Linearization problem

Apr 2016
1
0
Brazil
Hi
If I have 2 different systems of coordinates: polar and cartesian.
And I want to linearize a function f(x,y) in cartesian coordinate:

f(x,y)= (df/dx)x + (df/dy)y (I)

And in polar coordinate:

f(p,o)= (df/dp)p + (df/do)o (II)

The linearization in (I) is equal to (II)? if is equal, there is a way to proof (II) from (I)?
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
I am going to use "r" and "\(\displaystyle \theta\)" for the polar coordinates instead of "p" and "o".

No, the fact that the linearizations of the same function in Cartesian and in polar coordinates are not the same should be clear from the fact that a function can be linear in one coordinate system but not linear in the other.

f(r, \(\displaystyle \theta\))= r is already linear but \(\displaystyle f(x, y)= \sqrt{x^2+ y^2}\) is not. Similarly, f(x, y)= x is linear but \(\displaystyle f(r, \theta)= r cos(\theta)\) is not.