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Vector Analysis Identity derivation

  1. Feb 19, 2008 #1
    1. The problem statement, all variables and given/known data

    derive the identity:
    del((F)^2) = 2 F . del(F) + 2Fx (del x F)

    the dot is a dot product

    2. Relevant equations

    3. The attempt at a solution
    first i set F = <a,b,c>, making F^2 = a^2 + b^2 + c^2
    I took the partial derivatives with respect to x, y, and z (to get the necessary parts for the gradient), which gives me:

    d/dx(a^2 + b^2 + c^2) = 2a * da/dx + 2b*db/dx + 2c*db/dc
    d/dy(...) = ... 2a * da/dy + 2b*db/dy + 2c*db/dy
    d/dz(...) = ... 2a * da/dz + 2b*db/dz + 2c*db/dz

    But, because its a gradient of a scalar, the three components above are those of a vector. After putting together the components im not sure where to go. What actually confuses me is that in the derivation you are supposed to show that a vector equals a vector plus a scalar -- the ( 2F . del(F)) term is a scalar... . If someone could please help me out that would be greatly appreciated. Thank you
  2. jcsd
  3. Feb 19, 2008 #2


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    If F is a vector, then you can't interpret del(F) as a vector. It has to be a rank 2 tensor. It's ij component is del_i(F_j) or del_j(F_i). So the i component of F.del(F) would be F_j*del_i(F_j) (summed over j) or possibly F_j*del_j(F_i) (summed over k). It's kind of ambiguous. What does it mean??
  4. Mar 19, 2009 #3

    how can you do the derivation of d = Da/2b
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