Calculus 3 Lecture 15.3: How to Compute Line Integrals (Over Non-Conservative V.Fields)
Вставка
- Опубліковано 6 лют 2025
- Calculus 3 Lecture 15.3: How to Compute Line Integrals (and Over Non-Conservative V.Fields): An Introduction to Line Integrals and techniques of computation. Also, how to find Line Integrals over Vector Fields and an interpretation of their meaning.
Professor, I have to say thank you for making this free to watch. The education system currently has so many flaws that it is difficult to stay motivated. You would not believe how awful my professors are and how little they do. You are not just a great teacher but a great person for putting this online for others. Words cannot describe how thankful I am for you and I fully intend on paying you for your classes once I can afford to. Whatever you make as a professor it is not enough. On a side note, you're one hot piece of meat.
Same here dude
I wonder if he has a part time job on fashion show for men's clothing during vacations...
PREACH!
Lol the side note 😂🤣
@@subarnasubedi7938 on the side note?
28:14 - Example 1
35:24 - Example 2
46:22 - Example 3
1:03:09 - Example 4
1:17:03 - Example 5
1:27:10 - Example 6
1:46:32 - Example 7
2:03:08 - Example 8
👍💯
thanks
not all heroes wear capes
@@alvinnyakakweto8108 Sometimes they can be a buffed guy teaching Math on conceptual level on UA-cam
My calc 4 professor told me to drop his class after receiving a 66% on his midterm, just got my final back, 103%, highest score in the class. Thank you
Nate_396 foRrif different schools have different names
@@bigshaqsmathematicalinstit3318 For my Calc 1 course, I had an overall of 56% at week 13 of 16 and bumped it up to a 72% before the end. Not proud of the grade, but I was quite sick that semester, and I was able to pass. I ended up with a 98 in Calc 2 the following semester.
@kneeger
Calculus 4 can be things like "Differential Equations" or "Complex Calculus". I think that there is even a course called "Calculus 5", and that course seems to bring up Laplace Transforms and things like that.
Did you tell him it was because you were wasting time and coming to his class, all while you could have watched youtube instead, cause I would have.
ur professor is a walking L
The number of times I start crying in his videos because I finally, finally get it after spending hours in lecture, textbooks, and practicing problems is a little embarassing. He's so gentle in the way he walks us through these hard freaking concepts and its so different from the way I've been taught math - something rigorous, challenging, and honestly leaves me feeling sick to my stomach because I have to swallow down things that are not intuitive.
Thanks so much Professor Leonard. I hope this isn't inappropriate but you make my day and sometimes I really want a big hug from you and have you tell me everythings gonna be ok. Both my mom and dad have been sick and hearing an adult be so nice just makes me feel so safe, even if it is just during class.
Professor I am from South Africa and I am doing Bsc Civil Engineering at University of Kwa Zulu Natal.Your videos are helpful and they have helped me on calculus1,2 and three.You must know this know this here at south africa you are saving lives form failing math. maybe like 75% of 1&2 years student pass calculus with your videos. We have given you a name here which is Sgora it's a Zulu name which means FIT GUY. KEEP UP THE GOOD WORK SIR , all those writing negitive comment it's just that they needed your videos and you didnt post them when they needed them
lol,,, sunglekisa weh Mthandi :) :) :) :v :v :v
yini indaba
You are awesome
@Vhuthu Rodney na nne
Wow. Funny i come across this comment. Coz I am at UKZN TOO. AND DOING CIVIL TOO. This guy is a saviour. Wish i knew about him before classes even started to get ahead and get more under my belt.
Hi professor, I know you're probably never going to see this but I wanted to thank you anyway for these videos. I suffer from ADHD and I literally would be 100% incapable of passing this class if it weren't for these videos. The fact that they're free, 1-2 hours long, comprehensive, and taught so clearly is mind blowing to me. Thank you for uploading and giving me another shot at college math
I've learned more in first 12 minutes of this video than I have in the past two weeks of my professor trying to teach this stuff. Good work Professor!
