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When the gradient of a vector field is symmetric?

  1. Oct 4, 2011 #1
    1. The problem statement, all variables and given/known data

    "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function"
    Pure Strain Deformations of Surfaces
    Marek L. Szwabowicz
    J Elasticity (2008) 92:255–275
    DOI 10.1007/s10659-008-9161-5

    f=5x^3+3xy-15y^3
    So the gradient of this function is a vector field, right? Now the grad of this grad is a tensor which is symmetric and according to Marek it's always like that.
    Can you guys think of any reason or proof for it?

    2. Relevant equations



    3. The attempt at a solution
    Maybe it has something to do with double differentiation...but I can't figure out why...
     
    Last edited: Oct 4, 2011
  2. jcsd
  3. Oct 4, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    It's based on the "mixed partial derivative property":
    As long as the derivatives are continuous,
    [tex]\frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}[/tex]
     
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