evaluating a double integral using the Jacobian

[Ninja]

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.

[larryevans]

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

[scottmiller]

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).

[conanedojawa]

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

[joansgf]

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.

[cesarmorenoy]

You should do more higher level calc topics!

[Chrisuan]

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

[chessthejameswei]

What a coincidence that I just learned this today!

[wileyst.germain]

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

[ShanBojack]

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

[Yufanto]

What will happen if the V = - U, when the U is 2x - y and V is y - 2x, there is no Jacobian?

[louiejohncastillo]

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?? 🙀🙀

[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