Does the Divergence Theorem Apply to Complex Vector Fields and Hemispheres?

[lazyluke]

Homework Statement

2. Verify the divergence theorem for the vector field:
F =(r2cosθ) r +(r2cosφ) θ −(r2cosθsinφ) φ
using the upper hemisphere of radius R.

Homework Equations

Is this any close to be correct? The question marks indicate parts I am not sure about please help.

Anyone know what are the scale factors for spherical coordinate system, i cannot find them anywhere, i think the product of all of them is r^2sine(e) but I am not sure which ones are which (h1=h2=r, h3=sin(e)?pls help

The Attempt at a Solution

http://img522.imageshack.us/my.php?image=pictureop2.jpg
http://img204.imageshack.us/my.php?image=picture001yz9.jpg

 

[johnster08]

h1 = 1, h2 = r, h3 = rsin(θ)

 

[lazyluke]

Thank you, how do you derive that do u know?... is there a general formula for all coordinate systems to egt the scalar factors?... i don't need it for this part but the other question...and is my solution any close to be correct? (link to the file at the bottom of the post)

 

[johnster08]

I haven't studied Divergence Theorem in a Physics context but you can derive the scale factors using geometrical arguments. Here are my notes from Theoretical Methods for the scale factors. You can access my notes on these links

http://img407.imageshack.us/my.php?image=scan0019mm2.jpg"
http://img407.imageshack.us/my.php?image=scan0020jp4.jpg"

 

[lazyluke]

http://img262.imageshack.us/my.php?image=picture003dk1.jpg
http://img510.imageshack.us/my.php?image=picture004ew8.jpg

Thats what i came up with, i have a feeling its incorrect since the whole thing came to 0...pls can someone look over that?
thank You Luke

 

[lazyluke]

Thanks to johnster08 as only he answered to my 1 out of 3posts...thnx guys, i don't think ill be here too aften...cya