User:Saippuakauppias/Fréchet

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The total derivative (ou derivative in the au sense of Fréchet) exists and equals the gradient of f of a ( $\mathbf{\nabla f(a)}$ ) if $\lim_{h \to 0}{{f(\mathbf{a+h}) - f(\mathbf{a}) - \langle \mathbf{\nabla f(\mathbf{a})}, \mathbf{h} \rangle} \over \|\mathbf{h}\|}= 0$ .

[edit] Idea

$f(\mathbf{a+h}) = f(\mathbf{a}) + \langle \mathbf{\nabla f(\mathbf{a})}, \mathbf{h} \rangle + r(\mathbf{h})$

As example R², $f (a + h)$ , where h is a very small number, equals the sum:

$f (a)$
$g (h)$ where $g(x) = \frac{df}{dx}(a) *x$ (straight line with gradient of the function at the point $(a,f(a)) \,$ )
the rest $r (h)$ which depends only of h