Awesome video! I think it would be great if you do a video proving the change of variables theorem. I love when you do a video proving a cool theorem. Please do more of that!
Emmanuel Azadze Yep, he presented it really well. Jacobian basically calculates the area change. So we change it to a nicer shape and account for area change via jacobian.
Dr. Peyam you are the GOAT! My multivariable calculus and advanced calculus professors were unable to explain why the change of variables formula involves a Jacobian determinant and what that means. You are the best and keep doing what you are doing as you are f****ng amazing at it!
Realy good work. How would a physist solve this problem? He would directly formulate the problem in adequate coordinates, namely rotate the coordinate system and use elliptical cooridantes. It is intersting to see, the way to solve is nearly the same. Only matrix diagonalisation is not necessery using this way.
Good video! I have a couple questions though. 1. at 8:48, why do we need the absolute value sign? I know you said "to keep it positive," but why do we need to keep it positive? 2. I'm confused about how you went from dxdy/dudv to the Jacobian. Could someone explain it to me?
1) Basically the Jacobian measures a positive change in the area or volume, and you need that because if f is a positive function, then the integral after change of vars should be positive
2) dxdy/dudv is just notation, it doesn’t mean anything. The Jacobian is the correct formula to use, and again because the det measures a change in area/volume
Hi Dr Peyam, I'm confused about why r ranges from 0 to 1. If we fix a θ, r is always =1 since it's a unit circle, surely it's never 0. But again we would be integrating from 1 to 1 which is 0 which raises another problem...Thanks!
Dr Peyam just wondering why is it that when we make a single variable U-Sub (i.e.the single variable calculus analogue of this) we do not need to take the absolute value of the dx/du?
You don't need abs value basically because the x-axis is the "region" being considered, and the x-axis is one dimensional. So it only really has one orientation. With multiple variables, the region being considered in often two-dimensional, representing an area. So imagine the region as a lake of water. You can either be above the surface of the water or below. In other words, if the region is upside down, you get a different value. This doesn't happen with a line, because a line can't really be upside down.
Since polar coords is really a second change of variables in disguise, surely there must have been a more direct change of variables to choose at the beginning to skip that step. Also, I remember learning this vaguely, but I could have sworn I remembered the jacobian formula involving some kind of cross product. What’s that about?
@@hyperboloidofonesheet1036 i know that but, im curious what the systematic method would be (he mentioned diagonalizing a matrix, but that doesnt seem to give you the direct transformation)
I did a substitution of u = x -y and v = x+y and got the double integral of (3/4u^2 + 1/4v^2)/2 du dv and got a result of 8/sqrt(3) times pi. My value is twice Dr Peyam's value. Why? Since from the integral of the ellipse of u and v, a = sqrt(8/3) and b = sqrt(8); area = pi times ab. I think that Dr Peyam should have used u^2 + v^2 =1 in the integral and then integrate r dr d(theta), then the area of the unit circle would be pi and the total integral would be 8/sqrt(3) times pi, which gives my answer.
Me : *trying to sleep.
UA-cam : hey, we have a new video for you.
Me : Ok, here i am.
You don't need sleep, you need calculus
Beautifully done!
Awesome video! I think it would be great if you do a video proving the change of variables theorem. I love when you do a video proving a cool theorem. Please do more of that!
I never understood why we had to do the Jacobian until now. Also nice suit.
Emmanuel Azadze Yep, he presented it really well. Jacobian basically calculates the area change. So we change it to a nicer shape and account for area change via jacobian.
Dr. Peyam you are the GOAT! My multivariable calculus and advanced calculus professors were unable to explain why the change of variables formula involves a Jacobian determinant and what that means. You are the best and keep doing what you are doing as you are f****ng amazing at it!
Thanks so much!!!
It was nice. But only 0
I was awaiting to hear "Where is my BpRp" but you were close :).
Man, so glad you found that lost 16/sqrt(3) slice of pi! I bet it was delicious!
From india lots of love
Niceeeee
Realy good work. How would a physist solve this problem? He would directly formulate the problem in adequate coordinates, namely rotate the coordinate system and use elliptical cooridantes. It is intersting to see, the way to solve is nearly the same. Only matrix diagonalisation is not necessery using this way.
Which video is it that you were talking about at the start to show the linear algbra approach to calculating our change of variable
have you got a proof for the jacobian matrix technique? Why are we doing a matrix calculation?
If you integrate a function over a subset of the complex numbers: is that not similar to computing an integral of a subset of R^2
Good video! I have a couple questions though.
1. at 8:48, why do we need the absolute value sign? I know you said "to keep it positive," but why do we need to keep it positive?
2. I'm confused about how you went from dxdy/dudv to the Jacobian. Could someone explain it to me?
1) Basically the Jacobian measures a positive change in the area or volume, and you need that because if f is a positive function, then the integral after change of vars should be positive
2) dxdy/dudv is just notation, it doesn’t mean anything. The Jacobian is the correct formula to use, and again because the det measures a change in area/volume
Awesome. Thank you!
Awesome presentation! Would you be interested in making a video on differential forms? Cheers man
Eventually :)
How about now?
Hi Dr Peyam, I'm confused about why r ranges from 0 to 1. If we fix a θ, r is always =1 since it's a unit circle, surely it's never 0. But again we would be integrating from 1 to 1 which is 0 which raises another problem...Thanks!
But we’re ranging over the unit disk not the unit circle, so r goes from 0 to 1, which represents the inside of the disk
Dr Peyam just wondering why is it that when we make a single variable U-Sub (i.e.the single variable calculus analogue of this) we do not need to take the absolute value of the dx/du?
Because the way they teach you in calculus is wrong
The Jacobian ua-cam.com/video/SrYStw84T4o/v-deo.html
You don't need abs value basically because the x-axis is the "region" being considered, and the x-axis is one dimensional. So it only really has one orientation. With multiple variables, the region being considered in often two-dimensional, representing an area. So imagine the region as a lake of water. You can either be above the surface of the water or below. In other words, if the region is upside down, you get a different value. This doesn't happen with a line, because a line can't really be upside down.
Does this also work if u & v aren't linear combinations of x & y?
Of course!
@@drpeyam Cool, thanks
What does the region look like? A good drawing would be nice
Just a diagonal ellipse
Dr Peyam yes, the xy factor tells us so
Dr peyan i canr find the link of methods to find the convinient change of variable via diagonalization can you give it to me please?
I think it’s called quadratic forms!
@@drpeyam ah alright ! do you have any videos regarding this topic?
Since polar coords is really a second change of variables in disguise, surely there must have been a more direct change of variables to choose at the beginning to skip that step.
Also, I remember learning this vaguely, but I could have sworn I remembered the jacobian formula involving some kind of cross product. What’s that about?
Cross product? Not sure
Original equation x²-xy+y²=2 becomes r²(1-sin(Θ)cos(Θ))=2
@@drpeyam found my old notes, i guess this is what i was thinking of maybe? i.imgur.com/yj8nmss.jpg
@@hyperboloidofonesheet1036 i know that but, im curious what the systematic method would be (he mentioned diagonalizing a matrix, but that doesnt seem to give you the direct transformation)
Ah yes, that’s for the surface integral of a function! A kind of Jacobian, I guess, although I usually don’t think of it as one
*The Jacobian*
Shouldn't the original region be defined by an inequality?
Yeah, but it’s a small detail
I did a substitution of u = x -y and v = x+y and got the double integral of (3/4u^2 + 1/4v^2)/2 du dv and got a result of 8/sqrt(3) times pi. My value is twice Dr Peyam's value. Why? Since from the integral of the ellipse of u and v, a = sqrt(8/3) and b = sqrt(8); area = pi times ab. I think that Dr Peyam should have used u^2 + v^2 =1 in the integral and then integrate r dr d(theta), then the area of the unit circle would be pi and the total integral would be 8/sqrt(3) times pi, which gives my answer.
But that’s incorrect though
u^2 + v^2 = 1 only on the boundary of the region, it’s not equal to 1 inside the region
@@drpeyam I take your point. Then, will you kindly use my u, v substititutions and arrive at the correct answer? Thank you in anticipitation.