Contravariant components and spherical component of acceleration

AI Thread Summary
The discussion focuses on deriving the components of acceleration in spherical coordinates and the confusion surrounding contravariant vectors. The original poster successfully derives acceleration components but encounters discrepancies when applying the contravariant vector definition. A suggestion is made to utilize covariant unit vectors to resolve the inconsistencies, leading to unique results. The conversation emphasizes the difference between coordinate basis vectors in Cartesian and spherical systems, noting that spherical coordinate vectors are not unit vectors. The importance of correctly applying vector definitions in different coordinate systems is highlighted.
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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