Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Homework Helper

    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


    User Avatar
    Science Advisor

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook