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B An equation from terms of operator del to terms of sums

  1. Aug 20, 2016 #1
    https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9
    https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials
    How to write this formula in terms of sums and vector components?

    What is ##v\cdot\nabla## ? I think it is some spatial derivative of speed, but since speed is not a field, it can not be that.

    I think rest of the equation is ##F_x=q\cdot(-\frac{\partial\phi}{\partial x}-\frac{\partial A_x}{\partial t}+\sum(v_i \cdot \frac{\partial A_i}{\partial x})-???)## .
    Is it correct?
     
  2. jcsd
  3. Aug 20, 2016 #2

    pasmith

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    Homework Helper

    The Lorentz force is the electromagnetic force on an object. That object has a position and a velocity, both of which are vector-valued functions of time.

    [tex]((\mathbf{v} \cdot \nabla)\mathbf{A})_x = \sum_{j} v_j \frac{\partial A_x}{\partial x_j}[/tex]
     
  4. Aug 20, 2016 #3
    So the equation on Wikipedia page is
    ##F_x=q\cdot(-\frac{\partial\phi}{\partial x}-\frac{\partial A_x}{\partial t}+\sum_i(v_i \cdot (\frac{\partial A_i}{\partial x}-\frac{\partial A_x}{\partial x_i})))## ?

    Is it so?
     
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