A history of vector analysis : the evolution of the idea of by Michael J. Crowe PDF

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 examine of the differential and essential calculus of vector and tensor services of area and time. Worked-out difficulties and strategies. 1968 version

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Extra resources for A history of vector analysis : the evolution of the idea of a vectorial system

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25) We shall follow Christ [Ch] with very minor alterations. We make the following definition. 36 Ilfll~. 16. We call a sequence of positive, measurable functions on •m, w = {wk)keZ, Carleson if QeJ:r~ wk(x)dx {keZ;2- _

To prove this claim, for a given f E L 1 fq L 2 and ), > 0, we perform the Calder6n-Zygmund decomposition for f at the level A (cf. [St]). Hence, we can write f -- g + b, where the "bad" b part is decomposed further as b = ~ i bj, where suppbj C QJ e Jr, fQj bj = 0 and ~ j ]QJ[ ~ A-111fllL1. Consequently, Using the vanishing moment property and the precise localization of O's and bj's, we see that this sum has only zero terms for Q ~ QJ. In particular, suppT~(bj) C QJ which, in turn, implies that I{x; IT~(b)l > A}l ~ ~--~ IQil ~ A-111f]lL 1, J and the claim is proved.

33 P r o o f . The only thing that we still have to check is (5). , 2 m - 1, is a left-Riesz basis for X L uniformly in k E Z. Restricting our attention to one dyadic cube Q E ~k and using the explicit expressions of the | we readily see that XQ2 is spanned by XQ~ and 04,1 in the set of C(n)-valued functions on R'* with its natural structure as a left Clifford module. Continuing this inductively, we see that any characteristic function XQi is spanned by XQ1 and O~2,1, 9--, | Q,2,~-1" Now, if f is the restriction to Q of a function from X L, we have 2rn_x i=l The fact that B(f, 1) < ,~ since = 0 implies that /~1 ---- 0.

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