Change of Variables

The Jacobian

[rohithnarra]

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!

[tgx]

It was nice. But only 0<r<=1 (regular)

[TheMauror]

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!

[shandyverdyo]

Me : *trying to sleep.
Youtube : hey, we have a new video for you.
Me : Ok, here i am.

[emman]

I never understood why we had to do the Jacobian until now. Also nice suit.

[hamanahamana]

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?

[manfredwitzany]

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.

[ugursoydan]

have you got a proof for the jacobian matrix technique? Why are we doing a matrix calculation?

[tomatrix]

Which video is it that you were talking about at the start to show the linear algbra approach to calculating our change of variable

[frozenmoon]

I was awaiting to hear "Where is my BpRp" but you were close :).

[dhunt]

Man, so glad you found that lost 16/sqrt(3) slice of pi! I bet it was delicious!

[cameronspalding]

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

[cheny]

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!

[Alidaher]

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?

[MathwithMing]

Awesome presentation! Would you be interested in making a video on differential forms? Cheers man

[nathanisbored]

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?

[lindsaywaterman]

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.

[dougr.]

What does the region look like? A good drawing would be nice

[brendonreidvictor]

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?

[morphomorph]

Does this also work if u & v aren't linear combinations of x & y?