Derivatives of Unit Vectors in Polar Coordinates
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Theorem
Consider a particle $p$ moving in the plane.
Let the position of $p$ be given in polar coordinates as $\polar {r, \theta}$.
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 derivative of $\mathbf u_r$ and $\mathbf u_\theta$ with respect to $\theta$ can be expressed as:
\(\ds \dfrac {\d \mathbf u_r} {\d \theta}\) | \(=\) | \(\ds \mathbf u_\theta\) | ||||||||||||
\(\ds \dfrac {\d \mathbf u_\theta} {\d \theta}\) | \(=\) | \(\ds -\mathbf u_r\) |
Proof
By definition of sine and cosine:
\(\text {(1)}: \quad\) | \(\ds \mathbf u_r\) | \(=\) | \(\ds \mathbf i \cos \theta + \mathbf j \sin \theta\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \mathbf u_\theta\) | \(=\) | \(\ds -\mathbf i \sin \theta + \mathbf j \cos \theta\) |
where $\mathbf i$ and $\mathbf j$ are the unit vectors in the $x$-axis and $y$-axis respectively.
Differentiating $(1)$ and $(2)$ with respect to $\theta$ gives:
\(\ds \dfrac {\d \mathbf u_r} {\d \theta}\) | \(=\) | \(\ds -\mathbf i \sin \theta + \mathbf j \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf u_\theta\) | ||||||||||||
\(\ds \dfrac {\d \mathbf u_\theta} {\d \theta}\) | \(=\) | \(\ds -\mathbf i \cos \theta - \mathbf j \sin \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\mathbf u_r\) |
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: