Problem $2(4$ marks$)$

Consider a flow velocity vector vec$(V),$ which can be described by the sum of two vectors

vec$(V{)}_1=$Uhat$(i)+0$hat$(j)$ and vec$(V{)}_2=0$hat$(i)+$Vhat$(j).$ Here, vec$(V)$ is associated with the stream function, $,$ while

vec$(V{)}_1$ and vec$(V{)}_2$ have corresponding stream functions ${}_1$ and ${}_2,$ respectively. If vec$(V)=$vec$(V{)}_1+$vec$(V{)}_2,$

evaluate $,{}_1,$ and ${}_2,$ and hence show that $={}_1+{}_2.$