Let \(\mathbf{e}_r = \left\langle \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right\rangle\) be a unit radial vector, where \(r = \sqrt{x^2 + y^2 + z^2}\).
(a) Calculate the integral of \(\mathbf{F} = e^{-r}\mathbf{e}_r\) over the upper hemisphere of \(x^2 + y^2 + z^2 = 9\) with the normal pointing outward.
(Give your answer in exact form. Use symbolic notation and fractions where needed.)
\(\iint_S \mathbf{F} \cdot d\mathbf{S} = 18\pi e^{-3}\)
(b) Calculate the integral of \(\mathbf{F} = 3e^{-r}\mathbf{e}_r\) over the octant \(x \ge 0, y \ge 0, z \ge 0\) of the unit sphere centered at the origin.
(Give your answer in exact form. Use symbolic notation and fractions where needed.)
\(\iint_S \mathbf{F} \cdot d\mathbf{S} = 54\pi e^{-3}\)