I love this professor so much..He doesnt just give out formulas but explains concepts so deeply that one is able to understand exactly what is going on..As a mathematics and computer science student from Kenya,i am forever grateful to this legend
There are no dumb students;there are only proffessors who have no passion or understanding to what theyre doin.its amazing how you make math intuitive and interesting.mathematics should be given in the right way otherwise it loses meaning.thank you so much, i just adore your lectures;THEYRE SUPER HELPFUL
Bro, you're a beast. I just had a whole semester of being a lazy little bum and not really caring about my work. Suddenly had an epiphany and craving for learning in-depth. Your videos came in clutch man. Hopefully, I make a comeback and am able to do well in my courses following your videos next semester for differential equations. God bless you, and to anyone here early in the college or maybe high school career, this guy is a godsend.
I stopped the video at 1:30 because I just had to say that you are the best math teacher I have ever had. You really empower your students, including all of us who watch these videos, with your clear, systematic and comprehensive explanations. There is so much caring for the students in your approach to teaching. Saying this is the only way that I and the rest of us can give back to you.
The one thing that makes this professor so proficient in teaching is summarizing and reiterating concepts. Every time he talks about a new concept or goes over a problem, he reiterates what he said and summarizes what he just did. It will help embed the information in your mind.
You are so good at explaining this stuff it isn't even real. Relating the reparameterization/integrals into basic areas, volumes, and the stuff we've been doing all our lives is very important and a life saver on exams. Maybe I don't fully remember something all the time, but you help me know what I'm LOOKING for and I can go from there.
0:00 Introduction to line integrals (calculus of vector fields)
19:45 Changing two-variable integrals into single-variable integrals using parametrization
28:20 Example 1 (given parametric equation)
35:20 Example 2 (not given parametric equation)
46:20 Example 3 (no vector function, given segment)
1:03:05 Example 3 (segment in 3-space)
1:13:13 Notes about variations in line integrals
1:17:00 Example 4 (discontinuous segments-- not "smooth")
1:27:10 Example 5 (dx + dy + dz, instead of "ds")
1:37:23 Introduction to line integrals through vector fields
1:46:32 Example 1
2:03:05 Example 2
Sir, the more I watch your videos, the more I like math. :)
same
Because this guy actually explains math properly and _wants_ to teach it to people.
And the more I like him🥵 I mean math
I am honored to be your 69th like
how can somebody be so attractive yet so good at explaining calc 3, it is beyond me.
Well more than a decade watching UA-cam and this is the first channel
I've bothered subscribing to. You are a legend and a lifesaver. Finally
someone who teaches things slowly and thoroughly enough for people like
me that get distracted every half a second. Cheers no end.
7:49 this is why I love your videos. It isn't just formulas. You give those formulas meaning
God-level teaching needs naught but a marker & board. I went from a 46% on my first Calc 3 exam to 100% on my mid-term, and this channel is the ONLY reason. I will definitely remember to make my own little monetary support of this Professor as soon as I'm no longer living on loans and grants. Did way more for me than the Ivy-pedigreed PhD's in my Uni who could confuse you on how to make toast.
Jeez 46% is very bad
I'll do the same. Literally those PhD fellas are annoying as hell.
Professor honestly its so hard to find people like you, who know how to explain so well and make Calculus fun. A day doesn't go by when im in Calculus class and Im wishing for you to enter through that door and say "okay guys im your new professor for Calculus, lets have a blast!!!!!". I salute you Sir
What a great teacher. He refused to let his students lose focus and stop paying attention and he remained engaging and alive throughout the entire lecture. He could have easily just let his students drift off and as a result struggle later on but he didn't let that happen. May Allah guide you to Islam for your efforts.
