Green's Theorem: Calculus 3 Lecture 15.5: An explanation of Green's Theorem and how to apply it for Line Integrals of Simple Closed Curves on Non-Conservative Vector Fields.
Thank you Leonard!!! You Guided me through calc 1, calc 2, and calc 3. I was a high school drop out, I never could pass algebra 2. And thanks to you, I've aced all my Calc courses. Its nice to reflect back and see how far I've come, with hard work and a good professor(You!), math is not impossible. Thank you for putting these resources online. I am forever grateful and I'll promise to pay forward your deeds.
Thank you so much Prof. Leonard for getting me through Calc 1, 2, and 3. Greens Theorem was the last section covered in Calc 3 in my university (I was so relieved to find out flux and surface integrals were not a part of our Calc 3 curriculum 😂) Calc 1 Grade: 96% Calc 2 Grade: 84% Calc 3 Grade: 88% (Calc 2 was the hardest for me.) When I graduate Electrical Engineering and get a job I'm 100% going to become a patreon! Thank you so much Prof. Leonard!
Hey Inder, congragualtions! When you took Calc III, what was the expectation for "change-variables" (14.8). In your class, were you guys given transformations and completed the integral from there, or did you have to create the transformations for your tests / homework. Thanks!
The best set of math videos I've ever come across. Literally the entire engineering faculty of our university relies on these to get through the math modules. Thank you for making our lives easier..
You are simply thee BEST!!! I've been using your videos for my entire semester and I have been doing very well. All thanks to you! I enjoy watching your videos. Your jokes have me laughing alone in the study area LOL, and they just make learning calculus the most awesome thaaang! Thank you so much sir. I just felt like expressing some love and appreciation. Please do continue posting your videos. Your effect in our academics is MASSIVE! I'm from South Africa, and yet it feels like I know you personally. Thank you very much for your awesomeness!!!!
Mechanical engineering student here, I'm adding to everyone else's thankful comments, and when I get a well-paying job thanks to my performance in calculus 3 you betcha I'm becoming the $5,000 patron and having you cook for me and my other math friends. I know we all passed solely thanks to you, so someday mark my words I'll pay for our dinner with you and a few more for yourself because lord knows you deserve it. From a (finally) thriving student, thank you!
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I actually am already working and have graduated 5 years ago, though I never needed this knowledge. This Christmass I decided to fill up the holes in my knowledge of mathematics. Thank You very much for Your work, it's a ride! 🙂 If only I had more time to study besides working.
Prof Leonard, thank you--you are an outstanding teacher, and your youtube channel is genuinely a blessing! Anyone who watches must be extremely envious of your students, as they get to know you through more than just a screen--you are a blessing, thx goat
19:34 What is also seriously cool about this is that the conservative property of vector fields appears to be analogous to holomorphy of functions in complex analysis. Cauchy’s integral theorem states that the line integral of a holomorphic function taken along a simple closed curve inside a domain in the complex plane is zero. To show this, one uses Green’s theorem and demonstrates that the double integral evaluates to zero if the function is holomorphic. Hence, holomorphy in the complex plane creates an environment identical to conservation in vector fields. I never realised this - thank you very much, Professor Leonard!
Important point to note about the example shown at 1:13:00, if you want the inner area to be subtracted from the outer area, this corresponds to the inner circle being traversed clockwise (opposite to usual). You can interpret this two ways, either as the subtraction of the inner area integral from the outer, which corresponds to negating the line integral around the inner circle. Negating a line integral over a vector field corresponds to reversing the direction of the path. Alternatively, you can consider it as one single line integral, with a small bridge connecting the two circles. Tracing this path (including the bridge) will then force you to traverse the inner circle in the opposite direction.
Prof. Leonard, thank you so much for teaching Cal 3. You are such a great teacher. Your lectures helped crush my fear the moment campuses were closed due to Covid-19. I would not have understood this much even if I sat in my class. God bless you real good.
You're awesome! As an older student going back to school, your lectures have saved me from drowning in my cal3 class. I went from 64 on my test to 96. By the way, when I double the video velocity you become super intense!!!
I am literally not learning anything from my calc3 class. I am learning everything from you, and it is so easy. I wish I knew about you for calc 1 and 2. Thank you.
Hi , Prof. Leonard i am at metu and we have almost the same syllabus and i appreciate you, your videos are awesome thanks for your teaching ambition :D
I've watched Professor Leonard for almost all his Calc I videos (I skipped the 0. sections), all his Calc II videos, and now almost all his Calc III videos. I've done this while taking meticulous notes and I've learned so much. I'm going to be so happy once I finish this playlist and come out highly proficient in mathematics.
