What is dx, dy and dz in spherical coordinates

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SUMMARY

The discussion focuses on the differentiation of spherical coordinates, specifically the variables dx, dy, and dz. The equations for converting spherical coordinates to Cartesian coordinates are established as x = r sinΘ cosΦ, y = r sinΘ sinΦ, and z = r cosΘ. The total differential of x is defined in terms of the variables r, Θ, and Φ, emphasizing the need to differentiate with respect to all variables due to the lack of a 1:1 correspondence in polar coordinates. For further understanding, the discussion recommends reviewing the total differential and relevant Wikipedia pages on polar coordinates.

PREREQUISITES
  • Understanding of spherical coordinates and their relationship to Cartesian coordinates
  • Familiarity with differentiation and total differentials in calculus
  • Knowledge of polar coordinates and their properties
  • Basic grasp of trigonometric functions and their applications in coordinate transformations
NEXT STEPS
  • Research "total differential" in calculus for a deeper understanding of its application
  • Study "spherical coordinates" and their conversion to Cartesian coordinates
  • Explore the "polar coordinate system" on Wikipedia for comprehensive insights
  • Examine trigonometric identities and their role in coordinate transformations
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with coordinate transformations and calculus, particularly those focusing on spherical and polar coordinates.

LSMOG
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What is dx, dy and dz in spherical coordinates
 
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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
 

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