Sketch the small patch of the sphere whose spherical coordinates satisfy

\[
\frac{\pi}{2}-0.15 \leq \theta \leq \frac{\pi}{2}+0.15, \quad \frac{\pi}{4}-0.1 \leq \phi \leq \frac{\pi}{4}+0.1
\]

Use the normal vector computed in Exercise 9 to estimate its area.

Data From Exercise 9

Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point.

\(\Phi(\theta, \phi)=(\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi) ; \quad \theta=\frac{\pi}{2}, \quad \phi=\frac{\pi}{4}\)