the jacobian has an rate of scaling under transformation and jacobians are the true derivative and finding the correct scaling factors from determinants to make the explosion in Riemannian rectangles of the integrals the error converges with infinite sum so the scaling factor is there to rectify the error rate in convergence in rectangles under transformation the rectangles explode and contract and at miniature scale the each point under transformation has the scaling factor
Feels great to know why the Jacobian comes into the calculations when switching coordinate systems. I never learned that while doing multivariate calculus this past semester. Keep up the good work! Regards from a fellow math nerd from Sweden.
I'm shocked how you've packed many topics such as vector product, Jacobian, areas, and more into such a video, while clearly explaining Jacobian, the main topic. Even if I don't speak English well I can understand it and it is very interesting to watch the explanation and behavior as if you are transmitting energy to the viewer. I'm very satisfied.
I brushed across the Jacobian while learning statistics recently. It seemed reasonable that we'd need to scale by the change of space in that context, but this video made it concrete as to what was going on behind the scenes. Thanks, Tom!
Hey there! The second you explained the Jacobian as the stretch factor of converting from one coordinate system to another, I understood it so much better! This was so much better of an explanation than my textbook
Thanks for your exceptional work Tom. I've got a degree in maths and still learning little things like this really makes sure I keep lifting my knowledge. You're putting a load of effort into these videos. It is greatly appreciated.
I envy the ability to be good and understand math, I’m doing intermediate algebra right now in college and I’m having a hard time grasping the concept. Love your videos, keep it up!
Thank you so much. I am a first year Maths student from India, and these simple yet beautiful concepts are what keep mathematics in my heart. Keep up the great work Sir!!
This is such a fantastic video! I'm currently in year 13, thinking of doing a maths degree, im fascinated with calculus, its by far my favourite aspect of maths, not only did multivariable integration make sense but also the use of determinants. Amazing video!
This really should be taught at A-level rather than first-year undergrad courses. Jacobians act as a nice sliproad onto the main highway of tensors and differential geometry in general, whose introduction is in turn often delayed (or even omitted) at bachelor's level.
Best intuitive explanation that I've seen so far and for once , even with my weak maths knowledge , understood it for the 1st time. Other youtube presentations never clicked with me but this one did.
Thanks! That was explained in an intuitive way. I guess the key here is to think of the elemental rectangular areas changing in to rotated parallelograms during the coordinate transformation. The example you gave in the beginning with regard to the area of the circle makes the concept clearer.
Absolutely love this video, currently in the process of studying vector calculus (and some other stuff I also don't understand) for machine learning and struggled to wrap my head around jacobian's, this makes so much more sense now
You really are saving me in university... I feel like I can understand where things comes from and why they are the way they are when you explain it... much better than my university professor who is more interested in making us fail class
This is super funny, because this is literally just out of the textbook. Maybe if you oafs read the textbook, you'd learn something. I tutor math and physics, and people say the same thing to me. "You make it so much easier than the professor, and you actually explain where it comes from!" This jacobian 'proof' is straight out of any Calculus textbook
What a mesmerizing presentation. I had math through differential equations at university thirty-five years ago. If you had given lectures, such as you present here, perhaps the 4.0 GPA achieved would had met something. Grade Inflation was in full bloom. Thank you.
Whenever I encounter double integrals of some version of the unit circle I’ve always been frustrated by the sudden appearance of the r term in rdrdtheta. But thanks to your wonderful explanation It finally begins to make a little sense :))
Tom I really like your videos. You're taking complex ideas and really explaining them clearly and you're very good at presenting!. Thank you for taking the time in doing them! they're very helpful! I'd say you're very good at this so keep up the great work! :)
Hi Tom. I come from practically 0 background of mathematics. I enjoy these videos however as you’re concise with your explanations and breakdown the overall operation to the basics in a sense. I think I may dive into mathematics at some stage and see more what it’s all about. Take care my man ! With love from Australia
Defining basis vectors as the rate of change of position vector would make this clearer: i = dR / dx, j = dR / dy, dA = |(dx * i) x (dy * j)|. The Jacobian naturally springs up when considering change of coordinates under these definitions. You don't need to rely on cartesian and the area element is well defined.
I saw this video days later, and today I was studying about soil mechanics where related this video content. And I thought "Oh, I saw this in a video on RUclips". Regards from Ecuador!
I took calc 2 at my university my freshman year and never new where that rdrd0 came from when switching from Cartesian to polar coordinates. Brilliant visualization + explanation!
Literally best Jacobian video I've seen so far (and I've been searching for a *long* time about it), just have a few things I was wondering 1. Why do you do the u in the i direction and v in the j direction ? 2. The very last part of the Jacobian you were writing J = (Xu Yv - Xv Yu) del u del v, and the double integral was like -> J du dv So I didn't really get the very last approximation
1. He didn't. He set x in the direction of the vector i and y in the direction of the vector j. He then set u in some arbitrary direction made up of one component of i and one component of j. Watch again from around 18:30 and you'll see that he transforms the vector i.dx into the vector (∂x/∂u).du.i + (∂x/∂v).dv.j. Similarly he sets up v in another arbitrary direction with i and j components. That transforms the vector j.dy into (∂y/∂u).du.i + (∂y/∂v).dv.j. 2. For some reason, he needlessly switched to using the confusing notation Xu to mean ∂x/∂u, and similarly Xv=∂x.∂v, Yu=∂y/∂u, Yv=∂y/∂v. You won't be the only one confused by that.
Using the differential approximation of x,y as functions os r and theta I think of the Jacobian matrix as the linear transformation that acts upon the space of dr and dtheta and the determinant of it as the stretch factor, I don't know if this is the formal way but i like it 😂
Nice video. I remember studying the Jacobian and the conversion from cartographic to polar coordinates during my degree career, good times. I remember too that these concepts could be applied to Physics but that was another thing that I didn't engage with haha
Instead of giving a vague argument for approximating the curvy rectangle in polar as a "normal" rectangle, you could've simply derived the area for an annular sector: The area of an annulus is A = π(b² - a²), b>a So that the area of an annular sector is A = π(b² - a²) × θ/2π Now let a=r, b=r + dr, and θ -> dθ Which gives the area of an infinitesimal annular sector: dA = [(r + dr)² - r²] dθ /2 = (r² +2r dr + dr² - r²) dθ = r dr dθ
Is this related to the jacobian in robotics? What would we do differently if we knew the new coordinate frame was only a rotation of the previous and not a scale?
Great explanation. I'd want to ask a question about the area ( 23:40 ) , I think you miss to put absolute value, it means Area = | Xu δu Yv δv - Xv δv Yu δu |
8 years after taking calculus, I finally understand wtf a jacobian is. Teachers have so little empathy for that their students don't ALREADY know this stuff, that they forget to try and really explain it. "Oh just make it r dr dtheta because that's you transform from rectangular to polar". What?
The sign of a true master is humility. Those who feel the need to belittle students or obfuscate ideas are not intellectual heavyweights. The real masters are putting their efforts into solving serious problems and winning Fields Medals, not wasting their time flexing on undergrads.
26:32 I used to think that in 2x2 matrix, the 1st column represents the destination of original x vector, and 2nd col for the y vector. But it seems the transformed x and y vector can be either columns or rows respectively without changing its determinant.
At 17:06 you say that |a| |b| sin θ is the cross/vector product, but really it's the length of the vector product. The vector product itself is, as the name suggests, a vector (as opposed to the dot/scalar product, which is a scalar). That confused me for a minute, before looking up the formula. Similarly, I've heard that some sources saying the Jacobian is actually that matrix you create, and the scalar you get when you take the determinant is simply called "the Jacobian determinant". I know I'm just being picky.
Very cool video! I'm still a student so forgive me if I misunderstood, but when writing vectors in every math and physics class I've taken, they have normally been notated with an arrow drawn overtop of the varible, but in this video you underlined the varible instead. Is there actually any difference in meaning or is this just an example of different notation?
20;25 I think there is a little mistake when you represent the transformed vector in the form of Original Cartesian and also when you write the Jacobian matrix . The result is the same because you change the location of two elements on the diagonal .
They indicate (respectively) a derivative, a partial derivative and an infinitesimal change. Understanding the first and last of those is the starting point for learning calculus (assuming you already understand limits), for which you can find a number of online tutorials (I honestly can't advise on which might be the best for you, because it will depend on what you already know and how comfortable you are with it).
Check out the full 'Oxford Calculus' series here: ruclips.net/p/PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S
👍👍👍
the jacobian has an rate of scaling under transformation and jacobians are the true derivative and finding the correct scaling factors from determinants to make the explosion in Riemannian rectangles of the integrals the error converges with infinite sum so the scaling factor is there to rectify the error rate in convergence in rectangles under transformation the rectangles explode and contract and at miniature scale the each point under transformation has the scaling factor
Feels great to know why the Jacobian comes into the calculations when switching coordinate systems. I never learned that while doing multivariate calculus this past semester. Keep up the good work! Regards from a fellow math nerd from Sweden.
Jesus Christ is God Almighty, The everlasting Father !
