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So in the book Analysis on Manifolds by Munkres it's said that the directional derivative is not a good candidate for the generalization of the notion of the derivative of a function defined on $\mathbb{R}^m$.

Definition (1): Let $A \subseteq \mathbb{R}^m$ and let $f : A \to \mathbb{R}^n$. Suppose $A$ contains a neighborhood of $a$. Given $u \in \mathbb{R}^m$ with $u \neq 0$, define $${f'}_a(u) = \lim_{t \to 0}\frac{f(a+tu)-f(a)}{t}$$ provided the limit exists. This limit is called the directional derivative of $f$ at $a$ with respect to $u$.

Munkres in his book states that composites of differentiable functions need not be differentiable using this definition. He goes on to say that the right generalization is given by the following definition

Definition (2): Let $A \subseteq \mathbb{R}^m$, let $f : A \to \mathbb{R}^n$. Suppose $A$ contains a neighborhood of $a$. We say that $f$ is differentiable at $a$ is there exists a $n \times m$ matrix $B$ such that $$\lim_{h \to 0} \frac{f(a+h)-f(a) - B\cdot h}{|h|} = 0$$. The matrix $B$ is called the derivative of $f$ at $a$.

Now in the book Differential Topology by Guillemin and Pollack (and even Topology from the Differentiable Viewpoint by Milnor), the definition of the "derivative" that's used is Definition $(1)$ above. They basically go on to formulate Differential Topology using this definition of the derivative.

In particular they claim that the chain rules exists for Definition $(1)$ of the the derivative. I don't see how this can be possible though based on what Munkres said in his book.

Furthermore from what Munkres said, that Definition $(1)$ is not the correct generalization of the derivative, why have Milnor and Guillemin and Pollack taken the incorrect generalization and used it to (as it seems to me) formulate Differential Topology on it? (I hope that is not too strong a thing to say, because they define tangent spaces using this seemingly incorrect generalization of the derivative)

Who is correct then, Munkres, or G&P and Milnor?

Perturbative
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    You're misreading Guillemin and Pollack. They merely use the fact that if $f$ is differentiable, then you can calculate the derivative map by giving its value on an arbitrary tangent vector $v$ as a directional derivative and computing that. Indeed, they assume a knowledge of basic multivariable analysis as a prerequisite for the book. – Ted Shifrin Jun 22 '18 at 21:55

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The latter two books are working with smooth manifolds and maps (whereas Munkres isn't). For smooth maps, and, indeed, for functions with continuous partial derivatives, both the directional derivatives and the "full" derivative exist and have the expected relationship $f'{}_a(u) = B \cdot u$, but this is not the case for functions without the condition of continuity of partial derivatives.

Essentially, in two and higher dimensions there is more information in the derivative than just the limits found by computation on lines that the partial derivatives and the directional derivatives have, because there are many more paths to look at in taking the limit than just straight lines. A function can have partial derivatives at a point and not be differentiable, and it can also have directional derivatives at a point and not be differentiable there. See, for example, this question and its answers.

Chappers
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