Calculus 3 Lecture 12.2: Derivatives and Integrals of Vector Functions

[@ken6277]

This man is on a mission to save every engineering student from themselves... and for that, I salute you good sir.

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so real

[@eunuchcommunity]

0:00:00 Derivatives of Vector Functions 0:13:55 Example 1 0:19:15 Example 2 0:25:52 Example 3 0:37:47 Example 4 (Point of Tangency) 1:00:51 Example 5 (Tangent Vector) 1:09:48 Unit Tangent Vectors 1:28:35 Tangent Lines 1:32:05 Example 1 1:46:46 Example 2 1:55:53 Integrals of Vector Functions 1:56:28 Example 1 2:00:25 Example 2 2:12:13 Example 3 2:21:05 Example 4 (Initial Value Problem) 2:29:34 Example 5 (Initial Value Problem / 2nd Derivative)

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u da best

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you are a legend mate

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YAAAAAAAAAAAAAAAAAAAAh

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Thanks

[@Peter_1986]

Zoning out and looking at my phone for a few seconds. Professor Leonard: "Please look. >:-/" O_o

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Right?! xD I feel like I am in his class

[@tylerhurson8515]

Finally, someone who explains this stuff on the conceptual level. I can't tell you how many professors I've had who jump right into the examples without explaining _what_ the thing that we are studying actually is and why we use it. As a more abstract thinker, these lectures have helped tremendously. Is prof. Leonard planning on doing a Diff Eq course?

[@chad6288]

luckily many of those chump "teachers" will lose their job soon. Once we have VR and can all attend the best lectures on the planet. Won't need as many teachers. Should be able to clean out the ego big shots who royally suck at transferring knowledge. Or some company will create awesome entertaining videos to replace the boring lecturers yapping at the white board.

[@reemaalnafisi1793]

I totally agree. As soon as I understand the concept and where the equations come from, I can solve the examples easily. Prof. Leonard is the best!

[@shadanjaved8827]

the example of here the vector function will only be for positive value of x . since x=e^t it can't be negative... but you have drawn the vector function for negative x also.. how???

[@TheLastLoad]

LMAO I'm the complete opposite, just give me the formula and what to plug in. LOL!

[@mussarratanwar3545]

@@chad6288 online classes brought me here :(

[@robertjr8205]

One of the many things about this professor is that he keeps telling us the purpose for what we are doing. Even from things that were a couple sections back, he will give a brief explanation to jog your memory which I NEED and appreciate so much. Many professors do not do that which makes the students end up forgetting what they are even solving for and why they did what they did. Leonard is awesome

[@Kazu1193]

Words cannot describe how grateful I am to be able to view your lectures. You are an extremely talented professor. Please never stop teaching!

[@jellyman0053]

ditto!!!!!!!!!!!!

[@Iaconmatrox1]

1:33:00 These are not stupid videos. Your videos are getting me through university. Thank you

[@MrSmian]

By far the best way to learn calculus! Your videos got me through Calc 2 and i am using your videos to review for my accelerated multi variable calculus course and your videos are so easy to follow! You're Awesome! Now i just need discreet math and linear algebra

[@xolisa5096]

did you find a great teacher for those two?

[@ClumpypooCP]

you couldve just said "multivariable calc" but you HAD to specify "ACCELERATED"

[@Minexorek]

@@ClumpypooCP It was a jerked course, not just accelerated. If you know what I mean

[@danaingram9028]

Thank you so much for these videos. I'm having a hard time in Calc 3 at my university right now and your videos are what's getting me through! Honestly, my professor is so bad I bring my laptop and headphones to class and watch your videos instead of listening to his lecture.

[@wilurbean]

The moment you truly *_understand_* what a derivative is.... mind = blown. Its so simple, so brilliant, and changes _all_ of math. Prof Leonard brings it to the masses, brilliant.

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Someone make a "Trust the Leonard" shirt line now.

[@cedricvanhaverbeke5594]

Leonard, I wish i had a professor like you. Thanks for making me understand all of this!

[@jdrace12]

I can't tell you enough just how much I appreciate you. You are the only reason I will pass this class this semester. Thank you for taking the time to do this. I've watched hours of your videos and you make this genuinely interesting to me. I'm so glad that you exist.

