Understand Contravariant Transformations b/w Coordinate Systems

[space-time]

I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems.

The contravariant transformation formula is:

V^j = (∂y^j/∂xⁱ) * Vⁱ

where V^j is in the y- frame of reference and Vⁱ is in the x-frame of reference. Einstein summation convention is used here.

As an example of applying this formula, I tried to convert a Cartesian vector to a polar coordinates vector to see what the output would be. Here is how that process went:

Vⁱ = <x,y>
x¹ = x
x² = y (These were the coordinate axes in the x-frame)

y¹ = r
y² =θ (These were the axes in the y-frame)

r(x,y) = sqrt(x² + y²)
θ(x,y)= tan^-1(y/x)
(∂r/∂x) = x/sqrt(x² + y²)
(∂r/∂y) = y/sqrt(x² + y²)
(∂θ/∂x) = -y/(x² + y²)
(∂θ/∂x) = x/(x² + y²)Applying all of this information into the contravariant transformation formula, I get:

V¹ (in the y-frame) = [V¹_(x-frame) * (∂y¹/∂x¹)] + [V²_(x-frame) * (∂y¹/∂x²)] =[ x * (∂r/∂x) ] + [y * (∂r/∂y)]

= (x² + y²) / sqrt(x² + y²) = r²/r = r

V² (in the y-frame) = [V¹_(x-frame) * (∂y²/∂x¹)] + [V²_(x-frame) * (∂y²/∂x²)] =[ x * (∂θ/∂x) ] + [y * (∂θ/∂y)]

= -xy/r² + xy/r² = 0

In short, my contravariant transformation from Cartesian to polar coordinates turned the vector <x,y> to <r,0>.

Is this the correct result. Am I appropriately and correctly applying the contravariant transformation formula or do you use this transformation in some different way?

 

[Orodruin]

Yes, you have just taken the position vector and it should be a familiar result that this vector is $r \vec e_r$ in polar coordinates in the plane.

A suggestion regarding notation, the more standard notation would be to use a prime to denote the components in one of the systems. It is also easier to keep track of things if you use the variable names instead of numbers to label the different components, i.e., $V^r$, $V^\theta$, etc.