If we have a multivariate function \( f(x,y) \) where \( x=x(t,s) \) and \( y=y(t,s) \) are functions of the variables \( t \) and \( s \), we have that the partial derivatives
$$ \frac{\partial f}{\partial s}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s}, $$and
$$ \frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}. $$the gradient of \( f \) with respect to \( t \) and \( s \) (without the explicit unit vector components)
$$ \frac{df}{d(s,t)} = \begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix} \begin{bmatrix}\frac{\partial x}{\partial s} &\frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} \end{bmatrix}. $$