Chapter 4.3, Problem 1P

Consider a function $f(x,y)$ which can be expanded in a two-variable power series,

(2.3) or $(2.7).$ Let $x-a=h=\Delta x,y-b=k=\Delta y;$ then $x=a+\Delta x,y=b+\Delta y$ so that $f(x,y)$ becomes $f(a+\Delta x,b+\Delta y).$ The change $\Delta z$ in $z=f(x,y)$ when $x$ changes from $a$ to $a+\Delta x$ and $y$ changes from $b$ to $b+\Delta y$ is then $\Delta z=f(a+\Delta x,b+\Delta y)-f(a,b)$.

Use the series ( 2.7 ) to obtain ( 3.11 ) and to see explicitly what ${ϵ}_1$ and ${ϵ}_2$ are and that they approach zero as $\Delta x$ and $\Delta y\to 0.$