What is the vector cross product in an oblique coordinate system?

[KleZMeR]

Homework Statement

Find vector product of [itex]C = A \times B [/itex] of two vectors in oblique coord. system. Give explicit expressions of components of C in covariant and contravariant components (constructing reciprocal basis from direct basis will be useful).

Homework Equations

I am basically just crossing two vectors, one product is that of two arbitrary contravariant vectors, and one is a product of two arbitrary covariant vectors. I understand this part, I just am always confused by the word "explicit."

The Attempt at a Solution

[/B]
I take this determinant (contravariant)
[itex] \begin{array}{ccc} a_1 & a_2 & a_3 \\ A^1 & A^2 & A^3 \\ B^1 & B^2 & B^3 \end{array} [/itex]

and this one as well (covariant)
[itex] \begin{array}{ccc} a^1 & a^2 & a^3 \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{array} [/itex]

and for my covariant components I can set [itex] a_i = \frac{e_i}{sin(\alpha)} [/itex]

I am not sure what else is being asked, if anything at all.

I have this relation for components:

[itex] x_1 = x^1 + x^2cos(\alpha) [/itex]

but not sure if I should apply it to get vector terms as [itex]a_1A_1 +...[/itex]:

 

[jedishrfu]

I found this via Google and thought it may help you resolve your problem:

http://www.mc.maricopa.edu/~kev2077170/i256/Nonortho_math.pdf

I couldn't find any other reference on vector cross products in oblique 3D coordinates except for this book reference:

http://books.google.com/books?id=gJ...v=onepage&q=oblique coordinate system&f=false