The integral

$a={\displaystyle \underset{S}{\overset{\phantom{}}{}}}da$

$is$ called the vector area $of$ the surface $S.$ The vector area $of$ a circle, radius $R,$ lying $in$

the $xy$ plane $is$ then obviously $a=R^2$hat$(z).$

$(a)$ Using spherical coordinates, write down $an$ explicit expression for the infinitesimal

vector area $da$ lying $on$ a sphere $of$ radius $Rat$ point $(r,,).$

$(b)$ What are the $x,y,$ and $z$ components $of$ this $da?$

$(c)$ Using the above results, find the vector area $of$ a hemispherical bowl $of$ radius $R$

whose perimeter $is$ the curve defined $by$ equation $x^2+y^2=R^2.$