Flux of a vector field through the boundary of a closed surface

Question

I am supposed to calculate the flux of the vector field $G(r) = x \hat e_x + y \hat e_y + z \hat e_z$ through the boundary of the volume $V$ defined by $ x^2+y^2+z^2 \leq 1, z\geq 0$ (the northern hemisphere of the sphere of radius 1 centred at the origin. The formula would be $$\iint \vec G(r) \cdot d\vec S $$for a sphere, the vectorial surface area element is: $R^2 \sin \theta d\theta d \phi$. I don't understand how I can compute this, as $d \vec S$ is given in spherical polar coordinates but $\vec G(r)$ is given in cartesian coordinates. I am also given the following hint: 'first answer the question why the disc $x^2 + y^2 \leq 1 $ does not contribute, then compute the flux through the curved part of the sphere. Any help would be great!

Answer 1

Claim: the flux of $\mathbf G$ over the unit disk $\mathscr D = \{(x, y, z) \,|\, x^2 + y^2 \leq 1 \text{ and } z = 0\}$ is zero.

Proof. Using the usual polar coordinates $x = r \cos \theta$ and $y = r \sin \theta,$ we parametrize the unit disk $\mathscr D$ by $D(r, \theta) = \langle r \cos \theta, r \sin \theta, 0 \rangle$ over the domain $U = \{(r, \theta) \,|\, 0 \leq r \leq 1 \text{ and } 0 \leq \theta \leq 2 \pi \}.$ We have therefore that $D_r(r, \theta) = \langle \cos \theta, \sin \theta, 0 \rangle$ and $D_\theta(r, \theta) = \langle -r \sin \theta, r \cos \theta, 0 \rangle,$ from which it follows that $(D_r \times D_\theta)(r, \theta) = \langle 0, 0, r \rangle.$ We conclude that $\mathbf G(D(r, \theta)) \cdot (D_r \times D_\theta)(r, \theta) = \langle r \cos \theta, r \sin \theta, 0 \rangle \cdot \langle 0, 0, r \rangle = 0$ so that $\iint_{\mathscr D} \mathbf G \cdot d \mathbf S = \iint_U \mathbf G(D(r, \theta)) \cdot (D_r \times D_\theta)(r, \theta) \, dr \, d \theta = 0.$ QED.

Considering that flux is additive, the flux you seek is equal to the sum of the flux on the unit disk $\mathscr D$ and the flux on the top of the northern hemisphere, i.e., the flux on $\mathscr S = \{(x, y, z) \,|\, x^2 + y^2 + z^2 = 1 \text{ and } z \geq 0 \}.$ Can you compute this flux and finish the problem? (Hint: use spherical coordinates to parametrize $\mathscr S.$)