The Role of the Jacobian in Change of Variables Integration

[philosophking]

Hey everyone,

What is the purpose of the Jacobian in change of variables integration? Does it have something to do with the fact that you are basically performing a linear transformation on a set that you are integrating over?

There's no rush on this, I was just wondering. Any websites or books you could guide me to would probably be of most help. I was looking at mathworld, but they really don't explain very well.

I guess the main thing that I'm confused about is the differential matrix, and how the determinant (which is the jacobian) relates to the change of variables.

Thanks again for your help :)

 

[Hurkyl]

The geometric way of looking at it is that the Jacobian tells you the relative size between two infinitely small parallelpipeds. (One with sides in the first set of variables, the other with sides in the second set of variables)


The algebraic way is to simply write out the differentials, and use the algebraic rule that dx dy = -dy dx, and dx dx = 0 to work out what the new integrand should be.

e.g. if x = 2uv and y = u^2 + v^2, then:
dx = 2v du + 2u dv
and
dy = 2u du + 2v dv
Then...
dx dy = (2v du + 2u dv) (2u du + 2v dv) = ...

 

[dextercioby]

You know,these things are very neatly described in differential geometry,so my advice is to put your hands on Spivak's compendium really soon.

As for why differentials in the Jacobian,remember that it is just a n-dimensional generalization of the 1d case in which

$$I=\int f(x) \ dx$$ under

$$x\rightarrow y$$ through a diffeomorphism

$$I=\int f(y(x)) \frac{dy}{dx} \ dx =\int f(y) \ dy$$

the HS-learnt substitution method...


Daniel.

 

[philosophking]

I'll take a gander at differential geometry someday, thanks. Are there any prerequisites for this course, or can I pretty much dive right in? I've taken through linear algebra and also real/complex analysis.

 

[HallsofIvy]

Dextercioby: do you get a commission from Spivak?

(Yes, I agree: Spivak's Differential Geometry is excellent.)

 

[dextercioby]

Nope,but i liked his books..."Calculus on Manifolds" was really excellent.

Daniel.