Transcription of DIFFERENTIAL FORMS AND INTEGRATION
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DIFFERENTIAL FORMS AND INTEGRATION
DIFFERENTIAL FORMS AND INTEGRATIONTERENCE TAOThe concept of INTEGRATION is of course fundamental in single-variable , there arethreeconcepts of INTEGRATION which appear in the subject: theindefinite integral f(also known as theanti-derivative), theunsigned definiteintegral [a,b]f(x)dx(which one would use to find area under a curve, or the massof a one-dimensional object of varying density), and thesigned definite integral baf(x)dx(which one would use for instance to compute the work required to movea particle fromatob). For simplicity we shall restrict attention here to functionsf:R Rwhich are continuous on the entire real line (and similarly, when wecome to DIFFERENTIAL FORMS , we shall only discuss FORMS which are continuous on theentire domain). We shall also informally use terminology such as infinitesimal inorder to avoid having to discuss the (routine) epsilon-delta analytical issues thatone must resolve in order to make these INTEGRATION concepts fully three INTEGRATION concepts are of course closely related toeach other in single-variable calculus; indeed, the fundamental theorem of calculus relates the signeddefinite integral baf(x)dxto any one of the indefinite integralsF= fby theformula baf(x)dx=F(b) F(a)(1)while the signed and unsigned integral are related by the simple identity baf(x)dx= abf(x)dx= [a,b]f(x)dx(2)which is valid whenevera one moves from single-variable calcu
generalises to the Lebesgue integral, or more generally to integration on a measure space. Finally, the signed definite integral generalises to the integration of forms, which will be our focus here. While these three concepts still have some relation to each other, they are not as interchangeable as they are in the single-variable setting.
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