Line Integrals and Surface Integrals

The previous chapters integrated over regions; this chapter turns to integrating over curves and surfaces. The definitions look as though they have suddenly become complicated — where does the $\mathrm{d}s$ in a line integral come from, and what does $\mathrm{d}S\,dx$ in a surface integral mean — but in essence they are all “accumulating over the object being integrated”.

The difference between the line integral of the first kind and that of the second kind lies in whether there is an orientation: the former accumulates only the values, whereas the latter accumulates the projections of a vector in a specified direction. The surface integrals split in the same way, into the first kind (with respect to area) and the second kind (with respect to coordinates), and both are built on parametrisation.

The core of this chapter is three formulas: Green’s theorem turns the integral along a plane closed curve into a double integral, Gauss’s theorem turns the integral over a closed surface into a triple integral, and Stokes’s theorem turns a surface integral into a line integral. They relate integrals of different dimensions, they are the most powerful tools in multivariable calculus, and they are the foundation for the study of electromagnetism and fluid mechanics.