Integral calculus on curves and surfaces
With this chapter we conclude the study of multivariable integral calculus. In the first part we define integrals along curves in ℝm 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.
KeywordsTangential Component Surface Integral Real Interval Integral Calculus Flux Integral
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