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Vector field operators

  1. Jan 7, 2009 #1
    Hello. I am stuck trying to find an understandable answer to this online:

    Carry out the following operations on the vector field A reducing the results to their simplest forms:

    a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k)
    b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k)

    I know this is a dot product and cross product thing, and I think that A should become "integral Anda" (div) but I'm not sure what to do with the curl for b.

    I can't find this worked out online. Can anyone direct me to a source?

    Thank you!
  2. jcsd
  3. Jan 7, 2009 #2
    The best to do when you get problems as this one, is to use the abstract index notation. your first expression is the divergence of a vectorfield, and the second one is the curl, of the same field. So I will show the problem, for an arbitrary [tex]\vec v [/tex] vector field, multiplied by some A scalar field. I will denote the differential operator as: [tex]\frac{\partial}{\partial x_j} \equiv \partial_j[/tex]. And we use the einstein summation convention. That is we sum automaticaly on double indices. So:

    [tex]\text{div}(A\vec{v}=\partial_k(Av_k)=v_k\partial_k A + A\partial_k v_k =(\vec{v}\nabla)A + A \text{div}\vec{v}[/tex]
    As we see the first part is the "substantive" part of the total time derivative, we use in hydrodynamics. The operator explicitly in cartesian coordinates:
    [tex]\vec{v}\nabla = v_x\frac{\partial}{\partial x}+v_y\frac{\partial}{\partial y}+v_z\frac{\partial}{\partial z}[/tex]
    Now in your problem: [tex]\vec{v}=(x,y,x)[/tex]. The A scalar field is not given explicitly. So we have for the divergence according to the previous:
    [tex]\text{div}(A\vec{v})=x\frac{\partial A}{\partial x}+y\frac{\partial A}{\partial y}+x\frac{\partial A}{\partial z}+A\left(\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial x}{\partial z}\right)=x\frac{\partial A}{\partial x}+y\frac{\partial A}{\partial y}+x\frac{\partial A}{\partial z}+2A[/tex]

    For the second part we have to calculate the curl. Using the same convention as above:

    [tex](\text{curl}(A\vec{v}))_k=\epsilon_{klm}\partial_lAv_m=A\epsilon_{klm}\partial_l v_m + \epsilon_{klm}(\partial_lA)v_m=[/tex]


    Where [tex]\epsilon_{klm}[/tex] is the three indice totaly antisymmetric tensor, the levi civita tensor.

    So now using the given field:


    And we are done.
  4. Jan 8, 2009 #3
    Thank you very much Thaakisfox!

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