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2016•Undergraduate texts in mathematicsOpen access

Integration of Differential Forms

Jerry Shurman

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Abstract

This chapter covers the integration of differential forms over surfaces, culminating in the general fundamental theorem of integral calculus. The general fundamental theorem is often called Stokes’s theorem, but along with generalizing the classical Stokes’s theorem, it also subsumes the divergence theorem (or Gauss’s theorem), Green’s theorem, and the one-variable fundamental theorem. Much of the chapter’s work is algebraic, in particular the result that differential forms innately pass through changes of variable.

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This chapter covers the integration of differential forms over surfaces, culminating in the general fundamental theorem of integral calculus. The general fundamental theorem is often called Stokes’s theorem, but along with generalizing the classical Stokes’s theorem, it also subsumes the divergence theorem (or Gauss’s theorem), Green’s theorem, and the one-variable fundamental theorem. Much of the chapter’s work is algebraic, in particular the result that differential forms innately pass through changes of variable.

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Available abstract

This chapter covers the integration of differential forms over surfaces, culminating in the general fundamental theorem of integral calculus. The general fundamental theorem is often called Stokes’s theorem, but along with generalizing the classical Stokes’s theorem, it also subsumes the divergence theorem (or Gauss’s theorem), Green’s theorem, and the one-variable fundamental theorem. Much of the chapter’s work is algebraic, in particular the result that differential forms innately pass through changes of variable.

Key concepts: Divergence theorem, Fundamental theorem, Fundamental theorem of calculus, Kelvin–Stokes theorem, Mathematics, Gauss, Differential (mechanical device), Variable (mathematics)

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