A unified view of Vector Calculus (Stoke's Theorem, Divergence Theorem & Green's Theorem)

In the final video of my vector calculus playlist (congrats to everyone for making it to the end!!!) I want to do a bit of an overview of the major theorems we have seen in this course - Stokes' Theorem, Divergence Theorem, Green's Theorem - all are part of a unified framework. Loosely, integrating a differential operator such as the curl or divergence along a region equates to a quantity computed just along the boundary. Exactly what quantity (flux vs circulation, say) changes depending on the set-up, but they all obey this principle. Indeed, this principle really goes back to the fundamental theorem of calculus II, which was also the root for the fundamental theorem of line integrals we saw earlier in the course.
Anyways, a HUGE thank you to everyone for following along, and I wish you best of luck in any exams!
MY VECTOR CALCULUS PLAYLIST:
►VECTOR CALCULUS (Calc IV) • Calculus IV: Vector Ca...
OTHER COURSE PLAYLISTS:
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS: • How to solve ODEs with...
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• 5 Tips To Make Math Pr...
►Cool Math Series:
• Cool Math Series
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[@j.o.5957]

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Only if the person speaks English

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[@juandaviduribe8267]

I'm so mad I didn't know this was a whole playlist. I would just search topics on their own if I felt like I needed clarifying something, but watching all would've helped my intuition immensely

[@davidcastillo4938]

Thank you so much Dr for helping me throughout my calc iii course. I got a 93% on my final and can finally get some rest this winter break. Hope this helps!

[@DrTrefor]

I totally missed this 8 months ago, so a delayed thank you!!

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One of the great playlist on YT for sure, thanks for all efforts. Kudos

[@DrTrefor]

Wow, thanks!

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[@JimMorikuni]

Thank you Dr. Trefor for such a wonderful playlist. As an engineer, I learned the equations of vector calculus many years ago but never really understood what they meant (the geometric interpretation). Your enthusiasm and your gift of teaching helped me understand them in a much more intuitive way.

[@evanwear5887]

I remember sitting in my high school Calc 1 class learning about the Fundamental Theorem of Calculus and thinking to myself "What makes that so fundamental? You integrate a derivative, you get the original function. What's so special about that?" Here I am now, sophomore year of university and finishing Calc 3. Looking back at it all, I finally get it. The Fundamental Theorem shows up *everywhere*. What a beautifully elegant and powerful theorem it is. I hated math until I took calculus, now it's one of my biggest passions as well as my minor. Huge thanks for helping me stay 2 weeks ahead of the class and get As on all my exams. Your explanations, visuals, and teaching style are excellent and I'll be spreading the word about your channel. Much love from Boulder!

[@DrTrefor]

That is so kind of you to say - thank you! So cool that you picked up the minor and enjoying everything. Congrats on the As:)

[@elias8356]

I had the exact same experience. I was going to major in polisci and philosophy before taking calculus. I’m now a physics major with a minor in math and am currently heading to my multivariable calc final. My two big breakthroughs came when I first read Euclid’s Elements and when I first learned about the Fundamental Theorem of Calculus. It’s a shame that we teach math in such a formulaic, number-crunchy way, because that’s not at all what it is, and the more I learn, the more I appreciate how much creativity and outside the box thinking it takes, as opposed to my old view of it.

[@evanwear5887]

@@elias8356 Good luck on your final dude! You got this!

[@ManojKumar-cj7oj]

this whole series is incomplete without this last video finally I understood the essence of vector calculus thank you very much, Dr. Trefor

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Need this type of content on yt.

[@francisc.coria_]

Just ended the playlist and I can only think that 'THANK YOU' isn't enough to say. You're such a great professor and made all this process of learning calculus amazing. Hope you receive all you have given to us! Thank you, Dr. Trefor Bazett! *Argentinian hug* 🇦🇷

[@DrTrefor]

Amazing, well done!

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Every thing is round it ends where it starts. so beautifully covered professor , loved it . respect

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[@DrTrefor]

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[@DrTrefor]

Congrats on making it to the end, and thank you so much for your kind words:)

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Professor Bazett. Take a bow. You are one of the best things that happened in KZbin

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2 playlists in one day has been a success. Thank you so much

[@DrTrefor]

Wow!

[@benberkebile2773]

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Dr Bazett you are a beacon of hope for every student of mathematics. You are a true teacher. You are capable of teaching complicated subjects in an intuitive and interesting manner. I want to sincerely thank you for everything you have taught me. I’m looking forwards to your future videos.

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[@DrTrefor]

Congrats on making it to the end!

