## Calculus 8th Edition

$s$
Since, $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial s}&\dfrac{\partial x}{\partial t}\\\dfrac{\partial y}{\partial s}&\dfrac{\partial y}{\partial t}\end{vmatrix}$ Here, we have $\dfrac{\partial x}{\partial s}=\cos t$ and $\dfrac{\partial x}{\partial t}=-s \sin t$ Also, $\dfrac{\partial y}{\partial s}=\sin t$ and $\dfrac{\partial y}{\partial t}= s\cos t$ Now, $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial s}&\dfrac{\partial x}{\partial t}\\\dfrac{\partial y}{\partial s}&\dfrac{\partial y}{\partial t}\end{vmatrix}=\begin{vmatrix} \cos t&- s \sin t\\\sin t&s \cos t\end{vmatrix}$ or, $=(\cos t) \times (s \cos t) -(\sin t) \times (-s \sin t)$ or, $=s \cos^2 t+ s \sin^2 t$ or, $=s(\cos^2t+\sin^2t)$ Hence, $Jacobian=s$