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Definition 1:

From https://www.math.hmc.edu/calculus/tutorials/tangentplanes/differentiability.pdf or Differentiability for a function of two variables

A function $f(x, y)$ is differentiable at the point $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\Delta f = f(x_0 +\Delta x, y_0 +\Delta y) − f(x_0,y_0)$ can be written in the form $$\Delta f=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y$$ where $\epsilon_1$ and $\epsilon_2$ are functions of $\Delta x$ and $\Delta y$ such that $$\lim\limits_{(\Delta x,\Delta y)\to(0,0)}\epsilon_1=\lim\limits_{(\Delta x,\Delta y)\to(0,0)}\epsilon_2=0.$$

Definition 2: From http://higheredbcs.wiley.com/legacy/college/hugheshallett/0471484822/theory/hh_focusontheory_sectioni.pdf

A function $f(x, y)$ is differentiable at the point $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\Delta f = f(x_0 +\Delta x, y_0 +\Delta y) − f(x_0,y_0)$ can be written in the form $$\Delta f=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+o(\sqrt{\Delta x^2+\Delta y^2}).$$

I wonder whether two above definitions of differential function are equivalent. Any explain are highly appreciated.

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