I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems. The contravariant transformation formula is: Vj = (∂yj/∂xi) * Vi where Vj 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?