Change of Variables & The Jacobian | Multi-variable Integration

You've reached the end of Multi-variable Calculus! In this video we generalized the good old "u-subs" of first year calculus to multivariable case with a multivariable change of variables. The trick is to set up a new coordinate system where the messy region before is now very nice, perhaps just a rectangle. We have to be careful though as we have to multiply our integrands by a scaling factors called the Jacobian. In this video I sketch the intuition behind the Jacobian and compute it out in an example.
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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
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When you're a visual learner, videos like this are an absolute godsend

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When you are any* learner

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[@pbillings808]

Greatly appreciated the second half as to WHY the Jacobian comes into play!

[@Chomusuke71]

The Jacobian part in the end is pure gold

[@treynoe4934]

Hey can someone explain how to find the individual parts of the jacobian? I’m using this vid as a study guide for a test and I’m struggling to find the multiplicative factor

[@treynoe4934]

help please someone

[@Chomusuke71]

@@treynoe4934 what do you exactly mean by the individual parts, elaborate a bit ill try to help

[@Chomusuke71]

do u mean the functions inside the determinant?

[@abhinavhampiholi54]

I had taken multivariable calculus a couple of years ago and needed a refresher. This was great!

[@useit97]

Exactly!

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dude!

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I'm taking multivariable calculus now

[@lancelovecraft5913]

Excellent vid my friend. I took multivariable calc years ago and never really understood the jacobian or its purpose. Your explanation made it so simple and obvious. Thank u

[@yogitshankar6348]

+1

[@GoutamDAS-ls1wb]

Thank you Professor Bazett for an excellent video on the Jacobian. I am taking Multivariable Calculus (II) right now on edX offered by MIT but I always return to your videos because of your clear and lively teaching style. I look forward to many more. You make the videos entertaining as well!

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no one can dislike this video

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Thanks a lot Dr. Bazett! I searched many materials and so far to me you are the only one that made the vectors transform part clear. Subscribed.

[@PinakCborty]

This is one of the most beautiful explanations on the intuition behind Jacobian I have come across on youtube. Thank you.

[@tomatrix7525]

This was absolutely top class as far as an explanation on an advanced topic can be. Great

[@walac]

Man, I was struggling to understanding the reasoning behind the Jacobian, but your video provided me the insight thanks a lot!

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That was the last one of the whole series on Calculus 3. I watched every one and they were brilliant. Thanks for all the insights.

[@AmitKumar-ud7yq]

Never found mathematics this much interesting.. Thank you for all the efforts you have made for us sir.# respect

[@jonasfarbo2661]

The motivation for the Jacobian explained with the u-substitution method from calculus 2 was very helpful for my intuition behind this topic. Thank you.

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I was about to google to find out the reason why we needed the Jacobian, your video presented is so beautifully it can't be beaten!!. Thank you so much !!

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This channel doesn't make me feel that I'm quarantined at home.:)

[@proxy8918]

You are so good at explaining the intuition of things. You are a great teacher, thank you.

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This is the best tutorial on change of variables I've watched!

[@AJ-et3vf]

Awesome video! Thank you! Interesting how the Jacobian features into integration when I first learned it for using newton's method for vector functions and vector root-finding, needing to get the partial derivatives of the vector function F.

[@Rogue_Art]

That part at the end was mindblowing. That literally tied everything together and I can't believe they never showed that between my professor's lectures and two textbooks!

[@DrTrefor]

isn't that part cool!?

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That one dislike is given by a man who didn't even know the single variable calculus. Awesome explanation Sir!❤

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This channel is awesome 👌 You bridged the gap of the required imagination to understand the concepts by the brilliant use of animation. Explaining something that can be seen brings the beauty of Math. I am following all your playlist just to learn and see again this beauty which I missed before . Thank you 😊 Moving on to Calculus IV 👋

[@DrTrefor]

Thank you! Have fun in calc4:)

[@maxpercer7119]

Jacobians tie together powerful ideas from linear algebra and calculus.

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That explanation about the origin of the Jacobian factor is very helpful!!! Thank you!!!!!

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I love your attitude! You make Calculus so much more enjoyable to learn :)

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This is the absolute best explanation on the topic I've watched so far!

[@DrTrefor]

Great to hear!

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Thanks sir after exploring whole day in youtube to understand the jacobian your single lecture gave me a tremendous touch of the concept of jacobian Really sir you are genius and also have very good experience as compared to other professionals 🥰🥰🥰🥰

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This is just soooo beautiful and mind-blowing! Thank you so much for the great explanation :)

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Have my calc 3 final today and this is such an amazing review!! Thank you!

