On the Definitions of Line and Surface Integrals

Question

When motivating the definitions of line and surface integrals, one usually defines the length and area elements \begin{align*} ds &:= \| \vec{r}^{\, '}(t) \| dt, \\ dA &:= \| \vec{\Sigma}_{u} \times \vec{\Sigma}_{v} \| du dv. \end{align*} The "justification" given is that both expressions approximate the actual real length and area elements. So, one gives geometric justifications (lines and parallelograms). However, there are many ways to approximate the actual real length and area elements. One must generally be careful when working with such "infitesimals", e.g. Is value of $\pi = 4$?.

Question : So, I would like to know if there is a more rigorous way of justifying that the definitions of line and surface integrals are really what we want. For example, why not take $$ ds := 0.9999 \cdot\| \vec{r}^{\, '}(t) \| dt $$ or approximate the length element by a slightly curved segment ?

Answer 1

The simplest case is that of curves. When $$\gamma:\quad[a,b]\to{\mathbb R}^n, \qquad t\mapsto{\bf f}(t)$$ is a curve in ${\mathbb R}^n$ parametrized by a continuous map ${\bf f}$ then for any subdivision $${\cal P}:\quad a=t_0<t_1<t_2<\ldots<t_N=b$$ of the parameter interval $[a,b]$ we can compute the elementary euclidean length of the polygonal path through the points ${\bf x}_k:={\bf f}(t_k)$ $\>(0\leq k\leq N)$. This length is given by $$L_{\cal P}(\gamma)=\sum_{k=1}^N|{\bf x}_k-{\bf x}_{k-1}|=\sum_{k=1}^N|{\bf f}(t_k)-{\bf f}(t_{k-1}|\ .$$ It is then natural to define the "real" length of $\gamma$ by putting $$L(\gamma)=\sup_{\cal P}L_{\cal P}(\gamma)\ .$$ The sup here intuitively corresponds to our geometric idea, and is the simplest kinf of "limit" under the given circumstances.

It is then a theorem requiring about two pages for proof that when ${\bf f}$ is in fact continuously differentiable then this length $L(\gamma)$ can be computed as an integral: $$L(\gamma)=\int_a^b|{\bf f}'(t)|\>dt\ .\tag{1}$$ A "differential" form of equation $(1)$ is the formula $$ds=|{\bf f}'(t)|\>dt\ ,$$ it reminds one of the fact that $L(\gamma)$ should not depend on the chosen parametrization. I won't go into this here.

Answer 2

These elements satisfy two conditions:

(1) When changing coordinates (i.e. reparametrizing the curve or surface), both of these quantities transform in such a way that the values of integral quantities computed using these elements does not depend on the choice of coordinates (parametrization) used.

(2) They give the usual formulas for lengths of lines and areas of rectangles (with any parametrization, though the standard ones will certainly do, provided (1) holds).

Note that your proposed definition for arc length satisfies (1) but not (2); indeed "lengths" are independent of your choice of parametrization but your line segments will not have the "right" length.