I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems.(adsbygoogle = window.adsbygoogle || []).push({});

The contravariant transformation formula is:

V^{j}= (∂y^{j}/∂x^{i}) * V^{i}

where V^{j}is in the y- frame of reference and V^{i}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:

V^{i}= <x,y>

x^{1}= x

x^{2}= y (These were the coordinate axes in the x-frame)

y^{1}= r

y^{2}=θ (These were the axes in the y-frame)

r(x,y) = sqrt(x^{2}+ y^{2})

θ(x,y)= tan^{-1}(y/x)

(∂r/∂x) = x/sqrt(x^{2}+ y^{2})

(∂r/∂y) = y/sqrt(x^{2}+ y^{2})

(∂θ/∂x) = -y/(x^{2}+ y^{2})

(∂θ/∂x) = x/(x^{2}+ y^{2})

Applying all of this information into the contravariant transformation formula, I get:

V^{1}(in the y-frame) = [V^{1}_{(x-frame)}* (∂y^{1}/∂x^{1})] + [V^{2}_{(x-frame)}* (∂y^{1}/∂x^{2})] =[ x * (∂r/∂x) ] + [y * (∂r/∂y)]

= (x^{2}+ y^{2}) / sqrt(x^{2}+ y^{2}) = r^{2}/r = r

V^{2}(in the y-frame) = [V^{1}_{(x-frame)}* (∂y^{2}/∂x^{1})] + [V^{2}_{(x-frame)}* (∂y^{2}/∂x^{2})] =[ x * (∂θ/∂x) ] + [y * (∂θ/∂y)]

= -xy/r^{2}+ xy/r^{2}= 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?

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# Contravariant transformation between coordinate systems

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