Its multivariable calculus, not multivariate calculus.
there is a difference ...
rest everything is affirmative ...
@@sachin-mavi multivariable and multivariate calculus are the same thing yo uoaf
🤓
@@SquidBeats amen
You're exactly like how Machine Gun Kelly would have looked if he taught Calculus
😂
Machine Gun Tom(my)
...and if he didn't try to diss Eminem.
RIP
@@rafaelfreitas6159 😂😂😂
I just recently called him “the MGK of mathematics”.
I'm shocked how you've packed many topics such as vector product, Jacobian, areas, and more into such a video, while clearly explaining Jacobian, the main topic. Even if I don't speak English well I can understand it and it is very interesting to watch the explanation and behavior as if you are transmitting energy to the viewer. I'm very satisfied.
I brushed across the Jacobian while learning statistics recently. It seemed reasonable that we'd need to scale by the change of space in that context, but this video made it concrete as to what was going on behind the scenes. Thanks, Tom!
you're very welcome :)
Hey there! The second you explained the Jacobian as the stretch factor of converting from one coordinate system to another, I understood it so much better! This was so much better of an explanation than my textbook
Thanks for your exceptional work Tom. I've got a degree in maths and still learning little things like this really makes sure I keep lifting my knowledge.
You're putting a load of effort into these videos. It is greatly appreciated.
you're very welcome :)
I envy the ability to be good and understand math, I’m doing intermediate algebra right now in college and I’m having a hard time grasping the concept. Love your videos, keep it up!
You sir are a very valuable math resource for students and perhaps even teachers. Thank you!
You're very welcome!
This is by far the most comprehensible explanation of the Jacobian I've ever found. Nice work!
glad it was helpful!
Tom never fails to explain what seem as hard mathematical concepts, in really easy way. Thank You!
Thank you so much. I am a first year Maths student from India, and these simple yet beautiful concepts are what keep mathematics in my heart. Keep up the great work Sir!!
This is such a fantastic video! I'm currently in year 13, thinking of doing a maths degree, im fascinated with calculus, its by far my favourite aspect of maths, not only did multivariable integration make sense but also the use of determinants. Amazing video!
This really should be taught at A-level rather than first-year undergrad courses. Jacobians act as a nice sliproad onto the main highway of tensors and differential geometry in general, whose introduction is in turn often delayed (or even omitted) at bachelor's level.
Bro, for real. As one of your generation I am happy to see that you stood consistent with the style of our generation.
Best intuitive explanation that I've seen so far and for once , even with my weak maths knowledge , understood it for the 1st time. Other youtube presentations never clicked with me but this one did.
Thanks! That was explained in an intuitive way. I guess the key here is to think of the elemental rectangular areas changing in to rotated parallelograms during the coordinate transformation. The example you gave in the beginning with regard to the area of the circle makes the concept clearer.
Absolutely love this video, currently in the process of studying vector calculus (and some other stuff I also don't understand) for machine learning and struggled to wrap my head around jacobian's, this makes so much more sense now
Glad it was helpful!
I already knew how to use change of coordinates and Jacobian. But it is actually the first time I understand the geometric meaning of it :)
Thank you
you're welcome :)
You really are saving me in university... I feel like I can understand where things comes from and why they are the way they are when you explain it... much better than my university professor who is more interested in making us fail class
This is super funny, because this is literally just out of the textbook. Maybe if you oafs read the textbook, you'd learn something. I tutor math and physics, and people say the same thing to me. "You make it so much easier than the professor, and you actually explain where it comes from!"
This jacobian 'proof' is straight out of any Calculus textbook
OMG! You are the best teacher to explain complex subjects.
glad you found it helpful :)
I love your explainations, I now have a better understanding of what I’ve learned in the past 😊 thanks so much for your videos
you're very welcome :)
I always feel grateful for sharing your high-level lectures on RUclips. you are cool.
My pleasure!
This video is too good. So informative and he explained such a difficult calculation so easily. Hats off and keep it up.Thanks Tom👍❤
you're very welcome :)
Congratulation to Tom for introducing the geometrical concept of Jacobian in a very clear manner.(Brazil).
What a mesmerizing presentation. I had math through differential equations at university thirty-five years ago. If you had given lectures, such as you present here, perhaps the 4.0 GPA achieved would had met something. Grade Inflation was in full bloom. Thank you.
Thank you for always providing such valuable learning content!
You're very welcome :)
@@TomRocksMaths Cheers!
Thank you. I always wondered what jacobian was. The geometric explanation was beautiful.
Today I understood what Jacobian really means. Thank you.
Yes finally your video that i watch for college, not for leisure!!!
