Line integral of a vector field

[realcomfy]

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.

 

[AvroArrow]

Yes, you are correct. The transformation matrix is orthogonal, which means that its inverse is the transpose. So, you can divide by the transformation matrix to obtain the transformation from Cartesian to cylindrical. Once you have done that, you can write your Cartesian vector in terms of the cylindrical coordinates. For example, if you have the vector \overline{A}=x^2\hat{i}+xy^2\hat{j}, then its cylindrical components would be:\overline{A}=r^2\cos^2(\theta)\hat{\rho} + r^2\sin(\theta)\cos(\theta)y^2\hat{\phi} + r\sin^2(\theta)y^2\hat{z}