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Math Help - div, grad, curl

  1. #1
    Junior Member
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    div, grad, curl

    I have the following query:

    I need to verfiy:

    curl(AxB) = (B. \nabla)A - B(divA) - (A. \nabla)B + A(divB)

    for A=(1,0,0) and B=(x,y,z).

    I can do the left side and the div terms of the right side but I am unsure of how to go about working out the other two terms with the \nabla in them.

    Please could someone explain to me how to calculate these.

    Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jedoob
    I have the following query:

    I need to verfiy:

    curl(AxB) = (B. \nabla)A - B(divA) - (A. \nabla)B + A(divB)

    for A=(1,0,0) and B=(x,y,z).

    I can do the left side and the div terms of the right side but I am unsure of how to go about working out the other two terms with the \nabla in them.

    Please could someone explain to me how to calculate these.

    Thanks.
    By definition \nabla is a vector. So:
    \nabla \equiv \left ( \partial _x , \, \partial _y , \, \partial _z \right )

    That means:
    A \cdot \nabla = \left ( 1 \partial _x + 0 \partial _y + 0 \partial_z \right ) = \partial _x
    (Recall that the dot product always produces a scalar quantity.)

    and
    B \cdot \nabla = \left ( x \partial _x + y \partial _y + z \partial_z \right )

    Note that the order of these are important: A \cdot \nabla \neq \nabla \cdot A.

    That means that
    \left ( A \cdot \nabla \right ) B = \partial _x (x,y,z) = (1,0,0)

    and
    \left ( B \cdot \nabla \right ) A = \left ( x \partial _x + y \partial _y + z \partial_z \right ) (1,0,0) = 0
    since A is a constant vector.

    -Dan
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