7:37-7:49 is what my professor needs to understand... Thank you for being a great teacher
I wish I made more of an effort to watch all of your videos as my semester progressed, I'm so behind and may have to retake this class again. You are so wonderful Professor Leonard!
omg i looked away and in that same sec he goes like "are you listening right now" hahaha whoppps sorry
Are u pakistani or bangladeshi
The same happened to me right now :D I was just gonna write it then I saw your comment lol
He got me both times
He once told me to "stop texting" when I was texting on my phone, lol - he even looked directly into the camera when he said that, so that made it even more creepy (one of the students joked about the viewers texting instead of watching).
I think that it was in one of the Calculus 2 videos on Geometric Series.
As someone who deals with ADHD, i find your videos to be incredibly helpful. Very engaging. It's harder to not pay attention than it is to drift off. I would not have passed Calc 2 without you. Today i'm finishing up Calc 3 and doing great. Thank you. Thanks for caring about what you do so much.
YESS!
This guy should get a medal for explaining this so easily (and a raise).
thank you so so much Prof Leonard. I got over 80% for my exam this week!! thanks
I can not praise and thank you in words professor ......... I found you when my professor made me crazy and desperate ... and when I was at the verge of losing all of my determination. I have abandoned my classes and just come here...... the way you try to visualize the geometrical concepts is amazing .... in future I would remember you as my Cal3 instructor not the instructor in my uni. I would love to contribute to your efforts.
Prof Leonard is the very best--period!!!!!!!!!!!! He should teach every calc course in every university
Professor Leonard, you have taught me more in the first 15 minutes of this video than my current Calc 3 professor. I just want to thank you so, so much. You just saved me for my math finals!!!!!!
Almost done with this whole playlist, took 18 creds this semester so it was tough nailing down every concept but im just glad to be passing calc3 because of you Prof Leonard
I am a current materials engineering major at Imperial College London, and I have to say the way you explained this topic is much clearer than the way my lecturer explained it. Well done Prof.
absolutely the greatest professor to watch if you're needing to supplement your lectures during these online times!!
The greatest professor of all time in college Calculus course.
You're help has made such a difference, as a student who has taken this class before at the community college then transferring to University and having to take it again, I found the professors at University basically suck at explaining the theory. Professor Leonard your explanations are easy to understand straight to the point, and then you run through some examples bringing attention to important points. I can't thank you enough.
You are hands-down the best calculus teacher on the internet -- back in college I had uninterested professors, impatient TAs and a textbook -- you are making these complicated subjects understandable and approachable for students around the world -- I hope your college appreciates your talents and rewards your special skills bc you are truly one-of-a-kind
Professor Leonard, you've made me like calculus classes even more, thank you so much for your great job
You're awesome man. It's so convenient to be able to turn to your videos when I'm having trouble understanding the concepts in class. Thank you so much!
I stopped reading my textbook and had a 50% rate of showing up to my Calc III lectures because I would just have Professor Leonard teach me it all. I got a 95% on my final and ended up with an 88% for the semester. He just explains things in a way that *clicks* and I can actually UNDERSTAND why we are doing things, without the rote memorization of formulas that universities like to shove at you.
Thank you so much for these videos, I wish i had found them earlier! I struggled so much in my earlier calc classes and had almost no hope of passing calc 3 when i reached it. Thanks to you, I have a fighting chance for once and feel actually capable of truly learning the material! I'll have to go back and watch the other videos when I have time so I can completely understand the basics before this as well, but this playlist is a LIFESAVER and the only reason im passing my class :)
Professor,Thank you for running these videos to youtube without payment.I liked your videos and it will be very useful because i got it what you said at video. My lecturer didn't teached very good like that , so i think i learned all the stuffs and ideas where they come from. I am thankful for videos Good luck for yours professional life.
Wow!, Matt Damon got really buff after leaving Hollywood to pursue his Math Ph.D.
did you not see goodwill hunting....he never left the university.