[@adhiamboalali]

same

[@Seyi1990]

If you forget the +C, trust me you would see a C. Hillarious! I really hope Professor Leonard does diff eq sometime in the future even though I'm taking it next semester. His videos are the only reason I survived Calc II & III. Keep up the good work sir!

[@jameskwonlee]

This is the best lecture on the topic I have ever seen.

[@CaptainDC06]

I love watching you teach. You're such a pleasant person and an amazing educator. Honestly, you're a big part of the reason why I want to become a math teacher. Thank you so much for your time and dedication to helping us all learn! Also, I'm sure you can upload just about anything and people will tune in. You're a lot funnier than you think haha.

[@juliannahummel6221]

Professor Leonard, you explained the derivative of vector functions so beautifully. Thank you for these lectures. You are an amazing teacher!

[@palak1997]

You are doing a great work my friend. I have never seen like this in my life. You are special and best thing is you are stitching concepts in series and one never lose the track.

[@georgesadler7830]

Professor Leonard ,thank you for another well organized video/lecture on Derivatives and Integrals of Vector Functions in Calculus Three. This material is easy to follow from start to finish, however the derivatives and integrals can be problematic. A solid review of Calculus one and Two is necessary before taking Calculus Three. I took this course many years ago in Catonsville ,Maryland.

[@bezveze8047]

1:03:50 for 1/x^2 in case anyone is confused Only the right chunk of that graph applies. Terminal point "starts" very high up for t= - infinity and slides down to far right x axis for t = + infinity, all in Q1.

[@jamesvrtis8863]

Im interested in the construction story now.

[@satellite_anomaly]

I always appreciate examples like the one at 1:46:46, because those e^t(trig fn) formatted expressions are important in differential equations and wave mechanics.

[@your_name_here2158]

I wish I had the intersect in my head like Chuck, so I could spend four or so days just absorbing all this man's teaching

[@jaysonjacob9042]

hahaha

[@senkuishigami1331]

im not the one to send comments in youtube but i gotta say, THANK YOU for emphasizing that the integral of a vector function is not just simply the area under space curve at 1:59:28 , that is what i've been stuck with for some hours now.

[@Ohhsoodead]

Your voice is very soothing to listen to.

[@handenursezgen3405]

I would like to thank you, Professor Leonard, for such a great effort. Your videos are absolutely helping me. Looking forward for the upcoming videos about arclength parametrization and so on...

[@tauceti8341]

1:10:45 1:19:30 ******************** 1:22:30 Rationalize, don't distribute if you don't have to. Tangent vector, tangent point. Vector || to a tangent line Every scalar multiple is ||. Brilliant. 1:40:18 ... ***Equation of Lines Planes, Parametric & Eq. Haha that trump joke. 1:44:10 SYMETRIC Eq.

[@mikenim9164]

Wow, this is mindbogglingly easy when Prof. Leonard teaches it. Good teachers are SOOOO important to have in life.

[@Carnezz]

I definitely need a lot of P.O.T when doing calculus :) Makes it more enjoyable.

[@juanmanuelmillansanchez8165]

what is that? i did not understand the joke.

[@DuncanTruittP]

@@juanmanuelmillansanchez8165 Mary Jane

[@hypersonic6649]

Doing drugs while doing maths seems unhealthy

I'm so happy I am finally understanding what these things are. PLEASE upload lectures of any other subjects you teach!!!

[@chiragadwani1875]

In the first five minutes of the video, I understood that the slope vector would be broken down in components, which would be given by the gradient. Professor Leonard, you are the best teacher i have ever got. Thank you so very much.

[@Horhew]

"Pro" tip: use 1.5x speed to watch this if you're just doing a review.

[@Oingoboingo710]

Even more pro, use it at 2.0x

[@ernishkacooper2352]

Thanks saved me 4 hours of video watching 😩😩😂

[@fleskenialation]

I do 1.25x when taking notes and it's fine

[@steamcode4441]

I use 2.0x speed for taking notes, but keep in mind that you probably have to build up to that. Start at normal speed then move up incrementally until you can understand and write down what's being said at 2.0x speed.

[@jorgeluisrodriguez3271]

2:18:27 the constant of integration maen a "family of vector functions" differs by a constant.

[@fredesch3158]

Simplesmente, o mais brabo, ta maluco

[@archerdev]

Math bless you Prof. Leonard for this amazing lecture, thank you.

[@m.s.7105]

That conversation at 1:33:15 though! Keep making videos professor you're one of the most talented teachers!

[@laureanoprottiarredondo8691]

Queeee?(With Costa Rican accent lol) Good to know the profesor who is helping me out has visited my country!Pura Vida!

[@mimomira8808]

he's my man in explaining Mathematics

[@JMac___]

I think the portion at 1:59:02 with the integration is wrong, u ended up with the right answer, but the answer would be incorrect with different bounds for the definite integral.

[@stevennyeo4148]

1:58:27 Shouldn't 1/2(1 - 0)^2 be 1/2(1^2 - 0^2) instead? I am confused by the step of "collecting" you mentioned.

[@Horhew]

Yup, it should definitely be 1/2(1^2 - 0^2)

[@josephyoung5204]

Yes, I was confused about that as well.

[@usmanalam23]

Boy am I glad you're doing Calculus 3 videos. Any chance you have some videos on multivariable calculus (vague, I know) in the pipelines?

[@ProfessorLeonard]

+Darth Bearder This is Multivariable. We will be getting to the function portion in Lecture 13.x and 14.x

[@stevekhan4063]

+Professor Leonard noo ! i just finished calc 3.. can you do videos on differential equations and linear algebra ?

[@usmanalam23]

+Professor Leonard Oh boy I feel silly now haha. Thanks for the reply, I look forward to those videos!

[@garrettgeffre1792]

I love the comment he made at 1:27:30. I really wish more professors were more enthusiastic about teaching as he is.

[@Reform3d]

2:05:38 This joke must’ve went over most of their heads. I love how he embraces being “corny”. If I heard him say this, I would’ve audibly laughed and facepalmed at the same time. 😂

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You changed my life, Professor Leonard.

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Thanks so much prof leo your lectures are really helping me out so much for that am super grateful

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Thank you Professor Leonard

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Charlie bit me and kitty playing with yarn 🤣🤣🤣🤣🤣

[@gianny2977]

Thank you so much Professor Leonard! You truly help me understand and grasp the concept and really stick it in my head. Also, you should definitely do story times relating to your career lol

[@HosRo4161]

Thank you. One small point: at (t=1:58:25 of the video) maybe it should read: (1/2)(1^2 - 0^2)(i) instead of (1/2)(1-0)^2(i) ... in other words: difference of square (and cubes, etc) instead of square of (cube of, ...) of differences. Thank you again.

[@Metallurgist47]

I picked up that too --had the limits been any numbers but 1 and 0 ,the answer would have been wrong

[@marinasalmon48]

I wish my calc teacher cared as much about our understanding of the concepts as you do....scratch that, I wish you were my calc teacher 😁

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thank you sir 😀😀😀😀😀😀

[@shivamyadav-ez7uv]

dhanyawaad guruji 🙏🙏🙏🙏🙏🙏🙏

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I want to come up with a good description for you but words are not enough to do so. you are the real hero.

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The Leonard has never steered me wrong!

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Imgur brought me here!

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+Gehab how

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+Onam King I saw a post on Imgur a few days ago praising this guy and how he teaches. Maybe that's what he's talking about.

[@Gehab]

FeRackle Yup! That's the post I saw!

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he explains better than my teacher !!!!

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Absolutely Awesome Prof Leonard!

[@abdullah-al-mahdi125]

Hey Professor , I came across only a few people who look as cool as you and teach as good as you... Love from Bangladesh...

[@DJ-kp1nb]

Professor Leonard I love your videos! Thank you for doing what you do. You are literally my life line this semester. But perhaps I could make a suggestion. Would you try using colors of greater contrast when you want to emphasize a point? Black and purple appear identical on a computer monitor. Maybe you could use Black and Blue or Black and Red? Thanks again.

[@ProfessorLeonard]

+DJ Yaz Hi, there, thanks for the suggestion. I have tried different colors, but they do not show up as well.

[@fancymachines6554]

I don't skip ahead despite finishing problems before him because I like the commentary

[@migtrewornan8085]

28:35 That "i" doesn't belong on the right hand side of that equation, the left hand side is just f'(t) not f'(t)i

[@alphiephoenix5303]

I hope you know these videos are still being used!!! iive used your videos from college algebra to this one, once I pass differentials I have no idea what ill do hahaha

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Very Well sir the way you satisfies the students its excellent love and respect from pakistan .

