Curves and line integrals
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).
KeywordsDifferential Form Parametric Representation Thermal System Line Integral Adiabatic Curve
Unable to display preview. Download preview PDF.