Calculate the flux of the field $F(x,y,z) = (yz, xz, xy)$ through the sphere: $ x,y,z > 0, \space x^2 + y^2 + z^2 = a^2 $ With outer normal.
Solution says: The normal is $N = \frac{1}{a}(x,y,z)$, hence $\langle F,N \rangle = \frac{3}{a}xyz$, and using spherical coordinates, we get: $ flux_F(M) = \int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}} \frac{3}{a}(a\cos(\varphi)\sin(\theta)\times a\sin(\varphi)\sin(\theta)\times a\cos(\theta))\times {\color{Red} \sin(\theta)} d\theta d\varphi = ... = \frac{3a^2}{8} $
My question is - shouldn't the part marked in red be $a^2\sin(\theta)$?
Because $r=(a\cos(\varphi)\sin(\theta), a\sin(\varphi)\sin(\theta), a\cos(\theta))$ is a mapping $r:\mathbb{R}^2 \rightarrow \mathbb{R}^3$, the formula for surface integrals is: $ \int_{M}f = \int_{\Omega}f\circ r \sqrt{det(D_r^T D_r)} $ and $\sqrt{det(D_r^T D_r)} = a^2\sin(\theta)$.
Thanks!