The two-variable version of the Lagrange mean-value theorem says that given a function $f(x,y)$,
$$f(\vec{p_o} + \vec h)-f(\vec {p_o})=df(\vec {p_{\theta}})$$
Where $\vec p_{\theta}=\vec p_o + \theta \vec h $
with $\theta \in (0,1).$
I don't understand this theorem, neither do I see the intuition behind it.
Is there a simple way to visualize it? If not, could you come up with a practical example where this theorem could be used?
Best Answer
For a function $f(x,y):\mathbb{R^2}\to \mathbb{R}$ the MVT
$$f(\vec{p_o} + \vec h)-f(\vec {p_o})=df(\vec {p_{\theta}})$$
has an interpretation very similar to MVT for functions of one variable.
Indeed we have that
$$df(\vec {p_{\theta}})=|\vec h|\frac{\partial f(\vec p_{\theta})}{\partial \vec v}$$
with $\vec v = \frac{\vec h}{|\vec h|}$ and thus
$$df(\vec {p_{\theta}})=|\vec h|\frac{\partial f(\vec p_{\theta})}{\partial \vec v}=|\vec h|\langle\nabla f(\vec p_{\theta}),\frac{\vec h}{|\vec h|}\rangle =\langle\nabla f(\vec p_{\theta}),\vec h\rangle$$
From the first expression the MVT can be expressed as
$$\frac{f(\vec{p_o} + \vec h)-f(\vec {p_o})}{|\vec h|}=\frac{\partial f(\vec p_{\theta})}{\partial \vec v}$$