Transcription of DIFFERENTIAL FORMS AND INTEGRATION - UCLA Mathematics
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DIFFERENTIAL FORMS AND INTEGRATION - UCLA Mathematics
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).
4 TERENCE TAO a linear relationship is a linear transformation. Thus, for each xi we shall need a linear transformation ωxi: R n → R that takes an (infinitesimal) displacement ∆xi ∈ Rn as input and returns an (infinitesimal) scalar ωx i(∆xi) ∈ Ras output, representing the infinitesimal “work” required to move from xi to xi+1. (In other
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