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Questions concerning the del operator.

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  • Fred
    Hey, I was looking for certain vector identities involving the del operator, gradient, curl and divergence when I saw the identity of del operator X ( A X B )
    Message 1 of 4 , Oct 29, 2006
      Hey,

      I was looking for certain vector identities involving the del operator, gradient, curl and divergence when I saw the identity of del operator X ( A X B ) or in other words, the curl of A cross B. http://mathworld.wolfram.com/VectorDerivative.html
      If you look at Equation number 3 from the following link you get a term, on the right hand side of the equations, that contains A dot del. I understand that del dot A is the divergence but I am not familiar with how A dot del is defined. I also came accross questions containing terms like A cross del as opposed to del cross A(which is the curl). if anybody could tell me what A dot del and A cross del it would be greatly appreciated.

      Fred



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    • Francois Rideau
      I suppose (A.del) is a differential operator acting on the vector field B. And this differential operator is the scalar product of the vector field A with the
      Message 2 of 4 , Oct 30, 2006
        I suppose (A.del) is a differential operator acting on the vector field B.
        And this differential operator is the scalar product of the vector field A
        with the del operator
        If A has components (A1, A2, A3) and del components (d1, d2, d3)
        Then A.del is the operator : A1 . d1 + A2 . d2 + A3 . d3
        Here d1 is the partial derivative wrt the first variable ( usually called x)
        and similarly for d2 and d3.
        So the operator A.del acts on a function f by:
        (A.del)f = A1.d1(f) + A2.d2(f) + A3.d3(f)
        So the symbol (A.del)B is the vector field with components:
        ( (A.del)B1, (A.del)B2, (A.del)B3)
        where (B1, B2, B3) are the components of the vector field B.
        Friendly
        Francois

        On 10/30/06, Fred <fkbd2002@...> wrote:
        >
        > Hey,
        >
        > I was looking for certain vector identities involving the del operator,
        > gradient, curl and divergence when I saw the identity of del operator X ( A
        > X B ) or in other words, the curl of A cross B.
        > http://mathworld.wolfram.com/VectorDerivative.html
        > If you look at Equation number 3 from the following link you get a term,
        > on the right hand side of the equations, that contains A dot del. I
        > understand that del dot A is the divergence but I am not familiar with how A
        > dot del is defined. I also came accross questions containing terms like A
        > cross del as opposed to del cross A(which is the curl). if anybody could
        > tell me what A dot del and A cross del it would be greatly appreciated.
        >
        > Fred
        >
        >
        > Send instant messages to your online friends http://uk.messenger.yahoo.com
        >
        > [Non-text portions of this message have been removed]
        >
        >
        >


        [Non-text portions of this message have been removed]
      • Francois Rideau
        I see I forget the (A x del) operator. I think that is the operator with components: (A2 . d3 - A3 . d2, A3 . d1 - A1 . d3, A1 . d2 - A2 . d1) with the same
        Message 3 of 4 , Oct 30, 2006
          I see I forget the (A x del) operator.
          I think that is the operator with components:
          (A2 . d3 - A3 . d2, A3 . d1 - A1 . d3, A1 . d2 - A2 . d1)
          with the same notations of my previous post.
          So acting on a function f, the symbol (A x del)f is the vector field with
          components:
          (A2 . d3(f) - A3 . d2(f), A3 . d1(f) - A1 . d3(f), A1 . d2(f) - A2 . d1(f))
          Friendly
          Fran├žois
          All these formulas are only mnemonic!
          I think to know the basic rules of differential calculus is far better!


          [Non-text portions of this message have been removed]
        • Fredrik
          Thanks, That helped a lot. I broke it into component form and it checks out. Fred ... with ... d1(f))
          Message 4 of 4 , Oct 31, 2006
            Thanks,
            That helped a lot. I broke it into component form and it checks out.

            Fred
            --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
            <francois.rideau@...> wrote:
            >
            > I see I forget the (A x del) operator.
            > I think that is the operator with components:
            > (A2 . d3 - A3 . d2, A3 . d1 - A1 . d3, A1 . d2 - A2 . d1)
            > with the same notations of my previous post.
            > So acting on a function f, the symbol (A x del)f is the vector field
            with
            > components:
            > (A2 . d3(f) - A3 . d2(f), A3 . d1(f) - A1 . d3(f), A1 . d2(f) - A2 .
            d1(f))
            > Friendly
            > Fran├žois
            > All these formulas are only mnemonic!
            > I think to know the basic rules of differential calculus is far better!
            >
            >
            > [Non-text portions of this message have been removed]
            >
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