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.
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..
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!
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!
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!!!!
My course follows his course pretty well so I decided to skip going to my lectures for Professor Leonard.. There's one more after this.. THANK YOU PROF LEONARD. I cannot thank this guy enough.
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.
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!
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.
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
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 CZcams.
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.
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.
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.
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!!!
"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.
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!!
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.
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!
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
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
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 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.
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.
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
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
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.
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.
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!
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.
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.
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.
I should pay you my tuition to be honest. Everything i know from cal3 is from your videos...
Exactly!!! Good work!!!!
I really don't understand why prof Leonard isn't in the top ten math YT channels.
because highschool kids can't watch 3 hour lectures lmao
At least he has pretty much perfect ratings on "RateMyProfessors".
@@GeorgeShum i am high school kid watching his 3 hour lectures... but i do put them on 2x speed lmao
@@annikanielsen740 same. Professor Leonard is awesome, but if you're gonna use him to cram, you have to leave your entire weekend open lol.
@@Peter_1986 What's his full name, I want to look him up
When this guy was interviewed for teaching:
- So what did you use to do in your last job?
(Him) - I worked as a journalist for Daily Planet...
is this a meme or what
@@user-zy5vw7fl9m he's implying that prof Leonard is Clark Kent/Superman.
Prof leonard, organic chem tutor, and patrickjmt are the secret of my straight A's in college. Thank you guys I appreciate all your efforts.
Wish I had you for my Calc 3 professor.
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..
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!
Lucky. My professor is horrifically difficult. We have learned everything in professor Leonard's
Cal 3 playlist in 5 weeks. Or about 10 hours.
@@p0intblAnkwaziT I don't know about Inder but mine doesn't cover it either, we also skipped 14.7 and 14.8
@@charlieherman6711 Appreicate you man!
currently studying vector calculus
got A's in previous 2
I feel like Professor Leonard cares about me more than my actual Calculus professor. Thanks for being so thorough and clear.
I wish my classes were this long. My prof gets 50 minutes to cram this into and no one keeps up
i think he usually combines 2 classes into one video
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!
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!!!!
ok i try to continue posting videos
@@tse278who are u lil bro 😭
saving my life right now! Wish you would do linear algebra too.
check out a channel names Technion, amazing LA videos
Zhao, check the MIT opencourseware. There you'll find Gilbert Strang teaching linear algebra...the best course hands down.
Wow, you are a life saver. Thanks!
Happy to be of some help. Believe me...once you'll go through his course you'll have a "Strang" understanding of linear algebra :)
check out maththebeautiful ran by a Drexel professor I was in his class he might seem boring but literally the best teacher I have ever had.
My course follows his course pretty well so I decided to skip going to my lectures for Professor Leonard..
There's one more after this.. THANK YOU PROF LEONARD. I cannot thank this guy enough.
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.
At first i thought this is going to be so boring(34 videos and each video is about 2 hours),but after watching ur lectures, u became my Guru.
I'm glad you stuck with it!!! Nicely done
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!
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.
Sometimes I raise my hand when he asks the class to raise their hands if they're okay with that one.
every time XD
same
I want to ask questions sometimes
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
Glad to be helpful !!
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 CZcams.
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.
omg! It´s actually Clark Kent. He is real!
and looks like he doesn't work for Daily Planet O.O
My good sir, you're the reason I have any hope of graduating. If only our professors were half as amazing as 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.
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.
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!!!
"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.
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!!
Been watching these videos for years. Prof Leonard is the best!!! in a class by himself!!
Thank you a million times. Your lectures helped me pass my statistics with a credit after poor course work. May God richly bless you
Professor: "do you have any idea how much headache greens is saving you here"
Me: NO, YOU are saving me headache professor!
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.
Thank you so much for your dedication. It is professors like you that motivate and inspire the innovators of this world.
I'm just another calculus student thanking you so much for your video. Thank you Professor Leonard
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 has made mathematics fun, especially calculus and vectors.I appreciate how his lectures are spreading knowledge to those in need .
Professor Leonard looks like a male model posing as a professor lol
Professor Leonard's "Center for kids who can't read good and want to learn how to do other stuff good, too"
But why male models?
Maybe there's more to life than being really, really, really ridiculously good looking?
I was certain there was no-one really in his class and it was all set up, but apparently it's actually a real class.
F
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 swear these videos are the only things getting me through calc, thank you so much!!!!
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
zordur 120
120...
120 yaz okulu...
120 corona semester...
same syllabus here too, from a turkish uni too.
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 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.
and that is by far the best teacher ever. thank god
Thank you, Professor Leonard! Very helpful you are.
Thank you for teaching and posting the awesome teaching!
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?
Maths is starting to become my fav. subject...Thank you Prof
After watching this guy for two years I find myself saying FANTASTIC a lot.
thank you for posting this video, I was waiting for it :D
This Is A Great Teacher!!
Wow....the video is extraordinarily clear
You are so good, Professor.
Best Math Professor EVER!!!
Awesome lecture, Professor Leonard.
Lovely lecture, Doctor Leonard.
The greatest of all time.
all the way from Kenya. Thank you
ARNOLD MATHENEGER!
Great Lecture, Sir.
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.
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
thank you so much. I know I've commented on several videos already, but I don't think they are enough.
OHHHHH....... THAT'S THE BEAUTY OF MATH I WAS SEARCHING FOR A LONG TIME..... THANKS ALOT 👍
Legendary professor!
Best Professor Ever
Thank you so much for your class!
the thumbnail is priceless honestly.
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
Personal “bookmarks”: 21:00, 58:10, 1:02:00, 1:18:30
Your lectures are the best
one of the best ever....
I was afraid of green's theorem but Professor made it so easy thank u sir
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.
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.
this is the best channel on yt, you have definitely earned wearing that superman shirt in your pfp
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 was waiting for this
You are a genius, thank you
the man is simply the best
Thank you sir for these superb calculus videos. Hello to guys who are struggling with MATH 119-120.
I study at coffee shops. I'm alway so tempted to raise my hand in public when he says raise your hand
It's been quite a journey. Thank you! With almost a million people subscribed for help what does that say about the courses we're being taught?
Professor Leonard is the GOAT
legendary prof …
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.
Really helpful stuff! Just to nitpick: 1:14:05 - should be set difference, not union...
And ur biceps 💪, m watching it.. 😳😳😊😊
I like the thumbnail for this video, Leonard is like "and there you have it, guys! isn't that awesome?".
how could you not get an A in this mans class
3:16 Underrated hulk joke, I laughed lol
Really Thank you!
i don't like any class more than 1 hour except your classes:)
1:16:50 Me when I have time
NC state here, normally we have great caucus teachers btw, but I got unlucky with mine this year😂 thanks prof!
literally helpful
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.
I salute you!