After having sat through a semester of Calculus, (Physics-oriented), I decided to work on my basic math and tried to learn Real Analysis (from "Analysis Course Volume II",Lima, Elon. L). I was surprised to see the differential of a function introduced between the notion of differentiability and the gradient. I thought to myself that this was rather weird since in my Calculus classes the differential had never been mentioned and instead the gradient had been presented as a sort of derivative for multivalued real functions. The author somewhat justifies this by saying that there is an isomorphism between the dual space of $\mathbb{R}^n$ and $\mathbb{R}^n$ itself, induced by the inner product, effectively favoring the more "geometrically intuitive" vector inner product as opposed to covectors acting on vectors, which makes sense.
However, I still feel uneasy. It seems like the differential is a much more central object, and I cannot seem to be able to interpret its role in Real Analysis, as opposed to the gradient and the derivative of a function. These are the definitions that Lima gives.
Differentiability
A function $f:\mathbb{R}^n \longrightarrow \mathbb{R}$ is said differentiable at a point $a$ if there exist constants $A_1,...,A_n$ such that for any vector $v=(\alpha_1,...,\alpha_n)$:
$f(a+v)=f(a)+\sum_i^n A_i \cdot\alpha_i+r(v)$
Where of course $A_i=\frac{\partial f}{\partial x_i}(a)$ and $\lim_{\,v\rightarrow \,0} \frac{r(v)}{|v|}=0$Differential
The differential of a differentiable function $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ at a point $a$ is the unique linear functional $df(a): \mathbb{R}^n \longrightarrow \mathbb{R}$ of whose value at the vector v is $$df(a) \cdot v=\sum_i^n \frac{\partial f}{\partial x_i}(a) \cdot \alpha_i$$ Now, we can easily see that the condition of differentiability can be written as $$f(a+v)=f(a)+df(a) \cdot v +r(v)$$ But I also know from studying differentiable $\mathbb{R}^m \longrightarrow \mathbb{R}^n$ functions that the broader sense of the derivative is the unique linear map that best approximates the function, at a point, to be linear, which in this case, would coincide with $df(a)$? So is the differential of a function actually the derivative, and not the gradient? I mean the gradient is not a linear map so I guess that's that, but then what of the Jacobian matrix? Was it not the derivative of a function?
In short:What is the differential? Is it the derivative?
If so, why are we not taught that way?