If i may add, one point I was surprised about was the fact that the first jacobian came out to be zero, though i beleive i have done it correctly
?
I think my method is correct here, but i must of gone wrong somewhere (note i cannot use divergence theorem)
I have a vector field:
Where a is a constant and a sphere:
R is the radius.
Now i have to do a surface integral i.e. not gauss's divergence theorem of a volume integral. Though i did this in order to match my answers and i got
Though i haven't yet found a method to get near to an answer for the surface integral. I know:
Though in this situation dx, dy and dz are all present:
then i sub in n into the flux do the dot products, cancel the 1/|N| terms and get:
Now i introduce spherical coordinates whereby:
Now it looks a little messy but is still all doable - albeit a little long winded but im not allowed to use the divergence theorem.
To cut it short i get
Which is not the obvious and the one i got from divergence theorem:
Any help is much appreciated and thanks for reading
I really dislike that way of doing a surface integral. Instead, I would do this:
Start by using spherical coordinates, with , a constant: , , so we can write the "position vector of any point on surface as
.
The derivatives of that vector with respect to the two parameters,
and
are tangent to the surface and their lengths contain information about the differentials of arc length. So their cross product
is normal to the surface and contains all differential information.
(I chose the order of multiplication to give the outward normal)
In other words, that is .
Now, so that
and your integral is
which does, in fact, give .