How Does the Jacobian Affect Coordinate Transformations?

Click For Summary

Discussion Overview

The discussion revolves around the role of the Jacobian in coordinate transformations, particularly how it relates to the change in area elements when transitioning between different coordinate systems. Participants explore the mathematical derivation of the Jacobian and its implications in both finite and infinitesimal cases, addressing conceptual clarity and potential ambiguities in textbook presentations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant explains that the Jacobian describes how surface or volume elements change under coordinate transformations, suggesting it characterizes differences in area measurement between coordinate systems.
  • Another participant notes that the statement ##dxdy=\lvert J(u,v)\rvert dudv## is more of a textual substitution rather than a strict mathematical equation, emphasizing the context of integrals.
  • A participant argues that the informal derivation of the Jacobian is intuitive but questions why textbooks express the infinitesimal area element as ##dxdy## instead of maintaining the relationship with the Jacobian in the limit.
  • Some participants discuss the implications of using orthonormal coordinate systems and how this affects the relationship between area elements in the limit, suggesting that the incremental area becomes rectangular.
  • There is a concern about whether the ##uv## coordinate system is curvilinear and how this affects the area calculations when mapped to the ##xy## coordinate system.
  • One participant expresses confusion regarding the clarity of certain notes on the topic, indicating that the coordinate system used may not be curvilinear, which could lead to misunderstandings.

Areas of Agreement / Disagreement

Participants express differing views on the clarity and correctness of the derivations and representations of the Jacobian in textbooks. There is no consensus on the best way to express the relationship between area elements in different coordinate systems, and some participants remain uncertain about the implications of curvilinear versus orthonormal systems.

Contextual Notes

Participants highlight that the derivation of the Jacobian and its application can vary based on the specific context and definitions used in different texts. There is an acknowledgment of potential ambiguities in how area elements are represented in the limit.

  • #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.
 
Last edited:

Similar threads

High School Solving for the Nth divergence in any coordinate system
  • · Replies 41 ·
2
Replies
41
Views
5K
Undergrad How to find the generator of this Lie group?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Matrix form of metric component transformation?
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Diffusion equation and a system of equations with reciprocal unknowns?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Infinitesimal Coordinate Transformation and Lie Derivative
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad Adding total time derivative to Lagrangian/Canonical Transformations
  • · Replies 2 ·
Replies
2
Views
843
Undergrad Lipschitz continuity of vector-valued function
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Surface waves and vorticity in 2D
  • · Replies 5 ·
Replies
5
Views
1K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
  • · Replies 1 ·
Replies
1
Views
3K
MIT OCW, 8.02 Electromagnetism: Potential for an Electric Dipole
  • · Replies 1 ·
Replies
1
Views
3K