Line integral of a vector field

I am attempting to calculate the line integral of the vector field [tex]\overline{A}= x^{2} \hat{i} + x y^{2} \hat{j}[/tex] around a circle of radius R ([tex]x^{2} + y^{2} = R^{2}[/tex]) using cylindrical coordinates.

It is simple enough to convert the x and y components to their cylindrical counterparts, but I am unsure what to do about the unit vectors. Since the line integral of a vector field contains a dot product between the vector field and the differential element, this requires the two to have the same unit vectors, right?

I found the conversion matrix for cylindrical to cartesian (http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates) but i'm not totally sure how to convert the other way around (from cartesian to cylindrical). Wikipedia says that the transform matrix is orthogonal, so does this mean I can simply divide it to the other side and end up with the transpose? Then write my cartesian vectors in terms of the cylindrical?

Thanks for your help.
 

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