6 - Introduction to Vector Analysis

Abstract

This chapter provides an overview of vector analysis. If Ω is a region in ℝ2, then F is a vector field in ℝ2 if F assigns a unique vector F(x) in ℝ2 to every x in Ω. If F is integrated along any piecewise smooth curve C joining two points, then the line integral is independent of the path. Assuming F to be continuous in a region Ω in ℝ2, then F is the gradient of a differentiable function f if and only if for any piecewise smooth curve C lying in Ω, the line integral is independent of the path. If F is the gradient of a differentiable function f, then F is said to be exact. Green's theorem shows how the line integral of a function around the boundary of a region is related to a double integral over that region. Like the computations of arc length and surface area, the direct evaluation of a surface integral will often be impossible as it will involve integrals of functions for which antiderivatives cannot be found.