Tag Archives: orientation of curve

Math2111: Chapter 3: Recommended reading: Line integrals and orientation

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We discuss in what sense the scalar line integrals do not depend on the orientation of the curve.

Let us start with integrals \int_a^b f(t)\,\mathrm{d} t where f is a Riemann integrable function. Let a \textless b and let P_n=\{x_0,\ldots, x_n\} be such that a=x_0 \textless x_1 \textless x_2 \textless \cdots \textless x_{n-1} \textless x_n =b, let x_{i-1} \le t_i \le x_{i} for 1\le i \le n and \delta(P_n)=\max_{1\le i\le n} x_{i+1}-x_i. We can now form the Riemann sum

\displaystyle \mathcal{S}(P_n) = \sum_{i=1}^n f(t_i) (x_i-x_{i-1}).

By considering the limit \lim \mathcal{S}(P_n) where n\to\infty such that \delta(P_n) \to 0, we obtain the Riemann integral \int_a^b f(t)\,\mathrm{d} t. Continue reading