a small doubt. Since both Normal and Binormal are perpendicular to tangent vector. How could we confirm that T' ( Tangent vector to the Tangent Vector) actually gives Normal (Turning) and not Binormal (Twisting)? Please help me out with this.
Thanks a lot BTW for the lecture series.
if i may suggest a thought process, try thinking of Plane curves, ie curve completely resting in 2-D plane, say a Paper sheet. You can easily visualize the tangent, the normal lying in the plane (can be obtained by rotating the tangent by 90 deg, in the plane), and a 3rd vector going out of the paper sheet. Now think of slowly pulling this curve out of the paper, and imagine the corresponding vectors. Mathematically, you can get a curve-Normal, by arc-length reparametrization followed by second differential. Then move on to biNormal by crossing the Tangent and curve-Normal. Hope it helps !!
Basically, i believe it is just a convention... we could define them the other way around, but we chose the way Professor Leonard explains.
My class is already on multiple integrals, do you know any other resources that can help me?
I love you so much leonard
36:00 smooth curve for vector function
dhanyawad guruji:)
"I'm a grown man! Every grown man needs a lightsaber, hello! Don't you know this?"
TNB starts at 1:35:12
why is he so much better than my prof
You are awesome, thanks!