$$ \newcommand{\qa}{P} \newcommand{\qb}{Q} \newcommand{\da}{dP} \newcommand{\db}{dQ} \newcommand{\positiverealnumbers}{\mathbb{R}_+} \newcommand{\realnumbers}{\mathbb{R}} \newcommand{\naturalnumbers}{\mathbb{N}} \newcommand{\positiveintegers}{\mathbb{Z}_+} $$
We frequently solve geometrical and physical problems by obtaining an approximate expression for differential $\da$ in terms of differential $\db$ and then integrating $\da$ to obtain $\qa$. We assume that the expression for $\qa$ is exact even though we used an approximate formula for $\da$. This is justified by saying that the differentials are infinitely small quantities. For example, when we derive an expression for the area of a circular disc (see example 1) we set $dA = 2 \pi r dr$ which is an approximate expression when the diffentials are interpreted as real numbers. In this article we try to define a method for computing $\qa$ so that we don't need approximate expressions in the derivation.
Theorem 1
Let $a,b \in \realnumbers$ and $a < b$. Suppose that $$ \Delta f = g(x) \Delta x + h(x, \Delta x) $$ for all $x \in \realnumbers$ and $\Delta x \in \positiverealnumbers$ for which $x, x + \Delta x \in [a, b]$. Suppose also that $g$ is Riemann integrable and for all $\varepsilon \in \positiverealnumbers$ there exists $R \in \positiverealnumbers$ so that $$ \left\vert \frac{h(x, \Delta x)}{\Delta x} \right\vert < \varepsilon \tag{1} $$ for all $x, x + \Delta x \in [a, b]$, $0 < \vert \Delta x \vert < R$. Let $t \in [a, b]$. Let $n \in \positiveintegers$, and define $\Delta' x := \frac{t - a}{n}$ and $x_i := a + \frac{i}{n} \Delta' x$ $i \in \naturalnumbers$, $i \leq n$. Define also $\Delta f_i := f(x_{i+1}) - f(x_i)$ where $i \in \naturalnumbers$, $i < n$. Define $$ f(t) := \lim_{n \to \infty} \sum_{i=0}^{n-1} \Delta f_i $$ Now $$ f(t) = \int_a^t g(x) dx $$ and $df = g(x) dx$.
Proof
Let $$ f(t) := \lim_{n \to \infty} \sum_{i=0}^{n-1} \Delta f_i = \lim_{n \to \infty} \sum_{i=0}^{n-1} g(x_i) \Delta'x + \lim_{n \to \infty} \sum_{i=0}^{n-1} h(x_i, \Delta'x). $$ Now $$ \lim_{n \to \infty} \sum_{i=0}^{n-1} g(x_i) \Delta'x = \int_a^t g(x) dx $$ by the definition of the Riemann integral.
Let $\varepsilon \in \positiverealnumbers$. Choose $R_1 \in \positiverealnumbers$ so that $$ \left\vert \frac{h(x, \Delta x)}{\Delta x} \right\vert < \frac{\varepsilon}{t-a} $$ for all $x, x + \Delta x \in [a, b]$, $0 < \vert \Delta x \vert < R_1$. Let $n \in \positiveintegers$ so that $$ n > \frac{t-a}{R_1} . $$ Now \begin{align*} \left\vert \sum_{i=0}^{n-1} h(x_i, \Delta'x) \right\vert & < \sum_{i=0}^{n-1} \frac{\varepsilon}{t-a} \vert \Delta'x \vert = n \frac{\varepsilon}{t-a} \frac{t-a}{n} = \varepsilon . \end{align*} Hence $$ \lim_{n \to \infty} \sum_{i=0}^{n-1} h(x_i, \Delta'x) = 0 . $$
$$\tag*{$\blacksquare$}$$
Theorem 2
A sufficient condition for inequality (1) is that there exist $S, C \in \positiverealnumbers$ so that $$ \vert h(x, \Delta x) \vert < C \vert \Delta x \vert^2 $$ for all $x, x + \Delta x \in [a, b]$ and $0 < \vert \Delta x \vert < S$.
Proof
Set $R := \min \{ S, \varepsilon / C \}$. Now $$ \left\vert \frac{h(x, \Delta x)}{\Delta x} \right\vert < C \vert \Delta x \vert < \varepsilon . $$
$$\tag*{$\blacksquare$}$$
Note that we can't prove Theorem 1 directly by the Fundamental Theorem of Calculus because we would need to assume that there exists a function $f : [a,b] \to \realnumbers$ for which $f(x + \Delta x) - f(x) = \Delta f(x, \Delta x)$ in order to use it.
Example 1
Derive an expression for the area of a disc whose inner radius is $r_a$ and outer radius $r_b$.
The area to be computed in example 1
Solution
Define $\Delta A$ to be the area of a disc with inner radius $r$ and width $\Delta r$. We have $$ 2 \pi r \Delta r \leq \Delta A \leq 2 \pi (r + \Delta r) \Delta r $$ By setting $g(r) := 2 \pi r$ and $h(r, \Delta r) := 2 \pi (\Delta r)^2$ we get $A = \pi r_b^2 - \pi r_a^2$ by Theorems 1 and 2.
Example 2
Suppose that a particle is moving under influence of a constant force $F = ma$ for time $T$ and the particle is initially at rest. Derive an expression for the kinetic energy of the particle. Assume that the work done by a constant force $F$ is $W = F s$ where $s$ is the distance that the particle moves in the direction of the force. Assume also that the kinetic energy of a particle at rest is $0$.
Solution
We define $\Delta s$ to be the distance that the particle moves in the time interval $[t, t + \Delta t]$. We have $v = at$, $$ a t \Delta t \leq \Delta s \leq a (t + \Delta t) \Delta t , $$ and $$ a (t + \Delta t) \Delta t = a t \Delta t + a (\Delta t)^2 . $$ Set $g(t) := a t$ and $h(t, \Delta t) := a (\Delta t)^2$ and it follows from Theorems 1 and 2 that the distance that the particle moves in time $T$ is $$ s = \int_0^T a t dt = \frac{1}{2} a T^2 $$ By setting $v_f = a T$ we obtain $$ E_k = W = \frac{1}{2} F a T^2 = \frac{1}{2} m a^2 T^2 = \frac{1}{2} m v_f^2 . $$
Do you find this formalism useful?
Tommi Höynälänmaa
\newcommand
s you have in the beginning of your post. Do you really find it more convenient to type\qa
than to typeP
? And is\Bbb R_+
(or even\mathbb R_+
) actually simplified by\positiverealnumbers
? $\endgroup$