User:Saippuakauppias/Fréchet

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 $$\lim_{h \to 0}{{f(\mathbf{a+h}) - f(\mathbf{a}) - \langle \mathbf{\nabla f(\mathbf{a})}, \mathbf{h} \rangle} \over     \|\mathbf{h}\|}= 0 $$.

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

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