"Do you understand Green's Theorem yet? No! But I'm about to show you" 3:12 Excellent. Writing my final year exam in two days and you sir have been an immense help. Thank you very very much.
I stopped going to my calculus 3 class and only show up to my exam days because my professor is the worst at explaining things. We have 3 hours of lecture every 3 days during the week and I feel like I learn nothing from him, especially since he takes half the time setting up his projector to show us examples on the projector using online graphs, but he still doesn't explain things he just graphs. I started watching professor Leonard and man, he's saved my life! Thanks Professor Leonard!
Professor Leonard is a fantastic teacher and is easily better and more easy to understand and learn than all of my professors. Not just pertaining to math. Thank you for the great work because university professors are incapable to help like you do. Thank you!!!
I've been in Calculus 3 for the past year for my senior year of high school and I just want to say thank you. The way you explain things has made it much easier to understand conceptually what is written in my textbook.
Thank you, Professor Leonard. Your lectures are like gold. I am so glad I was recommended your channel. I couldn't have passed Calc 3 without these. Looking foward to Differential Equations and Statistics knowing you have your lectures on RU-vid.
I took and finished Calculus 3 with an A before I looked at your lectures. It was one of my strengths throughout my physics and math throughout college, im still in college but I'm close to graduating. The course was brutal, my professor was truly an amazing professor but he was notoriously difficult. I revisited this to gain an understanding that I thought I had and because of your examples and the way you elaborated, ive gained a new found appreciation for Green's Theorem. Truly the maddening level of intelligence required to discover this math baffles me but this lecture gave me a much greater nsight into greens theorem
I took Calculus 1, but never take Calculus 2 and 3. You and Patrick JMT make it easier for me to understand. Now I can do all the problems in the Calculus book. Thanks! I'm hoping to take the Calculus 2, Calculus 3, and 16 units of child development courses to be a teacher's aide. I don't have the money for the classes yet. I wonder if you can help me. I need books(all subjects) from kindergarten to high school and I can't afford it. Beside, those books in school are easy to understand than the books in the bookstore. I need those books to be a tutor as well. I'm stuck in Accounting, Programming, Massage Therapy, and culinary field. Those cost more money and I'm not working lately. I only do survey to pay my credit card little by little. Someday I will go back to those fields that I'm interested in. I'm 45 and single. I don't have anyone except my family and my nieces are grown up. Now I need to get back to my career. It would be nice if you can get me those Accounting and Programming books with CD-rom. I studied faster by doing along with the book and those professors online. I hope to hear from you. Take care! Stay safe! Have a wonderful day!
I am finishing up my College math with this class. Thank you for all your help over the years. Saved me time and time again. Thank You, Professor Leonard.
These videos are really helpful for understanding the theory behind the math. It makes math so much more interesting when you actually know what you're calculating!!
0:00 Green's Theorem explanation 24:25 Example 1 35:30 Example 2 (fundamental example) 38:10 Example 3 43:20 Example 4 49:25 Example 5 (line integral) 57:58 Example 6 (conservative vector field) 57:45 Area of enclosed region 1:03:22 Area of an ellipse 1:10:50 Example 1 1:13:25 Example 2 1:20:27 Example 7 1:33:45 Example 8 (connecting different ways to solve line integrals; recognizing Green's Theorem)
Dear prof leonard, thank you so much. Today, I passed my calc3 exam thanks to you. I watched all of your calc videos. I love you so much. May good wishes be upon you. Here, I say farewell to you. I hope we'll meet one day somewhere. ❤❤❤❤❤
I just want to thank you, Professor Leonard, for posting these extremely helpful videos! Does anybody know what courses he's teaching in the next semester?
i cant thank you enough how much youve helped me. Though am not scoring great in maths but atleast am not clueless. i have my exams in 2 weeks hope i see some result this time.
You r making me ur biggest fan ever.,believe me sir, i love math and u made me love it more... U r just so passionate about math, i need u as my professor.. 🙏🙏
I thought calc was my kryptonite, however I found your channel towards the end of the semester and i just received a 92 on my final. You truly are the math superman, Thank you
At 9:45 you say dr=dx(i)+ dy(j), BUT " r " is in terms of "t" as you displayed, and thus r'(t) should be in terms of dt, not dx or dy. How do you get from dt to dx and dy?
