PX275 - B1f - spherical basis vectors

\vec{r} = r \sin θ \cos ϕ \hat{i} + r \sin θ \sin ϕ \hat{j} + r \cos θ \hat{k}

\begin{aligned} \frac{d \vec{r}}{d r} : {\hat{e}}_{r} & = \frac{\sin θ \cos ϕ \hat{i} + \sin θ \sin ϕ \hat{j} + \cos θ \hat{k}}{\sqrt{\sin^{2} θ \cos^{2} ϕ + \sin^{2} θ \sin^{2} ϕ + \cos^{2} θ}} \\ = \frac{\sin θ \cos ϕ \hat{i} + \sin θ \sin ϕ \hat{j} + \cos θ \hat{k}}{\sqrt{1}} \end{aligned}

∴ {\hat{e}}_{r} = \sin θ \cos ϕ \hat{i} + \sin θ \sin ϕ \hat{j} + \cos θ \hat{k}

\begin{aligned} \frac{d \vec{r}}{d ϕ} : {\hat{e}}_{ϕ} & = \frac{- r \sin θ \sin ϕ \hat{i} + r \sin θ \cos ϕ \hat{j}}{\sqrt{r^{2} \sin^{2} θ \sin^{2} ϕ + r^{2} \sin^{2} θ \cos^{2} ϕ}} \\ = \frac{- r \sin θ \sin ϕ \hat{i} + r \sin θ \cos ϕ \hat{j}}{r \sin θ} \end{aligned}

∴ {\hat{e}}_{ϕ} = - \sin ϕ \hat{i} + \cos ϕ \hat{j}

\frac{d \vec{r}}{d θ} : {\hat{e}}_{θ} = \frac{r \cos θ \cos ϕ \hat{i} + r \cos θ \sin ϕ \hat{j} - r \sin θ \hat{k}}{\sqrt{r^{2} \cos^{2} θ \cos^{2} ϕ + r^{2} \cos^{2} θ \sin^{2} ϕ - r^{2} \sin^{2} θ}}

∴ {\hat{e}}_{θ} = \cos θ \cos ϕ \hat{i} + \cos θ \sin ϕ \hat{j} - \sin θ \hat{k}

notation

\vec{r} = r {\hat{e}}_{r}

\vec{F} = F_{r} {\hat{e}}_{r} + F_{ϕ} {\hat{e}}_{ϕ} + F_{θ} {\hat{e}}_{θ}