@@markevans5011 LOL
I've commented on a video or 2 of yours before, but I need to say that you should be getting paid really well for the work you are doing here. For any one in his class, let me tell you that he is basically spoon feeding this information to you, and not every instructor does or even CAN do that. I basically don't even go to my class right now and I'm making A's while the other people in class are there every day and failing. I point them your way because the hours I work don't really allow for many study sessions to help other people.
You're a good Samaritan :D
Michael Lyman hey sorry for inconvenience but you have a problems then don't watch but why are telling sir that lectures is not use
there are many people are getting benefits of this lectures ,they want to learn calculus but can't afford classes
if you got a problems then keep it yourself
Same here, I don't remember the last time I've been to my Calc 3 lecture yet I outperform every single one of my classmates on tests thanks to Leonard.
Professor you made calculus fun again, i enjoyed it in highschool and since college started it has not been the same, you have invigorated my love for math and i can not thank you enough for that. Your videos have single handedly saved my Calc grade and ultimately my major
Thank you for making a well explained line integral video, I wouldn't have understood this without your explanation.
Thank you professor, I rarely comment on UA-cam videos, but you sir, you have my deepest respect.
This guy is fantastic. He makes this class look like Cobra Kai for calculus. Crush the integral, no mercy.
I really wish more professors taught the way that you did. My professor always seems to be able to somehow make the concepts much more complicated 😭 thanks a lot for your videos though! They help a lot!!!
i do not know how to express my gratitude in words - thank god for these lecture videos
There is a faster way to get the vector function for the line segments:
r( t ) = [ (1 - t) * r_0 ] + [ ( t ) * r_1 ]
r_0 is the first point ( a, b , c ) in the vector form < a, b, c >
r_1 is the second point ( d, e, f ) in the vector form < d, e, f >
Bounds for t are automatically from t = 0 to t = 1
It works and cuts down a lot of time
This is true. Im doing the same.
Dr. Leonard has the uncanny ability of making calc 3 seem easy. kudos
Math notation can be displayed at: try.websharper.com/embed/setr/0000Du
[If not rendering notation, refresh page]
CH 15.3
00:00:00 INFO: Line Integral Introduction
*) Essentially, the LINE INTEGRAL will give you the mass of a curved "wire"
*) mass = int_(c)f(x,y)ds, IFF f(x,y) is the mass density function
*) We need to define 'c' using parametric equations
*) We need to define 'ds' using arc length
*) f(x,y) gives a HEIGHT above each point on 'c'
*) 'ds' gives LENGTH of curve
*) HEIGHT[f(x,y)] * LENGTH[ds] = AREA
*) mass = int_(c)f(x,y)ds, adds up areas & gives area under f(x,y) ONLY over 'c'
*) [00:15:20] Visual aid
*) [00:20:33] ds explanation
*)*) ds = sqrt[(x'(t))^2+(y'(t))^2]
*) [00:23:40] c explanation
*) mass = int_(c)f(x,y)ds =
*)*) int_(t=a)^(t=b)[f(x(t),y(t)]||vec r'(t)||dt
00:28:00 INFO: Line Integral Example 1 (easy)
*) int_(c)[y]ds, c: vec r(t) = 2that i+t^3hat j, 0 (2,4) through V.F F(x,y) = xe^y*hat i +y*hat j
*) STEP1: Find x, y, z
*)*) Do trivial parametric equation
*)*) x = t ∴ y = t^2
*) STEP2: Find F(vec r(t))
*)*) F(x,y) = x*e^y*hat i+y*hat j (given)
*)*) F = te^((t)^2)*hat i + t^2*hat j (plug in x, y)
*) STEP3: Find vec r'(t)
*)*) RECALL: vec r(t) = x*hat i+y*hat j
*)*) vec r(t) = t*hat i+t^2*hat j (plug in x, y)
*)*) vec r'(t) = hat i + 2t*hat j
*) STEP4: Solve: F*vec r'(t)
*)*) F*vec r'(t) = te^(t^(2))+2t^3
02:02:20 INFO: Line Integral & Vector Field Example 2
*) F(x,y,z) = (x+2y)hat i+2z*hat j+(x-y)hat k, c:(-1,3,2)->(1,-2,4), 0
Kid sneezes @18:25, the professor is polite and unfazed. Smooth.
I appreciated it lol.
This is actually very good. It makes the the hardest part of mathematics understandable. My grandma can even understand Calculus if she understand English.
God bless you for having the whole class published online. I'm currently taking Electromagnetism and it is heavily based on Line Integrals, Surface Integrals, etc, all of which I was never taught in Calc 3. I have a huge uphill battle coming up, pray for me lmao.
The best teacher i have ever seen ❤ my concepts are being cleared.... My confidence is growing....saloot superman.....he is a blessing for us...a gift from almighty❤ love from bangladesh
3:08 - When he said gentleman I looked up from my work so fast
Professor leonard: "raise your hand if you are okay with that"
Me in my room on verge of tears at 3:44AM : *raises my hand*
Professor Leonard, thank you for another outstanding and long/lengthy video/lecture on How to Compute Line Integrals (Over Non-Conservative Fields) in Multivariable Calculus. The introduction to Line Integrals and its overall computation seems fairly simple to set up , however as the example gets a bit more complicated, I found the solution confusing. I will continue to review the examples for a deep and solid understanding of this topic.
I don't know how to thank you. I don't like math In general nor am I that good at it. But when it comes to your videos I just like to learn❤ thank you
professor, you are a breath of fresh air thank you for helping me learn calc 3.
Thanks Professor. Just thought you'd like to know your videos are still helping students out in 2022.
best teacher. best explanations
You are the best thing that ever happened to calculus
To start off, I suggest you to any and all of my friends that are or are not having math problems in any class that you have video's on. You helped me pass Calc II and have literally taught me everything that I know for Calc III. I enjoy watching your videos and you have reminded me why I enjoy math so much.
The real point of this comment though is to comment (duh) on your saying, "Always take advice that rhymes." I honestly didn't catch the issue with this statement the first time you said it but I noticed it in this video, and feel compelled to point it out.
The statement, "Always take advice that rhymes" is itself a contradiction. If you agree that the statement in question is in itself advice, (which I feel certain you would) and if you agree that the statement in question does not rhyme (yup), then the statement doesn't care if we always take into consideration it's advice (for lack of a better way to put it). Well, if we don't always take into consideration the advice of the statement in question, then we will not always take advice that rhymes but the statement says to "always take advice that rhymes" but the statement in itself doesn't rhyme so we don't always take its advice and thus we don't always take advice that ryhmes... and the circle continues.
I don't mean this to be rude, I mean this for fun and educational purposes. I personally enjoy thinking critically about things and sometimes annoy my friends with how often I do this to their statements.
If anyone disagrees with my thought process then please reply and I hope that I see it and we can discuss the issue with an educational argument.
I again thank you for the videos for they are a tremendous help and hope you have the best day that you can, every day.
Joseph Phillips I'm the same way. I think about the phase "everything in moderation". The word "everything" is every part of the world around you, not a moderate amount of things around you. So if you "do *everything* in moderation," you are not in fact following your own advice. Furthermore, it's generally assumed that things such as murder and stealing candy from babies is wrong and yet if we follow our own advice, we should steal from babies a moderate amount of times. On the other hand, there are things that are assumed we should do all the time such as driving on the correct side of the road. Again, if we follow logic, it is assumed that we are allowed to drive on the wrong side of the road a moderate amount of times. I submit my own phase of advice: "Do a moderate amount of things in moderation."
get a life both of you.. phew
It's like Zak Bagans if he was a math teacher. Jokes aside, this is really well done, and his enthusiasm for his craft is contagious.
I'm so happy I found this channel
0:00 Line integrals in scalar fields
1:27:10 Line integrals in vector fields
His excitement when he mentions force fields and star wars; love this guy
Thank you so much for your thorough explanation of how to parametrize a function. My professor skipped over that and just started talking about how to evaluate line integrals.
It makes me happy so see you and to learn from you! What an excellent professor!! Thank you sooooo much!
Drink a glass of water every time prof smiles
professor leonard you have great techniques for teaching and bisides this your teaching style so great therefore am a huge fan of you i never ever gorget you in life And i say you at last you remain happy in life forever.ok
You have been such a great help for my school career. i am really thankful to you how you take your time to explain every tiny detail. Wish you did some abstract algebra.:)
This is great! Thank you so much for this man. Your explanations are phenomenal.
Most entertaining professor ever.
this guys the goat. motivated me to do calc and hit the gym
Am I the only one wondering why they laugh at the beginnig of each video? Lectures are amazing but I feel incomplete as I miss the jokes :(
Let them have jokes. They're paying tuition for it
@@GoogleAccount-kj7bq was a joke
you saved my evening ! you are the greatest ever
You are the best Prof.
Dear Professor... You are the best.
At 2:14 -15 Dr. L was not clear about the work done. I believe the number we got from the line integral (-29) must be the work done BY the vector field ON the particle as it moved along our curve-segment. The fact that we got a negative number for work means 2 things -->1.The vector field had a net hampering effect on particle movement along C and 2. The particle was forced through the vector field along C by an outside force that did POSITIVE 29 units of work.
wooooow,.....you just nailed it, love from Sri Lanka
You are an amazing Prof! I easily understand you and you're also very enthusiastic and entertaining which helps a ton. Thank you so much!
You're welcome, I made him
Oh my God, Thank you soooo much for such a great explanation!!!!!!!!
Can't thank you ENOUGH
I understand integrals.thanx you lecturers on youtube
professor leonard, i owe this test i take tomorrow to you. thank you for your content. im going to start going to the gym.
Thanks a lot Professor Leonard.
It's an amazing class.It's very very helpful
Best math teacher
I just wish that in the future I have the passion for my work as this beautiful human being does
Your videos are extremely helpful. Thank you prof leonard! Any chance you'll post linear algebra? :)
I am so grateful for these videos. Thank you!
you make more sense at 4am than my professor does at 4pm
Professor Leonard, your videos have helped me in calculus 2 and calculus 3 greatly. I was wondering whether or not you are going to create videos for other courses, like Differential Equations or Discrete Mathematics? If you do, I would be extremely grateful and I'm sure others would be as well. Thank you.
Your help is really appreciated. Thanks from Zambia.
I'm at 27:10 and with you, understanding everything except the most important thing: how can an area be a mass? Area is 2 dimensional, mass is 3 dimensional, right? So how can they be equated? Maybe the answer is coming. I hope so. Great lecture so far.
Density is normally mass per unit volume, but for a thin wire, you can describe it as mass per unit length. The mass is then the density * length, which for a very small length gives F * ds, where F is the density function
@@gamerdio2503 thanks
"Mass" is just a common example of a unit in these kinds of situations;
the general idea is that you have a distribution of something, and then you add up all of the parts of that distribution. "Mass" is probably not a very intuitive unit for that scenario, but it makes more sense when it is used for volumes.
@@Peter_1986Thank you.
The vector field at the end of the lesson is F=-90*g k. You have the weight of the body as the force field not its mass. Fantastic work anyway Leonard!
Sir you deserve to be awarded
in 56:24 you can also just use the formula "r(t)=r1-r1t=r2t" where r1 and 2 are the coordinates and in this case it is -2 to 1 and after substuting these values you'll get the same answer
Professor Leonard, thank you so much for your lecture on calculus 3. very very helpful. I would've gone to American university when I was young. I eventually know why and how Americans love science , and we love business and money instead (too bad). Professor , Merry Christmas and happy new year!