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).
DIFFERENTIAL FORMS AND INTEGRATION 3 Thus if we reverse a path from a to b to form a path from b to a, the sign of the integral changes. This is in contrast to the unsigned definite integral R [a,b] f(x) dx, since the set [a,b] of numbers between a and b is exactly the same as the set of numbers between b and a.
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