[@user-jm6gp2qc8x]

@Kathe Schweinberg I've seen these messages in many places

[@djredrover]

a year after I graduating uni, I finally intuitively understand vector calc thanks to you, sir. During Uni, I saw all this stuff as symbol puzzles and never intuitively. Thank you thank you.

[@DrTrefor]

Glad to hear that!!

[@harsha015]

In short, this series is fantastic !!!. All the things you have covered here were exactly one-half of our calculus module this semester at my university. And guess what? I couldn't understand any of them because there wasn't any clear explanation about what was happening inside these theorems. And you've got all of them covered !. Now this is how you teach Mathematics !. I watched the whole series. A big thanks to Dr. Bazett. You are a lifesaver for sure.

[@NuclearPhysix]

Your graphical representations are so clear and cleverly selected. Makes it easier for me to visualize what you are explaining. Thank you so much

[@DrTrefor]

Thank you, and you’re most welcome!

[@richardneifeld7797]

my son was taking BC Calc in HS which includes vector Calc, so I decided to a take a refresher course, and yours was outstanding! I feel like I now know vector calculus at an intuitive level that I did not have back in the days of college and grad school. Thank you!

[@elatior]

The visuals and the way you presented make the ideas come together so clearly and helped with my understanding of vector calculus. Thank you so much for making these videos!

[@avinashsivaraman9682]

Thank you so much for helping me through vector calculus. Surface and line integrals didn’t make sense to me at all before I watched your videos

[@DrTrefor]

So glad they helped! Congrats on making it to the end:)

[@raghuramabl6729]

Thank You for this wonderful playlist. Initially, I saw two-three videos in between but did not get the concept, so decided to watch the full playlist it took me around 6:30 hr which really helped me to boost up my knowledge. Once again thank you. You are one of the wonderful teachers on youtube💕❤🙌👌

[@mitsosyt7853]

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I've got my exam in a couple of hours. This playlist and several long days of practice has set me up to do really well on the final! Thank you for making this Dr. Bazett

[@DrTrefor]

Good luck!!

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[@DrTrefor]

Thanks so much!

[@alexsaracene3095]

Hey Dr Trefor, I just want to thank for this playlist. I just finished my all nigher leading into my 7:30am exam and I went through the whole playlist and took notes the whole time. I may not pass this class but thank you for giving me some good content to study.

[@anthonyheak3479]

Your videos are great! You speak loud and clear, so it was easy to understand you.

[@naveennaidu9064]

I've learnt this course so fast!!! The best thing is, Prof u've made it so so simple and precise!!! I had never thought of them explained so simple . A real teacher indeed, hats off, ur CONTRIBUTION is helping all students across the world!! Global teacher, may God bless u and ur family. This really really helped me alot, I'm an engineering student , aspiring for GATE @INDIA , u've helped me build such strong base in calculus, I'm gonna rock on engineering subjects now , Lots of love from INDIA 😍😍😍😍😍😍😍

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I needed this last week before my final! But fr though, this playlist has been incredibly helpful

[@DrTrefor]

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Dr. Bazett, thank you so much for this series on vector calculus. You have made this potentially complicated subject into one that can be easily understood, an outcome I attribute to your exceptional intellect, your passion for helping others learn, your talent for communicating effectively online, and your dedication of time and energy to making this all work out all right so that one leaves with a sense of wonder, enjoyment, and enlightenment.

[@DrTrefor]

Thank you and congrats making it to the end!

[@therunkumar]

Your content is as wonderful as the content of guys like 3Blue1Brown and Zach Star. Your enthusiasm is wonderful and I hope I get to see more of your videos in my recommendation feed.

[@DrTrefor]

Thank you for such a lovely comparison!

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[@DrTrefor]

Congrats on making it to the end!

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Good afertnoon dr. Buzett, I have just finished this playlist and I would like to thank you for all these amazing videos. It’s really impressive how you can expose the theory so clearly and at the same time so fast. Your videos are so helpful for a lot of people. Thank you again for your exceptional job. Best wishes.

[@DrTrefor]

Congrats on making it through, that’s awesome!

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so digested the full playlist ,,and ready for my end semester exam tomorrow. thanks a lot dr Trefor..lots of love from India

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I seldom leave comments under videos but I feel like I MUST do this for the end of this legendary playlist. A big thank you to you for making these videos. They literally save me :D

[@DrTrefor]

Thank you! Congrats on making it to the end:)

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Just finished the series, I am so overwhelmed with gratitude that I am motivated to make my first youtube comment ever. You are a real one, I love your work and your passion for every topic you discuss, I can't wait to watch your series for future math classes. I am for sure going to buy some merch on the website. I love you and THANK YOU!!!!!!!!!!!!!!!!

