3.19.1 (Straight lines). Let (x(t),y(t))=(a+th,b+tk)(x(t),y(t))=(a+th,b+tk). Then (x(t),y(t))(x(t),y(t)) describes a straight line segment as tt varies. For any function f(x,y)f(x,y) we have
3.19.2 (Circle).Let x(t)=costx(t)=\cos t and y(t)=sinty(t)=\sin t. Determine dfdt\frac{df}{dt}.
Solution. We have dxdt=-sintanddydt=cost.\frac{dx}{dt}=\,{-\sin t}\;\mbox{and}\;\frac{dy}{dt}=\,{\cos t.}
So dfdt=-sint∂f∂x+cost∂f∂y=-yfx+xfy.\frac{df}{dt}=\,{-\sin t\frac{\partial f}{\partial x}+\cos t\frac{\partial f}{% \partial y}}\,{=-yf_{x}+xf_{y}.}