How Does the Jacobian Affect Coordinate Transformations?

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SUMMARY

The Jacobian is a mathematical tool that describes how area or volume elements change during coordinate transformations, specifically in the context of mapping between two coordinate systems, such as (u,v) and (x,y). The Jacobian determinant, defined as J(u,v) = (∂x/∂u)(∂y/∂v) - (∂y/∂u)(∂x/∂v), quantifies the distortion of area elements when transitioning from one coordinate system to another. In the infinitesimal limit, the relationship between the area elements is expressed as dx dy = |J(u,v)| dudv, indicating that the infinitesimal area in the (x,y) plane is proportional to the area in the (u,v) plane scaled by the Jacobian determinant. This relationship is crucial for understanding integrals over transformed coordinates and highlights the importance of the Jacobian in differential geometry.

PREREQUISITES
  • Understanding of coordinate transformations in calculus
  • Familiarity with partial derivatives and their notation
  • Basic knowledge of differential geometry concepts
  • Experience with integrals and area calculations in multiple dimensions
NEXT STEPS
  • Study the derivation of the Jacobian determinant in higher dimensions
  • Learn about differential forms and their application in coordinate transformations
  • Explore examples of coordinate transformations in polar and spherical coordinates
  • Investigate the implications of Jacobians in multivariable calculus and integration techniques
USEFUL FOR

Mathematicians, physicists, and engineers who work with multivariable calculus, coordinate transformations, and differential geometry will benefit from this discussion. It is particularly relevant for those involved in fields such as fluid dynamics, robotics, and computer graphics where transformations between coordinate systems are essential.

  • #31
zinq said:
Keep the concepts of a point, and its coordinates (which are just its name), separate. This usually clarifies what is going on.

Thanks for your detailed answer to my follow up comments.

If we are considering a mapping between two coordinate systems describing the same set of points making up an area in some space (manifold) doesn't the Jacobian (determinant) describe how the (coordinate) unit of area changes between the two coordinate systems?

Isn't it correct that we can use the inverse of a given coordinate map to parametrise the patch on the manifold (that is the preimage of the original coordinate map), and so each given coordinate system defines an (infinitesimal) unit of (coordinate) area that can be used to tile the area that we a considering on the manifold and subsequently integrate over to determine a value of area. The Jacobian determinant describes how these tiles are distorted as we map between two different coordinate systems, but of course we must obtain the same value for the area on the manifold (as the area exists independently of any coordinate system) and so both cases lead to the same result (upon integration).

Sorry, I feel I haven't really explained what I mean very well, but hopefully you get the gist of it.
 
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