[@DrTrefor]

So glad you enjoyed it!

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[@DrTrefor]

Hey congrats on making it to the end, that’s awesome!! Glad to hear about the applications in engineering:)

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700+ likes and ZERO dislikes ! Man, you are just brilliant !

[@colbyscott8545]

Hi Dr. Trefor, I want to thank you dearly for this playlist. My calculus 3 final exam was based entirely off of the final chapter of the course, which was all vector calculus. I realized my professor's teaching style wasn't helping me understand vector calculus once I realized I still didn't really understand what a line integral was even a couple weeks after we were introduced to it. I searched in the KZbin search bar, "visualizing line integrals" and that's how I found your channel. When you mentioned this playlist I decided to check it out and realized it was exactly what we were learning! I find your teaching style seems to match perfectly with how I learn - your visualizations are immaculate and I think they were the key to my understanding. Thanks to you, I earned a 94% on my final exam and, for the first time, secured an A as my final grade in a calculus class! I thank you from the bottom of my heart for allowing me to not only understand but also finally enjoy calculus 3. I quite like the Eat Sleep Math Repeat hoodie on the website, what are the dimensions of the medium and large sizes? I tend to flop between the two. Thank you so much!

[@DrTrefor]

Glad you enjoyed!! I actually just ordered that hoodie myself in the "Large" and I should be getting the first samples very soon. From what I can tell from the distribution folks these fit pretty standard, but I always like going slightly large as will shrink a little bit washes.

[@victorsalasgomez5097]

Exam is in 6 hours, I am confident thanks to your passion for teaching. Bravo maestro!

[@DrTrefor]

Good luck!

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[@gabriellarson6507]

youre the reason I passed my calc class, THANK YOU TREFOR :)

[@DrTrefor]

Congrats, that’s awesome!

[@rifatahamed7052]

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[@DrTrefor]

Glad you enjoyed it!

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Give this man a medal

[@carlmcgrath484]

Got my Vector Calculus exam next month. This will be my third time sitting it. Your videos have helped my understanding so much, just finished the playlist. Thankyou Dr Bazett!!!

[@carlmcgrath484]

Update. I passed! Trefor you are the best i love you! I have my maths degree now whooop

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My understanding of calculus has increased more in the last 3 days then it has in the last 3 months of school, thank you

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we need more professor like you, sir.

[@DrTrefor]

Thank you!

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Thankyou very much for this playlist... Perfectly explained.. easy language.. covered almost every topic... As a physics student this is very helpfull for me.. before studying electromagnetism one should go through this playlist once ..

[@ayanavroy8269]

Thank you for this course! I remember having to watch your videos for a Univ. of Cincinnati Discrete Math course, and found your way of teaching to be really engaging. And now, this playlist has been equally as helpful. I remember taking Multi in school and just mugging up all of the vector calculus formulas, now their actual meaning makes so much more sense. Hope that you can make more courses like this in the future!

[@DrTrefor]

Cool! Glad they helped, I miss my old UC days:)

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Waited for this😍

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I was having problem with physics calculus, just skimmed through your calculus series and it seems so clear now.... I just can't thank you enough

[@DrTrefor]

You are most welcome!

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kzbin.info/door/LiSQM08GcX38D8KmIi7izQ You can get videos with full concepts on this channel too.

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You are an awesome teacher. You just visualise vector calculus. Finally i ends this, before this i was struggling so much.

[@DrTrefor]

Congrats on making it to the end!

[@swaroopdewal4626]

Thank you so much !!

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completed this playlist in 1 day, a day before my endsems 💀 hope I do well thanks a lot prof, u taught very well

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Got a big test tomorrow and was having trouble understanding Green's and Stokes Theorem, this was super helpful. Thank you.

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This channel has been so incredibly helpful. I am officially a devoted follower. Would absolutely love to see some partial differential equations videos

[@DrTrefor]

Plan to do a series on those soon!

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[@DrTrefor]

THank you! And good luck in your exam in feb:)

[@markampatsidis8853]

Εxcellent work!!! "Unifying Principle": the "heart" of ANY explanation.

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I have seen every video more than three times,,,, amazinggggg

[@DrTrefor]

So glad they have been helping!

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finally finished an amazing playlist. Thank you so much Dr trefor. Lots of love from north west Pakistan.

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should we add some conditions for this field 7:30, such as being continuous and differentiable within the boundary?

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[@DrTrefor]

Glad it helped!

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This is the best math course I've gone through in my whole life. I now can feel every bit of that. Thanks a lot for such content. Wishing you a great life.

[@DrTrefor]

Congrats on making it to the end!

