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snowstorm69

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Can anyone explain to me how to expand this expression for curl which I find in the GR book I'm reading (by Hobson, Efstathiou and Lasenby, page 71)? In a section entitled Vector Operators in Component Form they state the curl as a "rank-2 antisymmetric tensor with components":

(curl)ab = DELaVb - DELbVa where a, b are subscripts, V is a vector(?, tensor?), and I used the admittedly poor notation "DEL" to indicate the nabla or del operator used to denote the grad of a scalar usually or the div of a vector.

I tried to write the components of this out with a going from 1 to 3 while keeping b at 1, then a going 1 to 3 with b at 2, and finally a going 1 to 3 with be at 3. In this way I got 9 pairs of terms, 18 terms in all (I don't have it in front of me but visualizing I think that's what I got). And ALL the terms cancel each other out. I believe my mistake is not taking into account the basis vectors and that if I had the terms would not all cancel out. They would yield the "curl" in 3-D.

What I get instead is for example the 1-3/3-1 pair of terms for the curl in 3-D, and then the same thing a bit later in the expansion, to cancel each other out.

Can you help me? I want to work the case out in 3-D so that I can next work it out in higher D. Obviously there are some things I don't understand about the basis vectors or perhaps the meaning of the notation itself.

Thanks.

(curl)ab = DELaVb - DELbVa where a, b are subscripts, V is a vector(?, tensor?), and I used the admittedly poor notation "DEL" to indicate the nabla or del operator used to denote the grad of a scalar usually or the div of a vector.

I tried to write the components of this out with a going from 1 to 3 while keeping b at 1, then a going 1 to 3 with b at 2, and finally a going 1 to 3 with be at 3. In this way I got 9 pairs of terms, 18 terms in all (I don't have it in front of me but visualizing I think that's what I got). And ALL the terms cancel each other out. I believe my mistake is not taking into account the basis vectors and that if I had the terms would not all cancel out. They would yield the "curl" in 3-D.

What I get instead is for example the 1-3/3-1 pair of terms for the curl in 3-D, and then the same thing a bit later in the expansion, to cancel each other out.

Can you help me? I want to work the case out in 3-D so that I can next work it out in higher D. Obviously there are some things I don't understand about the basis vectors or perhaps the meaning of the notation itself.

Thanks.

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