${\mathbb {R}^{p}}\rightarrow \mathbb {R}$ functions

Abstract

In mathematical analysis, points of the plane are associated with ordered pairs of real numbers, and the plane itself is associated with the set $\mathbb {R}\times \mathbb {R}=\mathbb {R}^2$. We will proceed analogously in representing three-dimensional space. The coordinate system in three-dimensional space can be described as follows. We consider three lines in space intersecting at a point that are mutually perpendicular, which we call the x-, y-, and z-axes. We call the plane spanned by the x- and y-axes the xy-plane, and we have similar definitions for the xz- and yz-planes. We assign an ordered triple (a, b, c) to every point P in space, in which a, b, and c denote the distance (with positive or negative sign) of the point from the yz-, xz-, and xy-planes, respectively. We call the numbers a, b, and c the coordinates of P. The geometric properties of space imply that the map $P\mapsto (a,b, c)$ that we obtain in this way is a bijection. This justifies our representation of three-dimensional space by ordered triples of real numbers.

1.11 Appendix: Tangent Lines and Tangent Planes

In our previous investigations we introduced the notions of tangent lines and tangent planes in connection with approximations by linear functions. However, the intuitive notion of tangent lines also involves the idea that tangents are the “limits of the secant lines.” Let, for example, f be a one-variable function differentiable at a. The slope of the line (the “secant”) intersecting the graph of f at the points (a, f(a)) and (x, f(x)) is ${(f(x)-f(a))/(x-a)}$. This slope converges to $f'(a)$ as $x\rightarrow a$, and thus the secant “converges” to the line with slope $f'(a)$ that contains point (a, f(a)), i.e., to the tangent line. More precisely, if x converges to a from the right or from the left, then the half-line with endpoint (a, f(a)) that intersects (x, f(x)) “converges” to one of the half-lines that are subsets of the tangent and lie above $[a,\infty )$ or $(-\infty , a]$, respectively. This property will be used for a more general definition of the tangent.

Let $x_0$ and x be different points of ${\mathbb {R}^{p}}$. The half-line $\overrightarrow{x_0 x}$ with endpoint $x_0$ and passing through x consists of the points $x_0 +t(x-x_0 )$ $(t\in \mathbb {R},\ t\ge 0)$. We say that the unit vector $(x-x_0 )/|x-x_0 |$ is the direction vector of this half-line. Let $x_n \rightarrow x_0$ and $x_n \ne x_0$, for every n, and let ${(x_n -x_0 )/|x_n -x_0 |} \rightarrow v$. In this case we say that the sequence of half-lines $\overrightarrow{x_0 x_n }$ converges to the half-line $\{ x_0 +tv:t\ge 0\}$.

Let $H\subset {\mathbb {R}^{p}}$, and let $x_0 \in H'$. If $x_n \in H\setminus \{ x_0 \}$ and $x_n \rightarrow x_0$, then by the Bolzano–Weierstrass theorem (Theorem 1.9), the sequence of unit vectors ${(x_n -x_0 )/|x_n -x_0 |}$ has a convergent subsequence. We say that the contingent of the set H at $x_0$ is the set of vectors v for which there exists a sequence $x_n \in H\setminus \{ x_0 \}$ such that $x_n \rightarrow x_0$ and ${(x_n -x_0 )/|x_n -x_0 |} \rightarrow v$. We denote the contingent of the set H at $x_0$ by $\mathrm{Cont}\, (H;x_0 )$. It is clear that $\mathrm{Cont}\, (H;x_0 )\ne \emptyset$ for every ${x_0 \in H'}$.

In the next three examples we investigate the contingents of curves. By a curve we mean a map $g:[a, b] \rightarrow {\mathbb {R}^{p}}$ (see [7, p. 380]).

Example 1.109.

1. If the single-variable function f is differentiable at a, then $\mathrm{Cont}\, (\mathrm{graph}~f ; (a, f(a)))$ contains exactly two unit vectors, namely the vector

$$\left( \frac{1}{\sqrt{1+(f'(a))^2 }} ,\frac{f'(a)}{\sqrt{1+(f'(a))^2 } } \right) $$

with slope $f'(a)$ and its negative.

