Clarification on Jacobians please

[Ocifer]

Hello,

I need to change some triple integrals from Cartesian coordinates to cylindrical coordinates. I understand the geometry used to derive the equations for doing so, but I have a question about the Jacobian. We need to adjust by the absolute value of the determinant of the Jacobian to evaluate the transformed integral.

My professor's notes are a bit spotty, so I have been trying to teach myself from the internet.

At this wikipedia link (http://en.wikipedia.org/wiki/List_o...ransformations#From_cylindrical_coordinates_2)

It gives me the Jacobian for moving FROM Cartesian TO cylindrical coordinates.

However, this link (http://www.math24.net/triple-integrals-in-cylindrical-coordinates.html)
gives a different Jacobian for converting FROM Cartesian TO cylindrical coordinates.

Which one is correct?

 

[lugita15]

Um, as far as I can tell they look identical. The wikipedia page lists the Jacobian matrix and the volume element. The other page lists the Jacobian determinant (the determinant of the Jacobian matrix) and the expression for triple integrals in cylindrical coordinates. The two pages are consistent with each other, and they're both right.

 

[Char. Limit]

No, they're definitely the same. The only difference I see is in terminology. Wikipedia uses r for the radius and the other website uses $\rho$.

 

[Ocifer]

Perhaps I didn't pose my question clearly enough.

The second link gives a Jacobian matrix where the determinant is rho. In that link, they are discussing moving FROM Cartesian TO cylindrical.

The Wikipedia link gives the very same matrix, having determinant rho, under the section "TO Cartesian ... FROM Cylindrical".

The matrices are the same, yes, but the two links seem to imply a different "direction". The wikipedia link says

Cylindrical ---> Cartesian

and the second link (non-Wikipedia) seems to imply they're talking about

Cartesian ---> Cylindrical

It doesn't seem right to me that the Jacobian is the same regardless of the "direction" (again using the word direction here in the vernacular sense) of the transformation. The two links seem to imply opposing directions. Can someone explain what I'm missing?

EDIT: I can deal with the differing usage of theta, r, rho, etc... but the direction issue is what's confusing me.
EDIT2: Here, for example, is a different wikipedia page on changing coordinate systems (for multiple integrals) http://en.wikipedia.org/wiki/Multiple_integral
In example 3-a on that page, they are going from Cartesian -> Cylindrical, but they use the Jacobian determinant of the opposing direction (if one were to compare against the other Wikpedia article)
EDIT3: In the Wikipedia article I originally posted, other sections also seem to be going in opposing directions (to my confused mind). For instance, when changing a multiple integral from Cartesian to Spherical,
it is customary to adjust by rho^2 * sin( phi )... in the Wikipedia article I first posted, they again go in the opposing direction, listing that Jacobian determinant under Spherical --> Cartesian.

The main confusion for me is that all the Jacobians listed in the Wikipedia article (first one) seem to be going in the opposite direction from what I would expect.

 

[Ocifer]

Think I may have figured out the source of my confusion. Please tell me if this makes sense.

Since the Jacobian is taken for some scalar-valued or vector-valued function with respect to another vector, we're actually considering the FUNCTION which takes one coordinate system to another.

So when I want to change an integral from Cartesian coordinates to cylindrical coordinates I have to use the following functions:

x = g( r, theta, z) = rcos(theta)
y = h(r, theta, z) = rsin(theta)
z = m(r, theta, z) = z

The functions g, h, m when aggregated into a vector are a map FROM (r, theta, z) TO (x,y,z). So even though I use said map to change my integral from Cartesian to cylindrical, the map itself is from cylindrical coordinates into (x,y,z) coordinates.

I think that's why everything seemed "reversed" to me. Have I got it now, though?

 

[lugita15]

Think I may have figured out the source of my confusion. Please tell me if this makes sense.

Since the Jacobian is taken for some scalar-valued or vector-valued function with respect to another vector, we're actually considering the FUNCTION which takes one coordinate system to another.

So when I want to change an integral from Cartesian coordinates to cylindrical coordinates I have to use the following functions:

x = g( r, theta, z) = rcos(theta)
y = h(r, theta, z) = rsin(theta)
z = m(r, theta, z) = z

The functions g, h, m when aggregated into a vector are a map FROM (r, theta, z) TO (x,y,z). So even though I use said map to change my integral from Cartesian to cylindrical, the map itself is from cylindrical coordinates into (x,y,z) coordinates.

I think that's why everything seemed "reversed" to me. Have I got it now, though?

Yes, you've got it. The mapping is FROM cylindrical coordinates TO Cartesian coordinates, and this mapping is used to translate FROM an integral in Cartesian coordinates TO an integral in cylindrical coordinates.

 

[Ocifer]

Thanks a lot! The confusion arose because I've only seen Jacobians in the context of coordinate transformations for multiple integrals, and I guess I just needed to stew on it to consider the Jacobians on their own.