$\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$)?

Question

In a section discussing global maxima/minima, my textbook says the following:

Simply stated, $\mathbf{x}_0$ and $\mathbf{x}_1$ are points where $f$ assumes its largest and smallest values. As in one-variable calculus, these points need not be uniquely determined. Suppose now that $D = U \cup \partial U$, where $U$ is open and $\partial U$ is its boundary. If $D \subset \mathbb{R}^2$, we suppose that $\partial U$ is a piecewise smooth curve; that is, $D$ is a region bounded by a collection of smooth curves -- for example, a square or the sets depicted in figure 3.3.7.

If $\mathbf{x}_0$ and $\mathbf{x}_1$ are in $U$, we know from Theorem 4 that they are critical points of $f$. If they are in $\partial U$, and $\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$), then they are maximum or minimum points of $f$ viewed as a function of $\partial U$. These observations provide a method of finding the absolute maximum and minimum values of $f$ on region $D$.

Page 181, Vector Calculus, by Marsden and Tromba.

I don't understand what is meant by

(i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$)

I understand that a "smooth" curve (or function) is a curve (or function) where derivatives of all orders exist and are continuous (in other words, the function is said to be of class $C^\infty$). However, I do not understand at all what is meant by the authors above statement.

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

A smooth path is a smooth map $c : [a,b] \to \mathbb{R}^2$. Its image is the set $c([a,b]) \subset \mathbb{R}^2$. $c' \ne 0$ is a little imprecise, it should be $c'(t) \ne 0$ for all $t \in [a,b]$.

I suppose that $D$ is assumed to be compact, hence also the boundary $\partial U$ must be compact. In general it may have more than one connected component, but here it is required that $\partial U$ is the image of a smooth path, hence has exactly one connected component. Moreover, it should be required that $c(a) = c(b)$ (i.e. that $c$ is a closed path) and $c(t_1) = c(t_2)$ implies $t_1 = t_2$ or $\{ t_1, t_2 \} = \{ a, b \}$ so that $\partial U = c([a,b])$ is a homeomorphic copy of a circle.

Answer 2

$\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}′ \neq 0$)

Here the authors explain what they mean by a smooth curve. This is what Paul already explained perfectly.

then they are maximum or minimum points of $f$ viewed as a function of $\partial U$.

Since the boundary curve $\partial U$ is smooth, we can find the extrema on the boundary by considering $f(\mathbf{c}(t))$. This function is smooth and therefore we can derive w.r.t. to the parameter $t$ in order to find the extrema.

For this the authors require that $\mathbf{c}'(t)$ is nowhere zero. Indeed, deriving $f(\mathbf{c}(t))$ gives $\nabla f(\mathbf{c}(t))\cdot\mathbf{c}'(t)$. Note that the derivative is automatically zero at points where $\mathbf{c}'(t)=0$, so these points are not automatically maxima or minima of $f$ along the curve. If you want to find the critical points of $f$, you have to separately at

• the parts where $\mathbf{c}'(t)\neq 0$,
• and the points where $\mathbf{c}'(t)=0$.

Geometrically points where $\mathbf{c}'(t)=0$ can be sharp edges of a curve, so these have to be dealt with separately. The authors basically assume there are no such pesky points.

These observations provide a method of finding the absolute maximum and minimum values of $f$ on region $D$.

Consider the extrema on the interior $U$ and also the extrema on the boundary curve $\partial U$. If you take the extrema on $U$ or $\partial U$ which has the biggest function value, then you have found the global maximum on the whole region $D$. Similarly, the extrema of $U$ or $\partial U$ where the value of $f$ is minimal is the global minima on $D$.