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Thank you so much for this video. I wish my Calculus 2 classes were this straightforward.

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Thank you so much Trefor, your series of videos really helped me clarify some questions I had from Calculus Adam's textbook.. Keep up the great work and thanks for teaching! :)

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SOOOOOOO helpful. Thank you for taking the time to make this!

[@TN6625]

A really informative explanation and presentation!

[@maulaahmad2542]

I was banging my head trying to understand the jacobian cuz it was introduced with the chain rule topic in my school. This video just made it very easy to undertand now. Thank you very much

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I am actually shook. Whenever I watch videos like these, I expect to be disappointed. But when you started discussing the cross product, everything suddenly clicked.

[@DrTrefor]

Haha nice!

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What an absolute legend, thank you so much for this.

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Your explanations have helped me a lot, I am very grateful! Thank you very much for creating context at the beginning of each of your videos and for making such clear points!

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dudee explained so well sire!!! thanksss a ton

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I never think that i understand this concept in such an easy manner 🙏🙏🙏

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That was amazing, visualisation hits different from a table at university. Thanks!

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Thanks you Dr Trefor! Absolutely amazing explaination :))

[@SuperDeadparrot]

This guy is doing exactly what I used to do and get wrong all the time! I eventually learned that you don’t define u= f( x, y ), v= g(x, y ) but you imagine the BACKWARD transformation, x = F( u, v ), y = G( u, v ).

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very very helpful and I love the way you explain!!

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Just beautiful!!!

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Finally i understand this J. Great video sir😊

[@samwhite4284]

at no point in my maths undergrad did any of notes/proffs explain the (actually simple) motivation for transforming with the Jacobian as well as this video - it was v enlightening... Thanks a lot!

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You make amazing videos, bless you!

[@sdsa007]

I’m a visual learner too… but being a stupid visual learner i had to watch this twice, because the first time i was watching it in small screen mode… small screen mode confused me because the dark red font used for J was no J enough, so i thought it was a close bracket… and this distracted me to no end looking for an open bracket in small screen mode. Jeeeez… I was so relieved after switching to a larger screen to make up for the idiosyncratic font style for depicting J… But i am definitely reaching my end goal of getting better intuition on cross product, curl, determinants and Jacobian! Thank You!

[@MrBurstfire99]

Excellently explained

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Wow... Simple and best explanation 👌

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finally this topic got crystal clear.....thanks a lot 😊😊

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AS SOON AS I SAW PARALLELOGRAM AND HEARD CROSS PRODUCT IT ALL CLICKED; THANKS MAN

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Sir, you are brilliant. Thanks a zillion.

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best explanation + nice visualization

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this literally saved me from failing in my exams

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this is such a good intuitive explanation, my textbook just derived it without any intuition lol

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Thank you so much sir 🔥🙏🔥

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Completed the multivariable calculus for the first time with intuition and visualisations. Thanks Professor. Looking forward for Ode series.

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Congrats on making it to the end!

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This blew my mind 🤯

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Amazing video man

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great content. excellent that you made the connection with change of variable and help see that connection. thanks

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Glad it was helpful!

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very nicely explained...

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I always feel you are underrated.

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thanks a lot sir.. that was a great explaination

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masterpiece, amazing

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Thank youuuuu! Got my 3rd year finals coming up and this really helped me get my head around it. :)

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Good luck!

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Great video ! very well structured

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Thank you!

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simply amazing...! this is such a wonderful explanation..!! I am really sick of the way they have been teaching this stuff at university..!!

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Brilliant explanation sir ! I got stuck on the subject trying to understand 3blue1brown's ideas..anyway these last minutes from this video made me to understand Grant's

[@DrTrefor]

Glad it helped!

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9:10 that explaination was excellent. It seems stupidly obvious now why we scale with the jacobian that I'm kind of dissapointed I didn't try to heuristically derive it myself. Damn.

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Thank you for the amazing content!

[@DrTrefor]

Thank you so much!

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Great video, thank you..

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Awesome explanation... Thank you sir...💜

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So nice of you!

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Well explained

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Thank you finally a rigorous explanation

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Glad it was helpful!

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Just finished playlist very good introduction to the material

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Congrats on making it to the end!

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I was really stuck on this concept for my class, but after this i now understand what i was doing wrong

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Awesome!

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Watching this at 2am studying for tomorrow’s final. Thank You!

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Good luck!!

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So good👍🏿

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