# Help w/ Seaborn's Mathematics for the Physical Sciences

1. Feb 15, 2009

### bcjochim07

1. The problem statement, all variables and given/known data
In spherical polar coordinates, a vector F is given by F=(r*costheta*cosphi)rhat + (r*costheta*sinphi)thetahat + (r*sintheta*cosphi)phihat

Check the validity of the divergence theorem for this vector and the volume that is one octant of a sphere radius b.

2. Relevant equations
$$\oint$$F $$\cdot$$ dA = $$\int$$Vgrad$$\cdot$$Fd3r

3. The attempt at a solutionThe solution for this problem is in my book, and I am having difficulty following it, I think because it is in polar coordinates, and I am still trying to get used to them.

The author proceeds to divide the surface into four area elements:

spherical surface : dA = rhat(b^2)*sintheta*dtheta*dphi

before I go on to any of the other elements, could someone please help me understand how the author derived this expression? Thanks.

2. Feb 15, 2009

### tiny-tim

Hi bcjochim07!

Mark a "rectangle" on the surface, with change in latitude dθ and change in longitude dφ.

Then its width is bsinθ dθ, and its height is bdφ, so its area is b2sinθdθdφ.

3. Feb 15, 2009

### bcjochim07

ok,

So I think I understand where the bdφ comes from; isn't that an arc length? Then I would think that the other dimension would be bdθ, so where does that sinθ come from?

Thanks.

4. Feb 15, 2009

### tiny-tim

Just look at a globe of the Earth …

a circle of latitude has radius bsinθ

(while a circle of longitude has radius b).

5. Feb 20, 2009

### bcjochim07

Thanks. Great explanation. Now I am trying to figure out how the book derives the other surfaces. For example, the area element in the xz plane is in the negative phi hat direction with dA = r dr dθ. Considering the area formula for a circular sector, I thought it should be (1/2)rdrdθ.

6. Feb 20, 2009

### bcjochim07

Ok... I think I see it now. I just need to divide the area up into little "rectangles" just like I did for the spherical surface. Each of these rectangles has dimensions dr by rdθ.

7. Feb 22, 2009

### tiny-tim

Yup! … it's all simple geometry …

just draw the right diagram, and it becomes obvious!