Seeking clarification on the difference between mean value theorem for vector functions as given in Stoker and Hormander.

Question

I'm studying Stoker's "Differential geometry" and Hormander's "The analysis of linear partial differential operators I, distribution theory and fourier analysis" on my own.

On p.10, Stoker says that, for a vector function $\vec x(t)$, we can apply mean value theorem for real valued functions to every component and arrive at the following: $$\vec x(t_1) - \vec x(t_0) = (x'_1(\xi_1),x'_2(\xi_2),x'_3(\xi_3)) \cdot (t_1-t_0) = \vec x'(\vec \xi_i) \cdot (t_1 - t_0)$$ For values $\xi_i$ between $t_0$ and $t_1$.

On p.6, Hormander says that, for a vector function $\vec f$, the mean value theorem must be replaced by the mean value inequality given by: $$||\vec f(\vec y) - \vec f(\vec x) || \le |\vec y - \vec x| sup \{ ||\vec f'(\vec x+t(\vec y-\vec x)||, 0 \le t \le 1 \} $$

I think the two expressions contradict each other. Stoker has vectors on both sides. The sides are equalized. By definition, two vectors are equal if their lengths and directions are equal. Therefore, it follows from Stoker's expression, that the norms of the left and right hand sides must be equal. Whereas, Hormander says that the norms of the right hand side and the left hand side are not necessarily equal.

Apart from that, Stoker's expression makes absolute sense to me and this example also makes sense to me. But the two must contradict each other as I explained above.

I'm wondering if anybody can clarify the issue for me, please.

Answer 1

Let us first observe that Hörmander considers a function $\vec f : I \to \mathbb R^n$ defined on an open interval $I \subset \mathbb R$. It is therefore misleading to write the arguments of $\vec f$ in the vectorial form $\vec x, \vec y$ because this suggest that $\vec f$ is defined on some subset of $\mathbb R^k$. Hörmander also writes $f$ instead of $\vec f$, but if want to use the vectorial notation, it is okay. Thus we have

$$||\vec f(y) - \vec f(x) || \le |y - x| \sup \{ ||\vec f'( x+t( y- x)|| : 0 \le t \le 1 \} . \tag{1}$$

There is no contradiction between this and Stoker's formula which we can rewrite as $$\vec f(y) - \vec f(x) = (f'_1(\xi_1),\ldots,f'_n(\xi_n)) \cdot (y-x) = \vec f'(\vec \xi_i) \cdot (y- x) \tag{2}$$ for values $\xi_i$ between $x$ and $y$.

The mean value theorem for real-valued functions $f : I \to \mathbb R$ says that $$f(y) - f(x) = f'(\xi)\cdot (y-x)$$ for some $\xi$ between $x$ and $y$. For vector valued functions $\vec f$ it is in general false that $$\vec f(y) - \vec f(x) = (f'_1(\xi),\ldots,f'_n(\xi)) \cdot (y-x) = \vec f'(\xi) \cdot (y-x) \tag{3}$$ for a single value $\xi$ between $x$ and $y$.

What could be a substitute for $(3)$?

Stoker introduces the vector $\vec f'(\vec \xi_i) = (f'_1(\xi_1),\ldots,f'_n(\xi_n))$. Personally I do not like this notation because $\vec f'(\vec \xi_i)$ is not the derivative of $\vec f$ at a single point between $x$ and $y$, but a vector composed of the derivatives of the coordinate functions $f_i$ at possibly distinct points $\xi_i$ between $x$ and $y$.

Hörmander only considers the vectors $\vec f'(\xi)$ with $\xi$ between $x$ and $y$. Then $$\sup \{ ||\vec f'( x+t( y- x)|| : 0 \le t \le 1 \} = \sup \{ ||\vec f'(\xi)|| : \xi \text{ between } x \text{ and } y \}$$ which gives a relationship in form of $(1)$ between the distance of $\vec f(x), \vec f(y)$ and the set of all intermediate values $\vec f'(\xi)$.

It depends on the specific situation whether $(1)$ or $(2)$ is more useful, but both are correct and have good reasons for their existence.

Answer 2

Maybe it's clearer to take an example where both theorems could apply. Consider the vector-valued function $x(t) = (\cos(t), \sin(t))$ on $[t_0, t_1] = [0, \frac{\pi}{2}]$, so that $x(t)$ sweeps out a quarter-circle.

Stoker's statement says that we can find $\xi_1, \xi_2$ so that $(0, 1) - (1, 0) = (\cos'(\xi_1), \sin'(\xi_2))\cdot \frac{\pi}{2}$. This is true, and the only solution is $\xi_1 = \arcsin(\frac{2}{\pi}), \xi_2 = \arccos(\frac{2}{\pi})$.

Hormander's statement says that $\sup(||x'||)$ on the domain is at least $\frac{||x(t_1) - x(t_0)||}{||t_1 - t_0||} = \frac{\sqrt{2}}{\pi}$. This is definitely true, because $||x'|| = 1$ everywhere in the domain.

The bound on $||x'||$ is not reached, and Stoker's statement can't be used to show that it is reached. Stoker's statement involves evaluating $(x_1'(\xi_1), x_2'(\xi_2))$ where $\xi_1, \xi_2$ are allowed to differ, but Hormander's statement involves evaluating $||x'(t)|| = ||(x_1'(t), x_2'(t))||$, where both coordinates are evaluated at the same $t$.

In general, Stoker's statement applies for functions $\mathbb{R} \to \mathbb{R}^n$, that take a single real parameter and give a vector value. Hormander's statement applies for a function $\mathbb{R}^m \to \mathbb{R}^n$ that take a vector parameter and give a vector result. Additionally, the object $\vec{x'}(\vec{\xi}_i)$ in Stoker's statement does not represent a value of $\vec{f}'$ in Hormander's statement even in cases where they both apply; it is just a short notation for $(x_i'(\xi_i))$.