anonymous
  • anonymous
Prove the following vector identities letting: V = u i + v j + w k and ∇= ∂╱∂x i + ∂╱∂y j + ∂╱∂z k that: ∇∙(fV̅) = f (∇∙V̅) + V̅ ∙∇f
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
What is fV ?
anonymous
  • anonymous
f is a scalar
anonymous
  • anonymous
v is the vector

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anonymous
  • anonymous
I am aware. I meant write down what it actually is.
anonymous
  • anonymous
Explicitly
anonymous
  • anonymous
f (u)i + f(v) j + f(w)k
anonymous
  • anonymous
f is multiply u, v, and w, right?
anonymous
  • anonymous
yes
anonymous
  • anonymous
So what's the divergence of that?
anonymous
  • anonymous
∂╱∂x i + ∂╱∂y j + ∂╱∂z k
anonymous
  • anonymous
No, what is the divergence of the expression you wrote for fV ?
anonymous
  • anonymous
i'm not quite following what your asking me
anonymous
  • anonymous
\[f\vec{V} = fu\vec{i} + fv\vec{j} + fw\vec{k} \] what is \[\vec{\nabla} \cdot (f\vec{V}) \] ?
anonymous
  • anonymous
d f(u)/dx i + d f(v)/dy j + d f(w)/dz k
anonymous
  • anonymous
the i, j, and k are gone.
anonymous
  • anonymous
Okay, so expand that out.
anonymous
  • anonymous
so just d f(u)/dx + d f(v)/dy + d f(w)/dz
anonymous
  • anonymous
That's right. Now show that that equals the identity that you were given above.
anonymous
  • anonymous
well the right hand side of the equation is where the problems start
anonymous
  • anonymous
\[ \frac{\partial (fu)}{\partial x} = \frac{ \partial f}{\partial x} u + f\frac{\partial u}{\partial x}\] etc....
anonymous
  • anonymous
those are equal what u just wrote above?
anonymous
  • anonymous
That's only the first term...
anonymous
  • anonymous
oh i think i just clicked what ur saying
anonymous
  • anonymous
okay..so the entire left side is now expanded
anonymous
  • anonymous
okay....i think i see where this is going now
anonymous
  • anonymous
if i expand the right side it's gonna come out the same way huh?
anonymous
  • anonymous
Yeah but it would probably be better if you just grouped the terms on the left. \[f \frac{\partial u}{\partial x} + f\frac{\partial v}{\partial y} + f\frac{\partial w}{\partial z} = f(\vec{\nabla} \cdot \vec{V})\] and so forth..
anonymous
  • anonymous
very good
anonymous
  • anonymous
thank you soo soo much

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