I'm looking for equivalent definitions for the differentiability in general . I'm really confused about the differentiation of multivariable functions . Also I want the proofs for equivalency of the definitions .
Here is the definition in the Stewart calculus but it doesn't make sense to me :
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Stewart's definition does seem a little confusing. I assume the notations $\Delta z,$ $\Delta x,$ and $\Delta y$ were defined earlier, since the definition is meaningless otherwise.
The essence of the derivative of a single-variable function $g$ is that in the neighborhood of an input value $x=a,$ the function is very well approximated by a first-degree polynomial function of $x,$ which is a function describing a line. That is, $$ g(x) \approx g(a) + m (x-a).\tag1$$ The derivative of $g$ at $x=a$ is then $g'(a) = m.$ The rest of the mechanics of the derivative are there to ensure that the linear function in $(1)$ actually is a good approximation, and if we need it to be better, we can just restrict ourselves to look at a smaller neighborhood of $x=a.$
The idea of the multivariate derivative is the same, but the input now is the pair of coordinates of a point $(x,y).$ The derivative of $f$ at $(x,y)=(a,b)$ should allow us to approximate $f$ near $(x,y)=(a,b)$ by first-degree polynomial in $x$ and $y$, which describes a plane. That is, $$ f(x,y)\approx f(a,b) + m(x-a) + n(y-b).$$ The parameters $m$ and $n$ in the approximate equation are the partial derivatives of $f$ at $(x,y)=(a,b)$, which are often written $f_x(a,b)$ and $f_y(a,b)$.
Stewart introduces the factors $\epsilon_1$ and $\epsilon_2$ to account for the fact that the approximation often is not exact and that it gets worse as you get farther from $(x,y)=(a,b).$ This is a little awkward, since (unlike in the single-variable case) you cannot determine both of these values at an arbitrary point $(x,y)$ just by looking at the values and derivative of $f$ at $(a,b)$ and the arbitrary $(x,y).$ You would have two unknowns but only one equation. Still, we could "guess" a relationship between arbitrary coordinates and the epsilons at those coordinates that would make everything in the definition true, provided that $f$ actually is well approximated by a plane at $(x,y)=(a,b).$
That part of the definition seems a bit messy. You have another answer that defines the derivative by looking at coordinates within a disk around $(x,y)=(a,b)$ and shrinking the radius of that disk to zero in the limit. It also lets us use just one "epsilon" instead of two. That seems to me to be a better way to define the limit than just writing $(\Delta x,\Delta y)\to(0,0).$ But I suspect that if you read the material before (or perhaps after) this definition carefully, the idea is the same.
There are plenty of definitions for differentiability. For example, within the context of a $\mathbb{R}^n$, we can suggest the definition to be:
A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable at $x \in \mathbb{R}^n$ provided there exists a linear mapping $A$ such that the limit $$\lim_{\epsilon \rightarrow 0, \ \ h \in B_{\epsilon}(x)} \frac{|| f(x + h) - (f(x) + h A(x)) ||}{||h||} = 0 $$
Clearly, in the context you are looking for, $n = 2, m = 1$.
It becomes a little complicated to prove equivalence however, simply by Analysis and it's ability to fit within a variety of different contexts and applications. An immediate question that is often discussed after the definition I wrote above would be how it relates to the "differential", which seems to be the approach to a derivative the excerpt you included implies. So, firstly, we would consider the differential of this function to simply be: $$df_x(h) = A(x) \cdot h$$ If we express $x = \pmatrix{a \\ b}$, $A(x) = \pmatrix{f_x(a,b) \\ f_y(a,b)}$ and $h = \pmatrix{\Delta x \\ \Delta y}$, then we see that: $$df_x(h) = f_x(a,b) \Delta x + f_y(a,b) \Delta y$$ Now giving ourselves a room of epsilon, we see that: $$\Delta z = \Delta f_x(h) = f_x(a,b) \Delta x + f_y(a,b) \Delta y + \Delta x \cdot \epsilon_1 + \Delta y \cdot \epsilon_2$$ where we ensure $\sqrt{\epsilon_1^2 + \epsilon_2^2} < \epsilon$ in the limit statement. So you see this statement is the same as the one you provided. It is rather specific, which is indivative of the source Stewart Calculus.