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I tried to prove
div(F x G) = G.(curlF) - F.curl(G)
and ended up getting the right hand side equaling twice the left hand side, with no idea where i'd gone wrong :(
can someone show me how to prove it correctly?
and, if you have time.. see if you can pick where i went wrong?
this is how I attempted to do it:
(I'm going to stop with the bolding for vectors because it's too annoying)
F x G = (f2g3 - f3g2)i + (f3g1 - f1g3)j + (f1g2 - f2g1)k
div(F x G) = d/dx(f2g3 - f3g2) + d/dy(f3g1 - f1g3) + d/dz(f1g2 - f2g1)
curlF = (df3/dy - df2/dz)i + (df1/dz - df3/dx)j + (df2/dx - df1/dy)k
G.(curlF) = d/dx(f2g3 - f3g2) + d/dy(f3g1 - g3f1) + d/dz(f1g2 - f2g1)
i can see I'm already in trouble here... this is the left hand side already...
then I used the same method for F.(curlG) which gets me the negative of G.(curlF) , so that when I take it from G.(curlF) I get twice the left hand side...
Any help will be greatly appreciated
div(F x G) = G.(curlF) - F.curl(G)
and ended up getting the right hand side equaling twice the left hand side, with no idea where i'd gone wrong :(
can someone show me how to prove it correctly?
and, if you have time.. see if you can pick where i went wrong?
this is how I attempted to do it:
(I'm going to stop with the bolding for vectors because it's too annoying)
F x G = (f2g3 - f3g2)i + (f3g1 - f1g3)j + (f1g2 - f2g1)k
div(F x G) = d/dx(f2g3 - f3g2) + d/dy(f3g1 - f1g3) + d/dz(f1g2 - f2g1)
curlF = (df3/dy - df2/dz)i + (df1/dz - df3/dx)j + (df2/dx - df1/dy)k
G.(curlF) = d/dx(f2g3 - f3g2) + d/dy(f3g1 - g3f1) + d/dz(f1g2 - f2g1)
i can see I'm already in trouble here... this is the left hand side already...
then I used the same method for F.(curlG) which gets me the negative of G.(curlF) , so that when I take it from G.(curlF) I get twice the left hand side...
Any help will be greatly appreciated