Divergence and Curl of Unit Vectors?

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The discussion revolves around understanding the divergence and curl of unit vectors, particularly in Cartesian and polar coordinates. The initial confusion stems from the belief that the divergence and curl of unit vectors like x_hat would yield zero. However, when considering r_hat in polar coordinates, it becomes clear that it can have non-trivial divergence and curl when expressed in Cartesian coordinates. The conversation also touches on the challenge of converting angular coordinates like theta and phi into cylindrical coordinates. Ultimately, the thread highlights the complexities involved in vector calculus and coordinate transformations.
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Homework Statement


http://img4.imageshack.us/img4/4218/divergenceandcurl.jpg

The Attempt at a Solution


Totally confused on what the question's asking. Wouldn't the divergence of say x_hat be the partial of x_hat over x which is just 0? So every answer would just be 0 or something? Same thing goes with the curl? Thanks
 
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In cartesian coordinates yes,

But, for example, the r_hat in polar coordinates, written in cartesian coordinates, is
r_hat = (x/sqrt(x^2 +y^2), y/sqrt(x^2 + y^2), 0)

which perhaps has a non-trivial divergence and curl.
 
oh i get it

edit: wait, not really. How can I make theta or phi into into something like r and z in cylindrical?
 
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