Whenever I encounter double integrals of some version of the unit circle I’ve always been frustrated by the sudden appearance of the r term in rdrdtheta. But thanks to your wonderful explanation It finally begins to make a little sense :))
The further you go out radially, the bigger the area you sweep for a given angle.
Very nice explanation sir,deserve more views and likes
And this works so well also for triple integrals and volume calculations. Nice video. Greets!
the best Jacobian explanation in the whole Universe
Tom I really like your videos. You're taking complex ideas and really explaining them clearly and you're very good at presenting!. Thank you for taking the time in doing them! they're very helpful!
I'd say you're very good at this so keep up the great work! :)
Thanks, will do!
Demystifying the Jacobian in 30 minutes. Nicely done.
glad it was helpful!
I'm finally learning at school the sort of material he talks about in this channel, feels a bit like a milestone haha.
This was a great video for self learning multivariable calculus, nice!
Nice explainings! Huge thanks and greetings from Spain!
sending love to Spain
Great visualization! That's how you make math accessible for a larger public. Good stuff.
thanks :)
Hi Tom. I come from practically 0 background of mathematics. I enjoy these videos however as you’re concise with your explanations and breakdown the overall operation to the basics in a sense.
I think I may dive into mathematics at some stage and see more what it’s all about.
Take care my man !
With love from Australia
with love to Aus
Thank you so much, theoretical physics is soooo much easier with your explanation for the mathematical concepts ♥️
Defining basis vectors as the rate of change of position vector would make this clearer: i = dR / dx, j = dR / dy, dA = |(dx * i) x (dy * j)|. The Jacobian naturally springs up when considering change of coordinates under these definitions. You don't need to rely on cartesian and the area element is well defined.
Excellent video. I wish all teachers were like you!
Don't judge this man by his attire and theme. He is pure genius.
ive never seen a scene mathematician but im digging it
I saw this video days later, and today I was studying about soil mechanics where related this video content. And I thought "Oh, I saw this in a video on RUclips". Regards from Ecuador!
haha amazing!
Best lecture about this subject I ever seen 👏👏👏
glad it was helpful :)
congratulatons, please make use of maths in simplifying the wonders of theoretical physics
Excelente explicação. Foi a primeira vez que vi Jacobiano explicado de forma tão simples.
I took calc 2 at my university my freshman year and never new where that rdrd0 came from when switching from Cartesian to polar coordinates. Brilliant visualization + explanation!
glad it was helpful :)
I wish I had been taught Jacobians this way many moons ago tbh. Well done Tom
glad it was helpful!
hi,professor,very helpful and very straightfoward, many thanks to you ,great expaination!!!
Literally best Jacobian video I've seen so far (and I've been searching for a *long* time about it), just have a few things I was wondering
1. Why do you do the u in the i direction and v in the j direction ?
2. The very last part of the Jacobian you were writing J = (Xu Yv - Xv Yu) del u del v, and the double integral was like -> J du dv
So I didn't really get the very last approximation
1. He didn't. He set x in the direction of the vector i and y in the direction of the vector j. He then set u in some arbitrary direction made up of one component of i and one component of j. Watch again from around 18:30 and you'll see that he transforms the vector i.dx into the vector (∂x/∂u).du.i + (∂x/∂v).dv.j. Similarly he sets up v in another arbitrary direction with i and j components. That transforms the vector j.dy into (∂y/∂u).du.i + (∂y/∂v).dv.j.
2. For some reason, he needlessly switched to using the confusing notation Xu to mean ∂x/∂u, and similarly Xv=∂x.∂v, Yu=∂y/∂u, Yv=∂y/∂v. You won't be the only one confused by that.
Excellent explanation. Thank you very much
Hey there, this has really helped me to make my concepts better, thanks for the work which u have done brother😊
I've already got the Maple Calculator! And very useful it is, too, especially as you say for visualisation.
It really is!
Wow I'm speechless this video is so amazing
Using the differential approximation of x,y as functions os r and theta I think of the Jacobian matrix as the linear transformation that acts upon the space of dr and dtheta and the determinant of it as the stretch factor, I don't know if this is the formal way but i like it 😂
best explaination ever seen of this topic
thanks, best explanation of Jacobian I found!
glad it helped!
