Show that the vector has zero divergence

[dizewarrior]

Homework Statement

Show that the vector
v = \frac{\hat{r}}{r²} (not sure why formatting isn't working?)

v = (r-hat) over (r squared)

has zero divergence (it is solenoidal) and zero curl (it is irrotational) for r not equal to 0

Homework Equations

div(V) = (d/dx)V_x + (d/dy)V_y + (d/dz)V_z

The Attempt at a Solution

I used del operator in spherical (the r component being (d/dr)) and it didn't seem to work?
for curl i was able to get curl(v) = 0
I've tried converting v to cartesian and using the cartesian del operator but it didn't work either
I'm stuck at this point =\

 

[Mute]

The r-component of the gradient operator in spherical coordinates is d/dr. The divergence operator, however, is different.

See http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordinates

 

[vela]

v = \frac{\hat{r}}{r^2} (not sure why formatting isn't working?)

Don't use the BBcode tags within LaTeX mark-up.

Also, if you're going to use LaTeX, use it for the whole equation instead of bits and pieces. It's easier to type, and it'll look better.

 

[HallsofIvy]

Also, you might need to refresh your screen to get the LaTeX to work.

 

[dizewarrior]

thanks everyone!

few questions though,
1) where can I find a guide on posting LaTeX code on the forums?

2) does the 1/(r^2) * (r^2 * v_r): does the r^2 come from metric coefficients?

 

[Mute]

thanks everyone!

few questions though,
1) where can I find a guide on posting LaTeX code on the forums?

2) does the 1/(r^2) * (r^2 * v_r): does the r^2 come from metric coefficients?

1) See https://www.physicsforums.com/showthread.php?t=386951 to get started.

2) I'm not sure what you mean by metric coefficients. These factors show up for similar reasons as to why the volume element in spherical coordinates is r² sin(theta). A vector calculus book should explain it in the section about grad, div and curl in different coordinate systems. Maybe even the wiki page I linked to earlier explains it?