# Unit vectors in Spherical Coordinates

by Starproj
Tags: coordinates, spherical, unit, vectors
 Sci Advisor HW Helper PF Gold P: 12,016 Unit vectors in Spherical Coordinates Ok, this is fairly trivial. Assume that some vector $\vec{u}$ (dependent on some independent variables) has unit size irrespective of the values of the independent variables, i.e: $$\vec{u}^{2}=1(1)$$ Then, labeling an independent variable as $x_{i}$, we get by differentiating (1) wrt. to that variable: $$2\frac{\partial\vec{u}}{\partial{x}_{i}}\cdot\vec{u}=0[/itex], i.e, the derivatives of the unit vector are orthogonal to it! Thus, starting out with the radial vector, [tex]\vec{i}_{r}=\sin\phi\cos\theta\vec{i}+\sin\phi\sin\theta\vec{j}+\cos\ph i\vec{k}$$, we perform the two differentiations here: $\frac{\partial\vec{i}_{r}}{\partial\phi}=\cos\phi\cos\theta\vec{i}+\cos \phi\sin\theta\vec{j}-\sin\phi\vec{k}=\vec{i}_{\phi}[/tex] and: [tex]\frac{\partial\vec{i}_{r}}{\partial\theta}=\sin\phi(-\sin\theta\vec{i}+\cos\theta\vec{j})=\sin\phi\vec{i}_{\theta}$ where the appropriate forms of the unit vectors $\vec{i}_{\phi},\vec{i}_{\theta}$ have been indicated.