(Solved): 1. (25 pts.) Consider the spherical coordinate system (R,,) shown below. Problem 1. The relatio ...

1. (25 pts.) Consider the spherical coordinate system $(R,?,?)$ shown below. Problem 1. The relation between the Cartesian and spherical coordinates are $\begin{array}{l}{x}_1=R\cos (?)\sin (?)\\ x_2=R\sin (?)\sin (?)\\ x_3=R\cos (?)\end{array}$ (a) Determine the scale factors $h_1,h_2$, and $h_3$ (b) Starting from the definition of the Curl of a Vector in generalized curvilinear coordinates, $?\times \vec{a}={?}_{i=1}^3{?}_{j=1}^3{?}_{k=1}^3\left(\frac{{?}_{ijk}}{h_j{h}_k}\frac{?}{?{?}^j}\left[a_{?k?}{h}_k\right]{\hat{e}}_i\right)$ show that the Curl of a Vector in spherical coordinates is $\begin{array}{rl}?\times \vec{a}=& \left[\frac{1}{R\sin (?)}\left(\frac{?}{??}\left[\sin \left(?a_{?}\right)\right]?\frac{?a_{?}}{??}\right)\right]{\hat{e}}_R\\ +& \left(\frac{1}{R\sin (?)}\frac{?a_R}{??}?\frac{1}{R}\frac{?}{?R}\left[R{a}_{?}\right]\right){\hat{e}}_{?}\\ +& \frac{1}{R}\left(\frac{?}{?R}\left[R{a}_{?}\right]?\frac{?a_R}{??}\right){\hat{e}}_{?}\end{array}$ where $a_R,a_{?}$, and $a_{?}$ are components of the vector in the spherical coordinate system.