Distributions

Chapter 10Line Integral of a Continuous Field

This chapter is dedicated to line integrals of continuous vector fields along a path, which is an essential tool for constructing primitives. Some classical properties for fields with values in a Banach space are extended here to values in a Neumann space.

After having defined the line integral of a field along a path we give the following properties.

1. — Line integrals concatenate, namely (Theorems 10.13 and 10.15).
2. — Every concatenation of ¹ paths, namely every piecewise ¹ path, can be reparameterized into a ¹ path (Theorem 10.16), without changing the line integral (Theorem 10.17).
3. — The line integral of a gradient along a closed path Γ is zero, that is (Theorem 10.8).
4. — The line integral of a local gradient, namely of a field q that is ...