- #1
QuanticEnigma
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Homework Statement
1. Establish the vector identity
[tex]
\nabla . (B[/tex] x [tex]A) = (\nabla[/tex] x [tex]A).B - A.(\nabla[/tex] x [tex]B)
[/tex]
2. Calculate the partial derivative with respect to [tex]x_{k}[/tex] of the quadratic form [tex]A_{rs}x_{r}x_{s}[/tex] with the [tex]A_{rs}[/tex] all constant, i.e. calculate [tex]A_{rs}x_{r}x_{s,k}[/tex]
Homework Equations
The Attempt at a Solution
1.
[tex]
\nabla . (B[/tex] x [tex]A) = \epsilon_{ijk}A_{j}B_{k,i}
[/tex]
Now I don't know what to do next.
2.
[tex]
A_{rs}x_{r}x_{s,k} = A_{rs}\partial_{k}(x_{r}x_{s}) = A_{rs}(x_{r}\partial_k x_{s} + x_{s}\partial_k x_{r})[/tex]
I have no idea if this is right or not.
I'm pretty good at proving vector identities (and Cartesian tensor notation in general), but I get lost when partial derivatives/nablas are involved. Any tips would be greatly appreciated!