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Conclusion
In this chapter, several-variable integration was developed from multiple integrals to the major theorems of vector calculus.
Double and triple integrals extend the one-variable integral to higher-dimensional regions, and Fubini’s theorem converts them into iterated integrals.
The change of variables theorem explains how integrals transform under coordinate maps, with the Jacobian determinant measuring local stretching.
Vector fields and differential forms provide a common language for divergence, curl, circulation, flux, pullbacks, and exterior derivatives.
Chains and orientations make it possible to define line integrals and surface integrals in a systematic way.
General Stokes’ theorem unifies the Fundamental Theorem of Calculus, Green’s theorem, the classical Stokes’ theorem, and the divergence theorem.
