Outer product of two one forms.

[ronblack2003]

Given two one forms f = (1,1,0,0,) and g=(-1,0,1,0): what are the components of f(x)g ... would appreciate any help.

 

[ronblack2003]

Outer Product of two One Forms

Given two one forms f = (1,1,0,0) and g = (-1,0,1,0):
What would be the Components of f(outer product)g be?
would appreciate any help.

 

[pervect]

I'm not quite sure if the "outer product" is the wedge product or the tensor product. Wikipedia seems to think it's the former. See

This Link

The wedge product of u ^ v is

u (tensor) v - v (tensor) u

The intent here is to anti-symmetrize the tensor product, The formula above works as written only for rank 1 tensors (but that's what you have).

The tensor product p = u (tensor) v in component notation for rank 1 tensors is:

p_ij = u_iv_j

Thus (1,2,3) (tensor) (4,5,6) is the second rank tensor (you can think of it as an array)

1*4 ,1*5, 1*6
2*4, 2*5, 2*6
3*4, 3*5, 3*6

Hope this helps

 

[pmb_phy]

I'm not quite sure if the "outer product" is the wedge product or the tensor product.

Yes. Its the tensor product.

Let "@" be the tensor product and let f = f_u w^v where f_u are components of f and w^u are a coordinate basis 1-forms. Same with g. Then let h = f@g. Then

h = (f_u e^u)@(g_u v e^v) = f_u g_v (w^u@w^v)

Therefore f_u g_v are the components of f@g. w^u@w^v are the covariant basis tensors for the outer product.

Pete