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What is the Divergence?

  1. Jul 27, 2008 #1
    What is the Divergence? is it only the Partial derivatives?

    Lets say I have a vector field: [tex]F=x^2i+y^2j+z^2k[/tex], the divergence is [tex]F=2xi+2yj+2zk[/tex]?

    And if it is, than what is the gradient?:confused:
    Last edited by a moderator: Jul 28, 2008
  2. jcsd
  3. Jul 28, 2008 #2


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    Re: Divergence

    A divergence is evaluated of a vector field, while the gradient (assuming you mean grad) is done for scalar fields. A related operation, the curl is performed on a vector field.

    So we have:
    curl: vector field -> vector field
    div: vector field -> scalar field
    grad: scalar field -> vector field

    I'm wondering if there is any defined operation such that we can get a scalar field from a scalar field?
  4. Jul 28, 2008 #3


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    Re: Divergence

    No. the diverence of this vecor field is the scalar function [itex]\nabla\cdot F= 2x+ 2y+ 2z[/itex]. The "[itex]\cdot[/itex]" in that notation is to remind you of a dot product: the result is a scalar.

    The gradient is, in effect, the "opposite" of the divergence: it changes a scalar function to a vector field: at each point [itex]\nabla f[/itex] points in the direction of fastest increase and its length is the derivative in that direction.

    Notice that if you start with a scalar function, the gradient gives a vector function and you can then apply the divergence to that going back to a scalar function:
    [tex]\nabla\cdot (\nabla f)= \nabla^2 f[/itex]
    called the "Laplacian" of f. That is a very important operator: it is the simplest second order differential operator that is "invariant under rigid motions".
  5. Jul 28, 2008 #4
    Re: Divergence

    Got it, thanks.
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