Verifying the divergence theorem for a given example

Question

I'm currently working on an exercise, in which at some point I'm asked to verify the divergence theorem for a given vector field and a sphere.
The Vector field is defined as $$\vec{w}=ax\vec{e}_x+2ay\vec{e}_y+3az\vec{e}_z,$$ and the sphere by $$\sqrt{x^2+y^2+z^2}\leq R^2.$$ I want to show $\int_S\vec{w}\cdot d\vec{S}=\int_V(\nabla\cdot\vec{w})\cdot dV$.
I started with the volume integral and got $$\int_V(\nabla\cdot\vec{w})\cdot dV=\int_V(\frac{\partial}{\partial x}ax+\frac{\partial}{\partial y}2ay+\frac{\partial}{\partial z}3az)\cdot dV$$ $$=6a\int_VdV$$ $$=6a\int_0^RR^2dr\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\phi$$ $$=6a\left(\frac{R^3}{3}\Big|_0^R\right)\left(-\cos\theta\Big|_0^\pi\left)\right(\phi\Big|_0^{2\pi}\right)$$ $$=8a\pi R^3$$ Now I have to calculate the surface integral, so that I can verify that $\int_S\vec{w}\cdot d\vec{S}=\int_V(\nabla\cdot\vec{w})\cdot dV$.
But I'm not quite sure on how to approach this. I tried to transform my vector field into spherical coordinates, but that seemed to complicated (taking in account that, for the volume integral, the vector field simplified to $6a$).
So my question is, do I really have to go the hard way and do it in spherical coordinates, or is there a shorter, maybe smarter way on how to approach this?
If I did some mistakes along the way, whether in calculations or notation, please let me know.
Thanks in advance.

Answer 1

I will calculate $$\int_S \vec{w}\cdot\hat{n}\ dA$$ on the surface of the sphere $x^2+y^2+z^2=R^2$. First, identify that $\hat{n}=\hat{r}$, the unit vector in the radial direction, a direction perpendicular to the surface of the sphere. $$\hat{n}=\sin\theta\cos\phi\ \vec{e_x}+ \sin\theta\sin\phi\ \vec{e_y}+ \cos\theta\ \vec{e_z}$$ and now write $$\int_S \vec{w}\cdot\hat{n}\ dA= R^2\int d\phi \int \sin\theta\ d\theta (ax \sin\theta\cos\phi+ 2ay \sin\theta\sin\phi+ 3az \cos\theta)$$ and recall that we are integrating over a spherical surface, so we are using spherical coordinates and $$\int_S \vec{w}\cdot\hat{n}\ dA= aR^3\int d\phi \int \sin\theta\ d\theta (\sin^2\theta\cos^2\phi+ 2\sin^2\theta\sin^2\phi+ 3\cos^2\theta)$$ and then you do the remaining integrals to determine the answer. If there is something left unclarified, please do not hesitate to ask. I hope this helps.