Turunan Vektor Basis Satuan Azimutal terhadap Koordinat Azimutal

$$\partial\hat{\phi}/\partial\phi = (\partial/\partial\phi)(\vec{e}_\phi/|\vec{e}_\phi|).$$
$$\vec{e}_\phi = \partial\vec{r}/\partial\phi.$$
$$\vec{r} = \hat{x}r\sin\theta\cos\phi + \hat{y}r\sin\theta\sin\phi + \hat{z}r\cos\theta.$$
$$\vec{e}_\phi = -\hat{x}r\sin\theta\sin\phi + \hat{y}r\sin\theta\cos\phi.$$
$$|\vec{e}_\phi| = r\sin\theta.$$
$$\hat{\phi} = \vec{e}_\phi/|\vec{e}_\phi| = -\hat{x}\sin\phi + \hat{y}\cos\phi.$$
$$\partial\hat{\phi}/\partial\phi = -(\hat{x}\cos\phi + \hat{y}\sin\phi).$$
$$\hat{r} = \hat{x}\sin\theta\cos\phi + \hat{y}\sin\theta\sin\phi + \hat{z}\cos\theta.$$
$$\hat{\theta} =  \vec{e}_\theta/|\vec{e}_\theta|.$$
$$\vec{e}_\theta = \partial\vec{r}/\partial\theta = \hat{x}r\cos\theta\cos\phi + \hat{y}r\cos\theta\sin\phi - \hat{z}r\sin\theta.$$
$$|\vec{e}_\theta| = r.$$
$$\hat{\theta} = \hat{x}\cos\theta\cos\phi + \hat{y}\cos\theta\sin\phi - \hat{z}\sin\theta.$$
$$\begin{pmatrix}\hat{r} \\ \hat{\theta} \\ \hat{\phi}\end{pmatrix} = \begin{pmatrix}\sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi & \cos\phi & 0\end{pmatrix} \begin{pmatrix}\hat{x} \\ \hat{y} \\ \hat{z}\end{pmatrix}.$$
$$\Delta = \begin{vmatrix}\sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi & \cos\phi & 0\end{vmatrix} = 1.$$
$$\hat{x} = \frac{1}{\Delta}\begin{vmatrix}\hat{r} & \sin\theta\sin\phi & \cos\theta \\ \hat{\theta} & \cos\theta\sin\phi & -\sin\theta \\ \hat{\phi} & \cos\phi & 0\end{vmatrix} = \hat{r}\sin\theta\cos\phi + \hat{\theta}\cos\theta\cos\phi - \hat{\phi}\sin\phi.$$
$$\hat{y} = \frac{1}{\Delta}\begin{vmatrix}\sin\theta\cos\phi & \hat{r} & \cos\theta \\ \cos\theta\cos\phi & \hat{\theta} & -\sin\theta \\ -\sin\phi & \hat{\phi} & 0\end{vmatrix} = \hat{r}\sin\theta\sin\phi + \hat{\theta}\cos\theta\sin\phi + \hat{\phi}\cos\phi.$$
$$\partial\hat{\phi}/\partial\phi = -((\hat{r}\sin\theta\cos\phi + \hat{\theta}\cos\theta\cos\phi - \hat{\phi}\sin\phi)\cos\phi + (\hat{r}\sin\theta\sin\phi + \hat{\theta}\cos\theta\sin\phi + \hat{\phi}\cos\phi)\sin\phi).$$
$$\partial\hat{\phi}/\partial\phi = -(\hat{r}\sin\theta + \hat{\theta}\cos\theta).$$