Coordinate transformation and multiplying with size of J

In summary, the conversation discusses the transformation of coordinates in multiple integrals and the use of the Jacobian, represented as |J|, in these transformations. The question arises on why it is necessary to multiply with the Jacobian and where to find the proof for this concept. The response explains that the volume of the transform is preserved when the size of |J| is one, and further clarifies the use of the Jacobian in calculating the volume of a parallelopiped formed from three vectors. The conversation concludes with a suggestion to google "Coordinate transformation multiple integrals" for more information on the topic.
  • #1
greisen
76
0
Hi,

I am using the book "Advanced Engineering Mathematics" by Erwin Kreyszig where I am reading on the transformation of coordinates - when changing from \int f(x,y) to \int f(v(x,y),v(x,y) it is necessary to multiply with the size of the jacobian, |J| - I cannot find the proof in the book and I don't quite understand why one should multiply with this?
Any help or advise where to locate the proof in order to better understand the multiplication with the Jacobian.

Thanks in advance

Best
 
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  • #2
if you google "Coordinate transformation multiple integrals", you get lots of hits :)
 
  • #3
Hi,

So if I transform and the volume of the transform is preserved the size of |J| is one?
 
  • #4
Yes. If you have a "parallelopiped" (3 d figure like a "tilted" retangle) formed from 3 vectors [itex]\vec{u}[/itex], [itex]\vec{v}[/itex], and [itex]\vec{w}[/itex], the volume is given by the length of the "triple" product [itex]\vec{u}\cdot(\vec{v}\times\vec{w})[/itex] which, in turn, can be calculated by the determinant having those vectors as rows. That's basically what the Jacobian is.
 

Related to Coordinate transformation and multiplying with size of J

What is coordinate transformation?

Coordinate transformation is the process of converting coordinates from one coordinate system to another. It involves using mathematical equations and transformations to map points from one system to another.

Why is coordinate transformation important in science?

Coordinate transformation is important in science because it allows for the comparison and integration of data from different coordinate systems. This is especially crucial in fields such as physics and engineering, where measurements and calculations are often done in different coordinate systems.

What is the purpose of multiplying with the size of J in coordinate transformation?

Multiplying with the size of J, also known as the Jacobian, is necessary in coordinate transformation because it accounts for the change in scale between the two coordinate systems. It is a crucial step in accurately transforming coordinates and ensuring that the relationships between points are preserved.

How do you determine the appropriate coordinate transformation for a specific problem?

The appropriate coordinate transformation for a specific problem depends on the type of data being transformed and the desired outcome. It is important to understand the characteristics and limitations of different coordinate systems and choose the transformation that best fits the problem at hand.

Are there any common mistakes to avoid when performing coordinate transformations?

Yes, there are a few common mistakes to avoid when performing coordinate transformations. These include forgetting to include the Jacobian in the transformation equations, using the wrong transformation for the data, and not accounting for differences in unit systems between the two coordinate systems.

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