Cross product of polar coordinates

[tiagobt]

When using cartesian coordinates, I use the following expressions to calculate the cross product of the basis vectors:

$$i \times j = k$$
$$j \times k = i$$
$$k \times i = j$$
$$j \times i = -k$$
$$k \times j = -i$$
$$i \times k = -j$$

Can I do the same in polar coordinates? How could I write the cross product for the vectors $$r$$, $$\theta$$ and $$z$$?

 

[arildno]

Yep, the right-hand version is $$\vec{i}_{r}\times\vec{i}_{\theta}=\vec{k}$$
and you can complete the cycle from there..

 

[dextercioby]

The cylidrical coordinates are orthogonal,which means that the basis vectors are orthogonal to each other,too.They can be made to form a rectangular trihedron,just like $\vec{i},\vec{j} \ \mbox{and} \ \vec{k}$.

Daniel.