Lagrange mean value theorem for two variables - visualization and intuition behind it 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?
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

$$\frac{f(\vec{p_o} + \vec h)-f(\vec {p_o})}{|\vec h|}=\frac{\partial f(\vec p_{\theta})}{\partial \vec v}$$
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

$$\frac{f(\vec{p_o} + \vec h)-f(\vec {p_o})}{|\vec h|}=\frac{\partial f(\vec p_{\theta})}{\partial \vec v}$$