Hey everyone!
The problem:
If f is a scalar-valued function and F is a vector field (both over R2),
then ∇∙(fF)=∇f∙F + f(∇∙F)
I'm preparing for a midterm and having some issues with this final problem, which I think ties together a lot of stuff and is really important. I think I should just look at this and know the answer, but I can't figure it out. If you could even just point out what concepts are involved, I would be super grateful! Thanks.
The problem:
If f is a scalar-valued function and F is a vector field (both over R2),
then ∇∙(fF)=∇f∙F + f(∇∙F)
I'm preparing for a midterm and having some issues with this final problem, which I think ties together a lot of stuff and is really important. I think I should just look at this and know the answer, but I can't figure it out. If you could even just point out what concepts are involved, I would be super grateful! Thanks.
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del(fF) = d/dx(fF) i +d/dy(fF) j + d/dx(fF) k
d/dx(fF) = f(d/dx F) + F(d/dx f), by product rule
same goes for d/dy(fF) and d/dz(fF), so
del(fF) = (f(dF/dx) + F(df/dx))) i + (f(dF/dy) + F(df/dy))) j + (f(dF/dz) + F(df/dz))) k
= del(f) * F + del(F) * f
d/dx(fF) = f(d/dx F) + F(d/dx f), by product rule
same goes for d/dy(fF) and d/dz(fF), so
del(fF) = (f(dF/dx) + F(df/dx))) i + (f(dF/dy) + F(df/dy))) j + (f(dF/dz) + F(df/dz))) k
= del(f) * F + del(F) * f