Contravariant components and spherical component of acceleration

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Discussion Overview

The discussion revolves around the derivation of acceleration components in spherical coordinates and the distinction between contravariant and covariant vectors. Participants explore the implications of using different coordinate systems and the resulting expressions for acceleration.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a derivation of acceleration components in spherical coordinates but encounters discrepancies when applying the definition of acceleration as a contravariant vector.
  • Another participant suggests that without seeing the derivation steps, it is challenging to identify the mistake made by the first participant.
  • A participant shares an example that illustrates a difference between their results and the expected spherical components, proposing that using covariant unit vectors could resolve the issue.
  • It is noted that coordinate basis vectors are generally not unit vectors, with a distinction made between Cartesian and spherical coordinate vectors.
  • There is a reiteration that using covariant unit vectors might lead to a unique result, but it remains unclear if this is universally correct.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to deriving acceleration components and whether the use of covariant unit vectors resolves the discrepancies. The discussion remains unresolved regarding the correctness of the proposed solutions.

Contextual Notes

There are limitations in the discussion, including missing derivation steps and assumptions regarding the definitions of contravariant and covariant vectors. The implications of using different coordinate systems are also not fully explored.

world line
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Hello
i know how to derive the components of acceleration in other coordinates like spherical
start here :
http://up.iranblog.com/images/0mbwuclckbu51bxt8jfa.jpg
and at last we have :
http://up.iranblog.com/images/geotowiaxdya2s6ewxk.jpg

also , i know that acceleration is a contravariant vector :
http://up.iranblog.com/images/fl7eosq4cieeoc1kroy.gif
but when i use that definition i don't derive the above result for acceleration !?
and reach to a different results
so what is my mistake ?
 
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world line said:
so what is my mistake ?

It is very difficult to find your mistake without seeing what you have done.
 
for example :
http://up.iranblog.com/images/90cm6lacnzsoxcju1v9.png
that it is different with the corresponding spherical component

if we use the covaraint unit vector the problem will be solved !
and both solution will have unique result
is it correct ?
 
world line said:
http://up.iranblog.com/images/fl7eosq4cieeoc1kroy.gif

This expression relates the components of a vector with respect to one coordinate basis to the components of the same vector with respect to a different coordinate basis. In general, coordinate basis vectors are not unit vectors. Standard Cartesian coordinate vectors are unit vectors, but spherical coordinate vectors are unit vectors.

world line said:
if we use the covaraint unit vector the problem will be solved !
and both solution will have unique result
is it correct ?

http://up.iranblog.com/images/geotowiaxdya2s6ewxk.jpg

is with respect to contravariant unit vectors.
 
Thank you
 

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