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Thread: div, grad, curl

  1. #1
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    div, grad, curl

    I have the following query:

    I need to verfiy:

    curl(AxB) = (B.$\displaystyle \nabla$)A - B(divA) - (A.$\displaystyle \nabla$)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 $\displaystyle \nabla$ in them.

    Please could someone explain to me how to calculate these.

    Thanks.
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  2. #2
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    Quote Originally Posted by jedoob
    I have the following query:

    I need to verfiy:

    curl(AxB) = (B.$\displaystyle \nabla$)A - B(divA) - (A.$\displaystyle \nabla$)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 $\displaystyle \nabla$ in them.

    Please could someone explain to me how to calculate these.

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

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

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

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

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

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

    -Dan
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