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

See also

• Gradient
• Fréchet derivative