Mathematical Analysis II pp 365-418 | Cite as

# Integral calculus on curves and surfaces

## Abstract

With this chapter we conclude the study of multivariable integral calculus. In the first part we define integrals along curves in ℝ^{m} and over surfaces in space, by considering first real-valued maps, then vector-valued functions. Integrating a vector field’s tangential component along a curve, or its normal component on a surface, defines line and flux integrals respectively: these are interpreted in Physics as the work done by a force along a path, or the flow across a membrane immersed in a fluid. Curvilinear integrals rely, de facto, on integrals over real intervals, in the same way as surface integrals are computed by integrating over domains in the plane. A certain attention is devoted to how integrals depend upon the parametrisations and orientations of the manifolds involved.

## Keywords

Tangential Component Surface Integral Real Interval Integral Calculus Flux Integral## Preview

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