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anonymous

  • 4 years ago

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

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  1. anonymous
    • 4 years ago
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    What is fV ?

  2. anonymous
    • 4 years ago
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    f is a scalar

  3. anonymous
    • 4 years ago
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    v is the vector

  4. anonymous
    • 4 years ago
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    I am aware. I meant write down what it actually is.

  5. anonymous
    • 4 years ago
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    Explicitly

  6. anonymous
    • 4 years ago
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    f (u)i + f(v) j + f(w)k

  7. anonymous
    • 4 years ago
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    f is multiply u, v, and w, right?

  8. anonymous
    • 4 years ago
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    yes

  9. anonymous
    • 4 years ago
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    So what's the divergence of that?

  10. anonymous
    • 4 years ago
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    ∂╱∂x i + ∂╱∂y j + ∂╱∂z k

  11. anonymous
    • 4 years ago
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    No, what is the divergence of the expression you wrote for fV ?

  12. anonymous
    • 4 years ago
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    i'm not quite following what your asking me

  13. anonymous
    • 4 years ago
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    \[f\vec{V} = fu\vec{i} + fv\vec{j} + fw\vec{k} \] what is \[\vec{\nabla} \cdot (f\vec{V}) \] ?

  14. anonymous
    • 4 years ago
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    d f(u)/dx i + d f(v)/dy j + d f(w)/dz k

  15. anonymous
    • 4 years ago
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    the i, j, and k are gone.

  16. anonymous
    • 4 years ago
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    Okay, so expand that out.

  17. anonymous
    • 4 years ago
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    so just d f(u)/dx + d f(v)/dy + d f(w)/dz

  18. anonymous
    • 4 years ago
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    That's right. Now show that that equals the identity that you were given above.

  19. anonymous
    • 4 years ago
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    well the right hand side of the equation is where the problems start

  20. anonymous
    • 4 years ago
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    \[ \frac{\partial (fu)}{\partial x} = \frac{ \partial f}{\partial x} u + f\frac{\partial u}{\partial x}\] etc....

  21. anonymous
    • 4 years ago
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    those are equal what u just wrote above?

  22. anonymous
    • 4 years ago
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    That's only the first term...

  23. anonymous
    • 4 years ago
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    oh i think i just clicked what ur saying

  24. anonymous
    • 4 years ago
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    okay..so the entire left side is now expanded

  25. anonymous
    • 4 years ago
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    okay....i think i see where this is going now

  26. anonymous
    • 4 years ago
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    if i expand the right side it's gonna come out the same way huh?

  27. anonymous
    • 4 years ago
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    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..

  28. anonymous
    • 4 years ago
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    very good

  29. anonymous
    • 4 years ago
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    thank you soo soo much

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