[@jasdrawssometimes4660]

huuuh he has such a way with words

[@saadhassan9469]

You are an amazing instructor

[@matthewkazen2584]

So happy to see that you were uploading Calculus 3 lectures. Will you be uploading Differential Equations at all in the future? Taking Discrete Structures now which is a whole new way of looking at math. Really cool. BTW, do you even lift bro?

[@rubinglen]

wow, i can't believe i had to watch all those lectures to finally get to some calculus

[@mugruberkyser8189]

Does anyone else get so involved in his lectures that they forget to like his videos ?

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I just go back and like all the videos when I finish a playlist.

[@81546mot]

Another great lucid explanation.

[@IqraAslam]

How does he make it look so easy!!!!! I never thought I would be having this much fun for learning calculus.

[@hairyasstruman2257]

I wish I had someone explain vector calculus this well about 25 years ago

[@thelonegerman2314]

VERY CONCEPTUAL EXPLANATION

[@TeesarMarvish]

Your videos are very good and satisfactory for my courses.Thank you so much

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Incredible videos

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Im very glad to hear Mr Leonard say that he loves his job

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These videos are great! Thank you so much for these professors.

[@Charles-bz8kk]

1:09:58 unit vectors 👏 *drops the pen*

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He's the kind of teacher that can make students who hate math enjoy it

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I sometimes do Professor Leonard's dance moves when I warm up at the gym.

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great work sir...pls keep uploading more videos.they are of great use..

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Thanks for the awesome video! Helps a lot for yr 1 physics and beyond!

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i would watch storytime with professor Leonard

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Get professor Leonard over to my school 😭, he's by far my best. So simple and communicative. Love you sir !

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thanks a lot!!!

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Thank you so much :)

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you are the best prof.Leonard :)

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Thanku Prof Leonard..may God bless you

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Thank you sir. I would have dropped if I didn't find your videos. I hope you get everything you deserve in life. God bless.

[@permantayev1988]

you are great, good luck in your future career and life:)

[@xkronos815]

Had to pause the video to say that I would definitely watch "Story Time w/ The Leonard"

[@Peter_1986]

If I get complicated and annoying constant factors in my calculations, then I usually just rename them into "A" or "B" or something.

[@Messi20200]

Where did you get pi/3 from as your point of tangency

[@Tobiky]

Professor Leonard invigorates my love for math

[@alis5893]

You are amazing. I wouldn't understand Plasma Physics if it was not for your lectures. Thank you

[@masterofmath449]

I like the way he explains it in his videos he's probably the best if not one of the best calculus teachers out there the only problem that I have though is his videos are super long and how much time he takes to explain 1 whole section I don't have that much time to watch all of his videos because I have to study for my other classes too. A perfect example of how long his videos can be is his calculus1 5.2 video is volume of solids by disks and washers methods and that video is 2 hours 47 minutes and 49 seconds long. I don't have that kind of time to watch the whole thing and then still have time to study for my other classes.

[@greense65]

I guarantee you would learn more just watching his videos (and taking notes and doing the example exercises, just like you were in his classroom) and making up the time by not going to your class, but unfortunately that is often not an option.

[@taekwondotime]

2:10:00 *There is a small but potentially fatal mistake here* . When you combine c1 i + c2 j + c3 k into C, it should be written with the HALF ARROW over top to denote the fact that it is a *constant VECTOR of integration* and not a scalar. The individual components are scalars C = but not the final C that you wind up with.

[@Syedmurtaza-n4k]

for the constant vector we have at the end of the integral, do we add a vector arrow on top of the C since it is a vector?

[@TU7OV]

you could if you wanted

[@JunkieJournalism]

How do you do calculus? I need help.

[@ashishKjr]

A minor error at 01:04:00. The plot to the left of y-axis should not be there since the range of exponential functions is always positive.

[@stevenz933]

Ashish, I noticed the same thing and agree with you completely. I guess converting from t parameter to x and y can be misleading as proven here! Professor Leonard should have put a restriction on the domain for x: x >0, agree?

[@rocksbox156]

Can confirm

[@wrednymaz]

These lectures rock. Thank you,