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Vector Analysis: Help needed

  1. Jul 3, 2012 #1
    Dear all,

    I have two vector fields [itex] \vec{B}[/itex] and [itex]\vec{A}[/itex] related by:

    [itex] \vec{B}=\nabla \times \vec{A}[/itex]

    How can I simplify the following term:

    [itex]\frac{\partial }{\partial \vec{A}} B^{2}[/itex]

    where [itex]\frac{\partial }{\partial \vec{A}}=(\frac{\partial }{\partial A_{x}} \frac{\partial }{\partial A_{y}} \frac{\partial }{\partial A_{z}} )[/itex]

    I would also like to know what are this kind of derivatives ( derivatives with respect to a vector field) called.

    Thanks.
     
  2. jcsd
  3. Jul 4, 2012 #2

    chiro

    User Avatar
    Science Advisor

    Hey Hassan2.

    Try expanding out the cross product of del and A first.

    Also when you say the vector derivative, are the elements of each vector mapped to the same corresponding element in the other? In other words if A = [x0,y0,z0] and B = [x1,y1,z1] then is x0 = f(x1), y0 = g(y1) and z0 = h(z1) (and the components are completely orthogonal)?

    If this is the case, you will be able to expand del X A using the determinant formulation and simplify terms depending on how you define your elements of your vector (even if they are more general than above).
     
  4. Jul 4, 2012 #3
    The elements of the vectors are NOT mapped correspondingly. In fact the first equation is the definition of B, thus, the components are intertwined.

    I couldn't simplify it by expanding the curl.It results in partial derivatives of second order multiplied by partial derivatives of first order.

    Thanks.
     
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