Nice video. I remember studying the Jacobian and the conversion from cartographic to polar coordinates during my degree career, good times. I remember too that these concepts could be applied to Physics but that was another thing that I didn't engage with haha
Instead of giving a vague argument for approximating the curvy rectangle in polar as a "normal" rectangle, you could've simply derived the area for an annular sector:
The area of an annulus is
A = π(b² - a²), b>a
So that the area of an annular sector is
A = π(b² - a²) × θ/2π
Now let a=r, b=r + dr, and θ -> dθ
Which gives the area of an infinitesimal annular sector:
dA = [(r + dr)² - r²] dθ /2 = (r² +2r dr + dr² - r²) dθ = r dr dθ
Is this related to the jacobian in robotics? What would we do differently if we knew the new coordinate frame was only a rotation of the previous and not a scale?
Thank you sir for creating such a brilliant lecture ☺️
my pleasure
Great explanation. I'd want to ask a question about the area ( 23:40 ) , I think you miss to put absolute value, it means
Area = | Xu δu Yv δv - Xv δv Yu δu |
No chance to an answer !
Thanks for this nice explanation. I remember I learned Jacobians at Univertisty 20 years ago, but I totallly forgot about them.
Glad it was helpful!
HOLY SHIT ITS SO SIMPLE YET SO COMPLICATED AT THE SAME TIME
... like pretty much all of maths
so intuitive explanation, thanks dude
Glad it was helpful!
8 years after taking calculus, I finally understand wtf a jacobian is. Teachers have so little empathy for that their students don't ALREADY know this stuff, that they forget to try and really explain it. "Oh just make it r dr dtheta because that's you transform from rectangular to polar". What?
The sign of a true master is humility. Those who feel the need to belittle students or obfuscate ideas are not intellectual heavyweights. The real masters are putting their efforts into solving serious problems and winning Fields Medals, not wasting their time flexing on undergrads.
@@tetrabromobisphenol I'm talking about high school. Nobody was flexing on anybody...
As an engineering student I can totally relate to this
26:32 I used to think that in 2x2 matrix, the 1st column represents the destination of original x vector, and 2nd col for the y vector. But it seems the transformed x and y vector can be either columns or rows respectively without changing its determinant.
Great discussion
Wow, realmente este canal......es mi mejor descubrimiento en RUclips. ..
Amazing lecture! Thank you so much...
really nice explanation!
Woah, I was just talking to a friend about Jacobians yesterday. What a coincidence!
google is listening...
Congratulations!!! It could extend to the Hessians without restriction and to the restricted.
great explanation i am speechless 🙇
Wow, that's a really clear explanation! Thanks so much!
glad it was helpful :)
Thanks a lot. An outstanding lecture.
Love this chap, i could easily learn from him.
Man I love these videos
what an wonderful explainantion by you .love you bro from india
You make math so beautiful.
This is the ultimate meaning of the determinant. Stretching space.
amen
Awesome video. Thank you
Glad you liked it!
Absolutely loved this...
At 17:06 you say that |a| |b| sin θ is the cross/vector product, but really it's the length of the vector product. The vector product itself is, as the name suggests, a vector (as opposed to the dot/scalar product, which is a scalar). That confused me for a minute, before looking up the formula.
Similarly, I've heard that some sources saying the Jacobian is actually that matrix you create, and the scalar you get when you take the determinant is simply called "the Jacobian determinant". I know I'm just being picky.
Now it all makes sense
I see MGK has had a career change, respect to Eminem. The gift that keeps on giving. Now we have a good math lecture.
great!!!! awesome explanations greetings from colombia
Great explanation!!
Glad it was helpful!
loving the hair my guy #MakeEmoGreatAgain
#emosnotdead
Great video. How you teach reminds me of Richard Feynman.
awesome, thanks!
That is so brilliant! Thank you so much❤️
Professor，your class about the jocobian is excellent，but I don't understand what does dx/du ×delta u i（j）means😢😢Can you explain it，thanks sincerely
Very cool video! I'm still a student so forgive me if I misunderstood, but when writing vectors in every math and physics class I've taken, they have normally been notated with an arrow drawn overtop of the varible, but in this video you underlined the varible instead. Is there actually any difference in meaning or is this just an example of different notation?
It’s just a notation thing - both can be used to mean a vector
@@TomRocksMaths Gotcha, thank you!
20;25 I think there is a little mistake when you represent the transformed vector in the form of Original Cartesian and also when you write the Jacobian matrix .
The result is the same because you change the location of two elements on the diagonal .
I got a feel of a Jacobian. That's good.
Can you recommend a good resource to explain the difference between 'd', 'cursive d', and 'delta' as they appear in these formulas?
They indicate (respectively) a derivative, a partial derivative and an infinitesimal change. Understanding the first and last of those is the starting point for learning calculus (assuming you already understand limits), for which you can find a number of online tutorials (I honestly can't advise on which might be the best for you, because it will depend on what you already know and how comfortable you are with it).