Chapter 1.4, Problem 1.38P

Express the unit vectors $\text{<mover accent="true"> <mo> r </mo> <mo> ^ </mo> </mover> ,}\hat{\theta }\text{,}\hat{\varphi }$ in terms of (that is, derive Eq. 1.64). Check your answers several ways $\left(\text{<mover accent="true"> <mo> r </mo> <mo> ^ </mo> </mover>}\cdot \text{<mover accent="true"> <mo> r </mo> <mo> ^ </mo> </mover>}\stackrel{?}{=}1,\hat{\theta }\cdot \hat{\varphi }\stackrel{?}{=}0,\text{<mover accent="true"> <mo> r </mo> <mo> ^ </mo> </mover>}\times \hat{\theta }\stackrel{?}{=}\hat{\varphi },\mathrm{...}\right).$ Also work out the inverse formulas, giving in terms of $\text{<mover accent="true"> <mo> r </mo> <mo> ^ </mo> </mover> ,}\hat{\theta }\text{,}\hat{\varphi }$ (and $\theta ,\varphi$ ).