- #1

Reshma

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I am having some trouble with this particular problem on Vector Calculus from Griffith's book.

The question is: Check the divergence theorem for the vector function(in spherical coordinates)

[tex]\vec v = r^2\cos\theta\hat r + r^2\cos\phi \hat \theta - r^2\cos\theta\sin\phi\hat \phi[/tex]

using your volume as

**one octant**of the sphere of radius R.

According to the Divergence Theorem:

[tex]\int_v (\nabla.\vec v)d\tau = \oint_s \vec v.d\vec a [/tex]

Here is how I went about it

**exclulsively using spherical coordinates**:

Part 1: Volume integral

I found the divergence of the given vector field according to spherical coordinates and integrated it over the volume of the octant. I got the answer for the left hand side as [tex]\frac{\pi R^4}{4}[/tex] which tallies with the solution provided.

Part 2: Surface integral

Here is where my problem lies. While calculating the volume integral, you are find the volume integral over

**4 surfaces of the octant**viz. the spherical surface, xy plane, yz plane and xz plane.

For the spherical surface:Radius R is constant

[tex]d\vec a_1 = r^2\sin\theta d\theta d\phi \hat r [/tex]

For the xy plane: [tex]\theta[/tex] is constant

[tex]d\vec a_2 = rdrd\phi \hat \theta[/tex]

Now I don't have a clue on how to calculate the surface integral for the other two surfaces. Although I get the required answer by calculating the surface integral just over the spherical surface, I still want to know the procedure for the other surfaces since it is specifically mentioned in the question that the surface integral has to calculated over the

**entire**surface.

Any help would be greatly appreciated.

Regards,

Reshma