By Michael J. Crowe
Concise and readable, this article levels from definition of vectors and dialogue of algebraic operations on vectors to the concept that of tensor and algebraic operations on tensors. It also includes a scientific research of the differential and fundamental calculus of vector and tensor services of house and time. Worked-out difficulties and ideas. 1968 version
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Extra info for A history of vector analysis : the evolution of the idea of a vectorial system
A These proofs do not apply to the square wave series of Chap. III, n◦ 2 and n◦ 11, but one can always examine what the results might mean in this case. To reduce to a Fourier series of period 1, one has to replace x by 2πx in the series cos x − cos 3x/3 + cos 5x/5 − . e. 8) = cos 2πx − cos(6πx)/3 + cos(10πx)/5 − . . = [e1 (x) + e−1 (x)] /2 − [e3 (x)/3 + e−3 (x)/3] /2 + . . 9) = −π/4 for 1/4 < |x| < 3/4, and by periodicity for the other values of x. 8), 1/4 ap = −1/4 = = = e−2πipx dx − 3/4 e−2πipx dx = 1/4 e−3πip/2 − e−πip/2 e−πip/2 − eπip/2 − = −2πip −2πip eπip/2 − e−πip/2 /2πip − e−πip eπip/2 − e−πip/2 /2πip = [1 − (−1)p ] sin(pπ/2)/πp, zero if p is even, and equal to 2(−1)(p−1)/2 /πp if p is odd; since we omitted a factor π/4, we ﬁnally have ap = 0 (p even) or (−1)(p−1)/2 /2p (p odd), which agrees with (8).
The essential tool is a famous theorem which would have been of great use to Cauchy: Dini’s Theorem. 16 Let (fn ) be a monotone sequence of continuous realvalued functions deﬁned on a compact set K ⊂ C and converging simply to a limit function f . Then f is continuous if and only if the fn converge uniformly on K. We can assume that the given sequence is increasing, whence f (x) = sup fn (x) for every x ∈ K. For every r > 0 and every a ∈ K, we then have f (a) ≥ fn (a) > f (a) − r for n large. If f is continuous, this relation is, for n given, again true on a neighbourhood of a.
One passes trivially from lsc to usc by remarking that ϕ is lsc ⇐⇒ −ϕ is usc. You may therefore, if it appeals to you, translate all the properties of the lsc functions into properties of the usc functions: it is enough to reverse the sense of all the inequalities and to replace the word “increasing” by the word “decreasing” everywhere. There is a theorem on the maximum, and not on the minimum, for usc functions on a compact set. Every usc function majorised by a continuous function is the lower envelope of the continuous functions which majorise it; this is always the case of a usc function on a compact interval by the maximum theorem.
A history of vector analysis : the evolution of the idea of a vectorial system by Michael J. Crowe