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[@MelodiCat753]

This video made me cry. It's so beautiful!!!

[@kirankumarpanda7551]

best ever playlist for vector calculus

[@jkgan4952]

This playlist was gold from start to finish! THANK YOU SO MUCHHH :)

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Completed the entire playlist in 2 days. That's how interesting & fun to understand this was. Take a bow Dr. Trefor

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I appreciate your teaching style and really like the way you are inducing interest in Students. Keep up the good work!!!

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I have seen the whole vector calculus series, enjoyed it a lot and the ending was great and a great thank you

[@suvkaka]

Thanks!

[@Superdoodle1000]

Thank you so much. Your course teaches vector calc in the same language as my school and follows the book exactly. This was a huge help and I appreciate it

[@nikhils216]

Awesome Playlist. Thank A Lot Prof. Trefor. Your beautiful explanations and picturisation made this Topic very simple and interesting. Thank you so much for making this wonderful content. Best Playlist on YT.

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Completed the entire course...❤️ Loved it! Love you from Uttarakhand State, India🇮🇳

[@DrTrefor]

Congrats on making it to the end!

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Thank you Dr. Bazett for helping me pass this course with ease

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I'm so glad I've found your channel. Thank you for all of this!

[@DrTrefor]

You are so welcome!

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big thank you!!!!!!!!!!!! This playlist has made my life som much easier!

[@404-j5s]

Completed the full series today. Thank you so much for the outstanding videos!

[@DrTrefor]

Congrats on making it to the end!!

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Thanks for your help from Australia hoping I get through next weeks exam!

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I haven't watched your channel for a while and you have done a lot of things 😍. Congrats and thank you for your great work! I'll watch the Calc IV playlist to help in my Electromagnetism course. I'll like and share all for sure! Please keep doing these wonderful videos.

[@DrTrefor]

Thank you and enjoy!

[@admiralhyperspace0015]

MashAllah, great bro. You explained it so crisply. I found this explanation in griffith E and M and you really outdid yourself by generalizing at the end. Really, really appreciate your work. And keep it up. You are helping alot of us. Thanks.

[@DrTrefor]

Thank you so much!

[@RAJANGUPTA-ce8lg]

today i ended with all the video of vector calculus ................a lot of Indian ,a lot of Iitian wacthing you ..............one thing i expect from u sir .....please provide the ppt of all the topic u covered daily in description ......thank u soo much ........luv u sir .............

[@DrTrefor]

Congrats on making it o the end!!

[@RAJANGUPTA-ce8lg]

@@DrTrefor Thank u sir ..........your visuliszation really help me understand math......sir please cover PDE vedio lecture .............a group of engineering student waiting for this

[@samborisuk9687]

This playlist has been amazing and fundamental for my studying. Thanks so much!!!

[@DrTrefor]

You're very welcome!

[@leonhkuhku]

谢谢！

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Quality and intuitive video collection you have. 🙏🙏

[@sachinu1986]

Very well structured videos with a gist of all important concepts

[@joshuaisemperor]

Many thanks for creating these kind of videos!

[@DrTrefor]

My pleasure!

[@Afrinn]

Thank you so much Dr. Bazett. Learnt a lot from this playlist.

[@SajjadAhmad-wv8mw]

"Equations are just the boring part of mathematics. I attempt to see things in term of geometry." Stephen Hawking This is what I'm always looking for, since I haven't been through all of these videos but watched most of them that helped me a lot in understanding calculas III. Thank you, @Dr. Trefor Bazett

[@gsejal1039]

I'm very happy to have found your channel. Helped me understand things in a very easier way. Great work sir!!

[@DrTrefor]

Glad to hear that!

[@dagadabhanu]

It was fun to learn with you. Got some great detailed insights. Appreciate your efforts Dr. Trefor

[@romanayon8483]

Thank you, Sir,though a 'thank' is not enough how much it helps me to understand vector calculus

[@yongmrchen]

finished watching all the videos from calculus 1-4 today.

[@DrTrefor]

Wow congrats!!!

[@trafo222]

Thank you very much for this great course Trefor.

[@matanshtepel1230]

beautiful wrap up 😍. Really gives the big picture.

[@DrTrefor]

Thank you! 😊

[@boidaemon8733]

I had previously watched the entire course on vector and multivariable calculus by Grant over on Khan academy, but i left very unfullfilled, it felt really overwhelming, and even though i learned alot of the key concepts, all of the pieces just weren't fitting for me. But with this playlist you have really stepped up to the plate and made everything super clear. I particularly enjoy how you make connections between all of the theorems. And show the computations, Well done, i feel that i have a much better grasp of the subject now.