1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Gradient of (1/r)

  1. Sep 11, 2014 #1
    1. The problem statement, all variables and given/known data

    gradient(1/r) = r(hat) / r^2

    2. Relevant equations
    r = (x-x')i + (y-y')j + (z-z')k
  2. jcsd
  3. Sep 11, 2014 #2


    User Avatar
    Gold Member

    gradient(1/r) = -r(hat) / r^2
  4. Sep 11, 2014 #3
    how do you prove that?
  5. Sep 11, 2014 #4


    User Avatar
    Gold Member

    In spherical coordinates, the gradient of a scalar function f is:
    [itex]\vec\nabla f(r, \theta, \phi) = \frac{\partial f}{\partial r}\hat r+ \frac{1}{r}\frac{\partial f}{\partial \theta}\hat\theta+ \frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi}\hat\phi [/itex].
    And we have [itex] \frac{d}{dr}\frac 1 r=-\frac{1}{r^2} [/itex].
  6. Sep 11, 2014 #5
    can this be done in cartesian coordinates?
  7. Sep 11, 2014 #6


    User Avatar
    Science Advisor

    Or, in Cartesian coordinates,
    [tex]\frac{1}{r}= \frac{1}{x^2+ y^2+ z^2}= (x^2+ y^2+ z^2)^{-1/2}[/tex]

    [tex]\left(\frac{1}{r}\right)_x= -\frac{1}{2}(x^2+ y^2+ z^2)^{-3/2}(2x)= -\frac{rcos(\theta)sin(\phi)}{r^3}= -\frac{1}{r^3} (rcos(\theta)sin(\phi))[/tex]

    [tex]\left(\frac{1}{r}\right)_y= -\frac{1}{2}(x^2+ y^2+ z^2)^{-3/2}(2y)= -\frac{rsin(\theta)sin(\phi)}{r^3}= -\frac{1}{r^3} (rsin(\theta)sin(\phi)([/tex]

    [tex]\left(\frac{1}{r}\right)_z= -\frac{1}{2}(x^2+ y^2+ z^2)^{-3/2}(2z)= -\frac{rcos(\phi)}{r^3}= -\frac{1}{r^3} (rcos(\phi))[/tex]

    So that [tex]\nabla \frac{1}{r}= -\frac{1}{r^3}\vec{r}= -\frac{1}{r^2}\frac{\vec{r}}{r}= -\frac{1}{r^2}\hat{r}[/tex]

    Where [itex]\hat{r}[/itex] is the unit vector in the direction of [itex]\vec{r}[/itex].
    Last edited by a moderator: Sep 11, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted