Functions of Several Variables pp 245-274 | Cite as

# Curves and line integrals

## 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## Preview

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