2. Let $g:[a, b]\rightarrow {\mathbb {R}^{p}}$ be a curve, and let g be differentiable at $t_0 \in (a, b)$ with $g'(t_0 )\ne 0$. The contingent of the set $\Gamma =g([a, b])$ at $g(t_0 )$ contains the unit vectors $\pm g'(t_0 )/|g'(t_0 )|$. Indeed, if $t_n \rightarrow t_0$, then

$$\frac{g(t_n )-g(t_0 )}{t_n -t_0 }\rightarrow g'(t_0 ).$$

We have

$$ \left| \frac{g(t_n )-g(t_0 )}{t_n -t_0 } \right| \rightarrow |g'(t_0 )|, $$

which implies

$$ \frac{g(t_n )-g(t_0 )}{|g(t_n )-g(t_0 )|} = \frac{(g(t_n )-g(t_0 ))/(t_n -t_0 )}{|g(t_n )-g(t_0 )| /(t_n -t_0 )} \rightarrow \frac{g'(t_0 )}{|g'(t_0 )|} $$

if $t_n >t_0$. Therefore, $g'(t_0 )/|g'(t_0 ) |\in \mathrm{Cont}\, (\Gamma , g(t_0 ))$. If $t_n$ converges to $t_0$ from the left-hand side, we get $-g'(t_0 )/|g'(t_0 )| \in \mathrm{Cont}\, (\Gamma , g(t_0 ))$ in the same way.

3. Let g be a curve that passes through the point $g(t_0 )$ only once, i.e., $g(t)\ne g(t_0 )$ for every $t\ne t_0$. It is easy to see that $g(t_n ) \rightarrow g(t_0 )$ is true only if $t_n \rightarrow t_0$. If we also assume that $g'(t_0 )\ne 0$, then we obtain that the contingent $\mathrm{Cont}\, (\Gamma , g(t_0 ))$ consists of the unit vectors $\pm g'(t_0 )/|g'(t_0 )|$.

The examples above motivate the following definition of the tangent.

Definition 1.110.

Let $x_0 \in H'$ , and let $|v|=1$. We say that the line $\{ x_0 +tv:t\in \mathbb {R}\}$ is the tangent line of the set H at the point $x_0$ if $\mathrm{Cont}\, ( H;x_0 )=\{ v,-v\}$.

By this definition, the graph of the function f has a tangent line at the point (a, f(a)) not only when f is differentiable at a, but also when $f'(a)=\infty$ or $f'(a)=-\infty$. On the other hand, if $f'_- (a)=-\infty$ and $f'_+ (a)=\infty$, then $\mathrm{graph}~f$ does not have a tangent line at (a, f(a)).

We can easily generalize Definition 1.110 to tangent planes.

Definition 1.111.

Let $x_0 \in H'$, and let S be a plane containing the origin (i.e., let S be a two-dimensional subspace). We say that a plane $\{x_0 +s:s\in S\}$ is the tangent plane of the set H at the point $x_0$ if $\mathrm{Cont}\, ( H;x_0 )$ consists of exactly the unit vectors of S.

Let the function $f:\mathbb {R}^2\rightarrow \mathbb {R}$ be differentiable at $(a, b)\in \mathbb {R}^2$. It is not very difficult to show (though some computation is involved) that the contingent of the set $\mathrm{graph}~f$ at the point (a, b, f(a, b)) consists of the unit vectors $(v_1 , v_2 , v_3 )\in \mathbb {R}^3$ for which $v_3 =D_1 f(a, b)v_1 +D_2 f(a, b)v_2$.

Then by Definition 1.74, the plane

$$ z=D_1 f(a, b) (x-a)+D_2 f(a,b) (y-b)+f(a, b) $$

is the tangent plane of the graph of f at (a, b, f(a, b)). One can see that this plane is the tangent plane of the graph of f according to Definition 1.111 as well.

We can define the notion of tangent hyperplanes in ${\mathbb {R}^{p}}$ similarly. One can show that for a graph of a function, the notion of tangent hyperplane according to Definition 1.75 corresponds to Definition 1.111, generalized to ${\mathbb {R}^{p}}$.