Cross Product Polar Coor

tekness

I would like to know how to perform a cross product on polar coordinates.

Thank You

tiny-tim

Hi tekness!

Can you give us an example of two vectors you're trying to cross-product?

tekness

Hi tim,

I am just looking for a general way to perform the operation. I will perform a cross product between E and H fields that are in polar coordinates. I don't want to go through the hassle of converting back and forth :).

I hope this explains it, if not please let me know what else I can add.

tiny-tim

Well, so long as the vectors are expressed in terms of perpendicular unit vectors such as ihat and jhat or rhat and thetahat, you just cross-product them the usual way.

The only problem might be converting into unit vectors.

tekness

so for example.
I have |i j k|
|rcos() rsin() Z1|
|r2cos()2 r2sin()2 Z2|

the 2 is for a different value/angle.
So just perform the same cross product operation as rectangular coordinates would require?

tiny-tim

Hi tekness!

(have a theta: θ )

I'm a little confused … those look like vectors from the origin.

You will generally want to cross-product the fields at a general point.

tekness

I will try to verify exactly what I need and respond back. Looks like I need to rethink my question.
Thank you for your help! I will be back asap.

Ben Niehoff

The confusion is that the cross product is an operation in the tangent space, not in the coordinate space. At a particular point, your field has components in the r-hat, phi-hat, and theta-hat directions. These three vectors constitute an orthonormal basis. So you simply take the cross product without any modification at all. For example,

[tex]\hat r \times \hat \theta = \hat \phi[/tex]

and the rest are similar.