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)
scalar. (Indeed, one should think of ∆xi as an infinitesimal tangent vector to the ambient space Rn at the point x i.) In the one-dimensional case, we converted the scalar displacement ∆xi into a new number f(xi)∆xi, which was linearly related to the original displacement by a proportionality constant f(xi) depending on the position xi ...
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