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In class we have defined what it means for a function of more variables to be differentiable as follows:

given $A \subset R^n, \ x \in Int(A), \ f:A \rightarrow R^m$ we will say that $f$ is differentiable in $x_0$ if there exists a function $L \in Hom(R^n,R^m)$ s.t. $$\frac{f(x_0 + h) - f(x_0) - L(h)}{||h||} \rightarrow 0$$ if $h \rightarrow 0$.

We have then shown that when $f : A\rightarrow R$ $$L(h) = \sum_{j = 1}^n \frac{\partial f (x_0)}{\partial x_j} h_j$$ and that $L(h)$ is unique.

Everything is fine except that I feel like I am missing some of the geometrical intuition behind this definition, also I struggle with the fact that the differential is a function of $h$. In single variable calculus the derivative in a point was a number that represented the infinitesimal rate of change and I don't understand how this generalizes well as a function of $h$.

If I have been clear enough could someone try to help me get a clearer picture of this definition and what $L(h)$ represents geometrically?

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    $\begingroup$ The derivative has always been a linear map. It's just that the linear maps $\mathbb{R} \to \mathbb{R}$ are given by multiplication by one number. $\endgroup$ – Chappers Jan 13 '16 at 18:56
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A differentiable function into $\mathbb{R}^m$ is entirely determined by its $m$ differentiable coordinate functions into $\mathbb{R}$ (but note that differentiability of the coordinate functions does not necessarily imply differentiability of the main function), so it suffices to understand the derivative of a map $\mathbb{R}^n \to \mathbb{R}$.

Think about the case $n = 2$. We can visualise $f:\mathbb{R}^2 \to \mathbb{R}$ through its graph, which will be some surface in $\mathbb{R}^3$. Fix a point $x_0 \in \mathbb{R}^2$ where we wish to understand the derivative geometrically. Imagine zooming in on the point $(x_0,f(x_0))$. As we get nearer, the graph of the surface becomes straighter, or less curved: it begins resembling more and more the tangent plane of the surface at $(x_0,f(x_0))$. Then, given a vector $v$ in $\mathbb{R}^2$, we can imagine it as sitting horizontally at $(x_0,f(x_0))$ and adjust its "$z$-coordinate" so that $v$ lies inside the tangent plane. This variation measured by the $z$-coordinate is precisely the value of $D_{x_0}(v)$ (the derivative at $x_0$, a linear map $\mathbb{R}^2 \to \mathbb{R}$, evaluated at $v$). As you can imagine, the smaller $v$ is, the closer it approximates the actual difference $f(x_0 + v) - f(x_0)$, for then the tangent plane lies very close to the graph of $f$. This is what we mean when we say that the derivative is the best linear approximation to $f$.

It should also help to understand that when we say that the derivative at $x_0$ is a linear map $\mathbb{R}^n \to \mathbb{R}^m$, we really mean that it is a linear map from the tangent space of $\mathbb{R}^n$ at $x_0$ to the tangent space of $\mathbb{R}^m$ at $f(x_0)$. For example, you should think of the derivative of a curve $f:\mathbb{R} \to \mathbb{R}^2$ at $x_0$ as a map that takes a small time $t$ and gives you your location if you followed the tangent line to that curve at $f(x_0)$ for the time $t$.

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