Cross product of unit vectors in cylindrical coordinates

Question

What are the cross products of the units vectors of the cylindrical coordinates $\hat{s}$,$\hat{\rho}$, and $\hat{\phi}$? I know the very familiar relationships for the Cartesian unit vectors, but I can't find the one for cylindrical polar coordinates. Thanks.

Answer 1

Any orthonormal set of vectors ${u_1, u_2, u_3}$ will satisfy

$u_1\times u_2 = u_3$ or $-u_3$, and

$u_2\times u_3 = u_1$ or $-u_1$ and

$u_1\times u_3 = u_3$ or $-u_3$.

Your coordinates in this case form an orthonormal basis, and they are right handed, so they will obey the rules above with all positive signs. Alternatively, you could just right out each of the vectors in cyclindrical coordinates and use the cross products of the standard basis to arrive at an answer.