1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Smarter way to solve a continuity equation?

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data
    The density in 3-D space of a certain kind of conserved substance is given by
    [tex]\[\rho (x,y,z, t) = At^{-\frac{3}{2}}e^{-\frac{r^2}{4kt}}\][/tex]

    where [tex]\mathbf r = x\mathbf i + y\mathbf j +z\mathbf k[/tex] and [tex] r = |\mathbf r|[/tex]. The corresponding flux vector is given by
    [tex]\mathbf J(\mathbf r, t) = Bt^{-\frac{5}{2}}e^{-\frac{r^2}{4kt}}\mathbf r[/tex]
    Here, A, B, k and positive constants.

    2. Relevant equations

    Show that [tex]$\rho, \mathbf J$[/tex] satisfy the conservation equation [tex]\frac{\partial \rho}{\partial t}[/tex][tex] + \nabla \cdot \mathbf J = 0[/tex] only if [tex]$ A = 2B$[/tex]

    3. The attempt at a solution
    So I've looked at this, found the derivative for the density function, had a fair play with the div function, i'm just wondering if there is a smarter way to solve this then actually deriving the partial derivative and the div function and re-arranging? I have a feeling there is something inherent, for example like the divergence theorum, that i can use?

    mind you in the time it took me to get the tex working i could have solved the thing, but i'm still curious
    Last edited: Sep 29, 2010
  2. jcsd
  3. Sep 29, 2010 #2
    One smarter way would be by using (first understanding the derivation, of course) Eqs. (49,50) from the entry "http://mathworld.wolfram.com/SphericalCoordinates.html" [Broken]".
    Last edited by a moderator: May 5, 2017
  4. Sep 29, 2010 #3
    Hmm. To be honest I dont know where to start. For example, how to i manage the position vector? Do i replace x, y, z with [tex] r cos \theta sin \phi[/tex] etc, then do the divergence? It starts to look very messy ... [tex] t^{-\frac{5}{2}}e^{-\frac{r}{4kt}} r cos \theta sin \phi[/tex] for the x component for example
  5. Sep 30, 2010 #4
    [tex]r[/tex] in the exponential is not the position vector. It is its length - one of the coordinates of the spherical system. It will stay as such. [tex]\theta,\phi[/tex] simply do not appear in the formula - which simplifies the calculations.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook