\[\mathbb{R}^2\]
is given parametrically by \[x=X( \alpha ), \; y=Y( \alpha ) \]
. In Cartesian coordinates the gradient of the curve at a point \[(x_0,y_0)\]
is given by \[\frac{dy}{dx} |_{(x_0, \; y_0)}\]
. In the parametric coordinates the curve is using we must write \[\frac{dy/d \alpha}{dx/d \alpha} |_{(\alpha_0)}\]
, where \[(x_0, \; y_0)=(X( \alpha_0), \; Y( \alpha_0))\]
.Example: The curve
\[C \subset \mathbb{R}^2\]
is defined parmetrically by \[x=cos 2 \alpha , \; y=2sin \alpha +3cos \alpha\]
. At \[(1,3)\]
, \[\alpha = 0\]
and\[\begin{equation} \begin{aligned} \frac{dy}{dx}_{1,3)} &= \frac{dy/d \alpha}{dx/d \alpha} |_{0} \\ &= \frac{-2sin \alpha}{2cos \alpha-3sin \alpha} |_0 \\ &= \frac{0}{2}=0\end{aligned} \end{equation}\]