evaluating a double integral using the Jacobian

Calculus tutorial on double integral with Jacobian. This is a highlight from 100 integrals part 2: • 100 integrals (almost ...
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[@Ninja20704]

I would really appreciate if you did more calculus 3 videos. I just finished high school and want to get a grasp of multivariable calculus and also more differential equation techniques before I start university.
Really appreciate your content.

[@matts2565]

I agree, I felt so lost and alone without (as many) BPRP videos during calc 3 lol

[@francompalmieri]

@@matts2565 Professor Leonard came in clutch though lmao

[@matts2565]

@francompalmieri yes and no; I love his videos but the whole point of CZcams for me is for when I explicitly *don't have time* to watch through an entire hour+ long lecture

[@francompalmieri]

@@matts2565 yeah in my case I used his videos as lectures pretty much so I get what you mean, I understood him much better than my professor.
bprp helped me massively in my first calculus classes in uni with his million examples of integrals and series though, forever grateful for that haha.

[@scottmiller2591]

I like to think of such problems like this:
1. You want to know the weight of a piece of rubber the shape of the hatched area on the curve. The rubber sheet weighs f(x,y) unit per area, that is, the areal density.
2. It would be much simpler if the area is transformed to a rectangular area in U,V coordinates.
3. In the process of transforming the shape of the area, the rubber sheet is stretched and/or compressed so the areal density changes; the Jacobian is the new areal density of the rubber sheet once it has been transformed into a rectangle.
Then it simply becomes a matter of figuring out how to transform to a rectangle (the substitution of variables), integrate the function using the new areal density from the Jacobian, and figure out the corner coordinates of the rectangle (the new limits).

[@larryevans2806]

Best damn math teacher I ever met and I've been around awhile!!!!

[@joansgf7515]

Note that d(u,v)/d(x,y)=(d(x,y)/d(u,v))^{-1}, so it'd have been easier to compute first d(u,v)/d(x,y), which is -2(y/x)=-2u and then to get d(x,y)/d(u,v) by "flipping" d(u,v)/d(x,y), therefore -1/2u.

[@torok2001kristofaron]

Wonderful derivation. 🙌

[@waziakbar9991]

I’m all for the calc 3 vids 😊

[@conanedojawa4538]

The vedio is very good . And I want to know how the Jacobian method works? I need a simple proof for this method

[@Ninja20704]

Heres a video by Dr Trefor Bazett where he gives a explanation on why the jacobian is what it is.
czcams.com/video/wUF-lyyWpUc/video.html
(Skip to 6:44)
It was very helpful to me, i hope its helpful to you too

[@chessthejameswei]

What a coincidence that I just learned this today!

[@cesarmoreno987y]

You should do more higher level calc topics!

[@Chrisuan]

Do you have a video on the basics of Jacobian, explaining why this process works?

[@othila9902]

I can't explain the whole process, but if you notice in the video he talks about "changing rectangles", basically rectangles that do not have the same width. The Jacobian allows you to integrate a rectangular space that's equivalent to the original curved one (like this example). If you notice, the jacobian is pretty similar to the u substitution that you use for integrals of one variable, but in this case it's like u substitutuon for many variables

[@ShanBojack]

I am wondering more about how did you get the first four relations between X and Y

[@wileyst.germain7374]

Would be much easier to do difference of integrals for each interval between intersections?

[@louiejohncastillo9822]

the answer in wolfram alpha is (ln 2)/2+ln 16 = 3.119

[@d.h.y]

This was such a wonderful video !!!
,,,and could somebody elaborate why the Jacobian determinant in 6:54 has to be an absolute value?? 🙀🙀

[@d.h.y]

@@joansgf7515 Thank you for the link! I will check it !!!

[@Bruh-bk6yo]

man😱

[@Zeke-_-Geekin]

Hey, im a high school student trying to learn about calculus, could you tag me in your introductory video for Calculus/Pre Calc