This took me some time to understand too, but this is what he did. We have r = xi + yj. As everything is in terms of t, we take the derivative with respect to t. So, we get dr/dt = (dx/dt)i + (dy/dt)j. Now, just cancel the dt on both sides (not rigorous, but it works) and you end up with dr = (dx)i + (dy)j
If you find a better prof than professor leonard hmu. Ps: you won’t be able to ! I skipped all my lectures and just started watching his videos. Simply amazing! Thank you so much for all the effort you put into teaching may god bless you ! I will surely support your channel through patreon. Professor Leonard + professor Jeff hanson = literally saved my life.
THANK YOU so much !!! your videos are so helpful , My first midterm was horrible I got a D =( after discovering your channel I got an A- on the 2nd midterm =) now I'm studying for my final
Mr Leonard thanks allot. Your incredibly perfect teaching stile really helped for Calc 3 this semester. Actually the Greens theorem is the last topic of our calc3 course and thanks that you have made video on this as well. Actually, Ive a question. I want to know your plans for the next semester; what classes would you start next( after finishing calc3)? Im really looking forward for ur new class. Im going to take INTRODUCTION TO DIFFERENTIAL EQUATION, AND MATHEMATIC FOR ENGINEERS courses next semester and hopefully could find your next semester classes helpful for my courses. Once again thank you Mr, and thanks God that we have such good people like Mr Leonard in our planet.
@@SNoCappidona Yes. Because if you travel the opposite way through a non-conservative force field, instead of work being done on you, you will do work on the agent of the force field. A conservative force field is a special case where no matter what path, and no matter what direction, the net work done on any object that moves on any closed path adds up to zero. In pure math terms, a conservative field, is a field whose line integral around any closed path is equal to zero, and a line integral around an open path only depends on the initial and final points. There are numerous other properties of these fields, such as the curl equals the zero vector, and it is the gradient of a scalar function called a potential function of the field.
For the parabola, I integrated first, with respect to x first and the with respect to y. Because the function is symmetrical, I did not have to break up the function. It turned out to be simpler with a little integration by parts when integrating with respect to y.
So quick question. Where does the concept of Flux come into all of this. I know in my math course we keep intermingling flux with the concept of line and surface integrals, and I'm unsure how that differs from flow and such. Appreciate the videos, they are definitely leading me to a new level of understanding.
At 54.57, since prof. Leonard is going from -1 to 1 on the top half plane, the angle theta goes from pi to 0. Else, he ought to use the bottom half plane for this semicircle - going from -pi to 0.. In fact he has integrated fin the cw direction and went from 1 to -1 when integrating with respect to theta.
Can someone help me with that question: calculate the line integral of this vector field along the path. The vector field F(X,Y)=(-Y,X) along closed path that is described by this polar equation : Rho=a .
If you set up the line integral, you get: ∮-y dx + x dy over the curve c. You may notice that this is 2 * the area of the region enclosed by the curve. The curve ρ = a traces out a circle of radius a. The integral is then 2(π*a²) = 2a²π
Professor Leonard ,thank you for another fantastic explanation of Green's Theorem and it's powerful impact on Line Integrals of Simple Closed Curves on Non-Conservative Vector Fields in Multivariable Calculus. The overall explanation of Green's Theorem is awesome from theory to practical. Pattern recognition is also helpful when doing examples and solving complex problems in Mathematics.
1:32:08 but wait.... you just found the work done along the right side of the lemniscate, but.... isn't the problem asking you to find the work done along the entire curve? what happens to the work done along the left side of the lemniscate? we know the total work isn't zero because the vector field is not conservative, but would the total work done along the lemniscate over that vector field be twice as much as the work done along the right side of the lemniscate (i.e. just one)?
Right. Our professor write in speed of light and explains while i am writing, then i try to understand myself and lesson becomes totally screwed up for me
"Speed of light are properties of magnetic field and electric field that lights propagate through and can be calculated from 1/sqrt( the sum of two constants)" If your teacher explains as fast as the speed of light, my teacher wouldn't even let you breath as that 'light' flashes by my classroom.
Sir, I watch your lectures on mobile. They're amazing but after I download them offline or even while watching online, I can't see what's written on the white bord. And you usually don't read out the problem because students sitting there can see board clearly. Could you please suggest me a solution if you can?? Your lectures are worth falling in love for... I watched your video on volume of solid revolution for fun just because I love your lectures... Can you believe that? ❤️❤️
