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I hold in fact: (1) That small portions of space are of a nature analogous to little hills on a surface which is on the average flat. (2) That this property of being curved or distorted is continually passed on from one portion of space to another after the manner of a wave. (3) That this variation of curvature of space is really what happens in that phenomenon which we call the motion of matter whether ponderable or ethereal. (4) That in this physical world nothing else takes place but this variation, subject, possibly, to the law of continuity.
—W. K. Clifford (1870)
Everyone who is seriously involved in the pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man.
—Albert Einstein
In Chapter 5 we studied integration over regions in \(\,{\mathbb R}^2\) and \({\mathbb R}^3\). In this chapter we study integration over paths and surfaces. This is basic to an understanding of Chapter 8, in which we discuss the basic relation between vector differential calculus (Chapter 4) and vector integral calculus (this chapter), a relation that generalizes the fundamental theorem of calculus to several variables. This generalization is summarized in the theorems of Green, Gauss, and Stokes.