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Symplify the expression.

  1. Jun 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Simplify the following two expressions:

    [tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}[/tex]

    [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}[/tex]


    3. The attempt at a solution

    for the first one: [tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}[/tex]

    [tex]\frac{\partial}{\partial z}z = 1[/tex]

    so therefore [tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}= y(1)\frac{\partial}{\partial x}= y\frac{\partial}{\partial x}[/tex]

    How come this is incorrect?




    For the second one: [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}[/tex]

    I cannot write that [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}= x\frac{\partial}{\partial z}z\frac{\partial}{\partial y}[/tex]

    so therefore [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}= zx\frac{\partial^2}{\partial y\partial z}[/tex]

    The second one I believe is correct though..
     
  2. jcsd
  3. Jun 8, 2009 #2

    CompuChip

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    You need to be a little more clear on your notation. Suppose that f is some arbitrary function that we can let the expressions act on.
    Then by
    [tex]
    \left( y\frac{\partial}{\partial z}z\frac{\partial}{\partial x} \right) f
    [/tex]
    do you mean
    [tex]
    y\frac{\partial}{\partial z}\left( z \frac{\partial f}{\partial x} \right)
    [/tex]
    or, as you imply in your first post,
    [tex]
    y\left( \frac{\partial}{\partial z}z\right)\frac{\partial f}{\partial x}
    [/tex]
    or yet something else?
     
  4. Jun 8, 2009 #3

    Dick

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    Depends on where you put the parentheses. If the first one means y*d/dz(z*d/dx) you need to use a product rule on the z*d/dx product. To make it clearer write it as y*d/dz(z*df/dx) where f is a test function.
     
  5. Jun 8, 2009 #4
    I don't believe there are any specific parentheses. The exact question is to find the following commutator [tex][Lx,Ly][/tex]

    where:

    [tex]Lx= (y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y})[/tex]

    [tex]Ly= (z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z})[/tex]
     
  6. Jun 8, 2009 #5

    Dick

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    In that case you should treat it as
    [tex]

    y\frac{\partial}{\partial z}\left( z \frac{\partial f}{\partial x} \right)

    [/tex]
     
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