# Curves and line integrals

• Wendell Fleming
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

Consider a vector-valued function g, from a 1-dimensional interval J into E n with derivative g’(t) ≠ 0. One can think of the point x = g(t) as traversing a curve while t traverses J from left to right. We do not call g itself a curve. Instead, any vector-valued function f obtained from g by a suitable parameter change is regarded as representing the same curve γ as g. In Section 6.3 we define the concept of differential form ω of degree 1. Then the line integral of a differential form ω along a curve γ is defined in Section 6.4. The differential df of a function f is called an exact differential form of degree 1. It is shown that the line integral of ω depends just on the endpoints of γ if and only if ω is exact (Theorem 6.1).

## Keywords

Differential Form Parametric Representation Thermal System Line Integral Adiabatic Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.