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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 asone octantof 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 itexclulsively 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 over4 surfaces of the octantviz. 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 theentiresurface.

Any help would be greatly appreciated.

Regards,

Reshma

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# Divergence theorem on Octant of a sphere

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