Derivatives of Unit Vectors in Polar Coordinates

Theorem
Consider a particle $p$ moving in the plane.

Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$.

Let:
 * $\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$
 * $\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$

Then the rate of change of $\mathbf u_r$ and $\mathbf u_\theta$ can be expressed as:

Proof
By definition of sine and cosine:

where $\mathbf i$ and $\mathbf j$ are the unit vectors in the $x$-axis and $y$-axis respectively.


 * DerivativesOfUnitPolarVectors.png

Differentiating $(1)$ and $(2)$ $\theta$ gives: