What is dx, dy and dz in spherical coordinates

Click For Summary

Discussion Overview

The discussion revolves around understanding the differentials dx, dy, and dz in the context of spherical coordinates, particularly how they relate to polar coordinates. Participants explore the differentiation of the Cartesian coordinates x, y, and z expressed in terms of spherical coordinates (r, Θ, Φ) and the implications of these transformations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants inquire about the expressions for x, y, and z in spherical coordinates and how to differentiate them.
  • There is a suggestion that differentiation should be performed with respect to multiple variables, as there is no one-to-one correspondence in polar coordinates.
  • One participant notes that the origin in polar coordinates is artificial and raises concerns about the valid intervals for the angles.
  • A later reply introduces the concept of the total differential of x with respect to r, Θ, and Φ, prompting others to look up relevant mathematical concepts.
  • Additional resources are provided for further reading on the topic, including links to Wikipedia pages that discuss polar coordinates and calculus.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the differentiation process and the implications of the coordinate transformations. Multiple competing views on how to approach the differentiation remain evident, and the discussion does not reach a consensus.

Contextual Notes

Participants highlight limitations related to the correspondence between Cartesian and polar coordinates, including the artificial nature of the origin and the need for restrictions on angle intervals. The discussion reflects a complex interplay of variables without resolving the mathematical steps involved.

LSMOG
Messages
62
Reaction score
0
What is dx, dy and dz in spherical coordinates
 
Physics news on Phys.org
What are ##x,y,z## in polar coordinates? Now differentiate.
 
fresh_42 said:
What are ##x,y,z## in polar coordinates? Now differentiate.
Okay. Because there are many variables. x = r sinΘ cosΦ , y = r sinΘ sinΦ and z = r cos Θ, I differebtiate with respect to what variable?
 
LSMOG said:
Okay. Because there are many variables. x = r sinΘ cosΦ , y = r sinΘ sinΦ and z = r cos Θ, I differebtiate with respect to what variable?
All. There is no 1:1 correspondence. E.g. the origin is artificial in polar coordinates, e.g. ##r=0##, and the angles? Then there has to be a restriction of valid intervals for the angles. We will have nine equations for (x,y,z) → (r,Θ,Φ).
 
Thanks, let me try
 
In this situation, dx is the total differential of x with respect to r, θ and Φ. So look up "total differential" and see what turns up...
 
Okay. Thanks
 

Similar threads

Undergrad Velocity Vector Transformation from Cartesian to Spherical Coordinates
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Volume element in Spherical Coordinates
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Integration over a part of a spherical shell in Cartesian coordinates
  • · Replies 1 ·
Replies
1
Views
3K
Graduate How to find the displacement vector in Spherical coordinate
  • · Replies 11 ·
Replies
11
Views
7K
Graduate Laplace transform in spherical coordinates
  • · Replies 3 ·
Replies
3
Views
3K
MHB Mass of a Torus in Spherical Coordinates
  • · Replies 1 ·
Replies
1
Views
2K
High School What Is the Meaning of $\frac{d}{dx} \frac{dy}{dx}$?
  • · Replies 10 ·
Replies
10
Views
2K
MHB Evaluate the spherical coordinate integrals
  • · Replies 8 ·
Replies
8
Views
2K
MHB Integrating (triple) over spherical coordinates
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad ## \int ~ dx dy dz ~ f(x,y,z)~ \delta (x+y+z-1)##
  • · Replies 11 ·
Replies
11
Views
4K