By Michael J. Crowe
Concise and readable, this article levels from definition of vectors and dialogue of algebraic operations on vectors to the idea that of tensor and algebraic operations on tensors. It also includes a scientific research of the differential and vital calculus of vector and tensor features of area and time. Worked-out difficulties and suggestions. 1968 version
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Extra resources for A history of vector analysis : the evolution of the idea of a vectorial system
Denote the numerator by P(z) and the denominator by Q(z). We assume that bm = 0 and an = 0. The degree of the numerator is n and the degree of the denominator is m. The zeros of Q(z) are called the poles of F(z). 2 we noted that, as a consequence of the fundamental theorem of algebra, in the complex plane each polynomial can be factorized entirely into linear factors. The denominator Q(z) can thus be written as Q(z) = bm (z − z 1 )ν1 (z − z 2 )ν2 · · · (z − z k )νk , Order of pole where z 1 , z 2 , .
The partial sum sn is 2 n equal to sn = 1 + z + z + · · · + z . Note that this is a polynomial of degree n. When z = 1, then we see by direct substitution that sn = n + 1. Hence, the geometric series diverges for z = 1, since limn→∞ sn = ∞. Multiplying sn by the factor 1 − z gives (1 − z)sn = 1 + z + z 2 + · · · + z n − z(1 + z + z 2 + · · · + z n ) = 1 − z n+1 . For z = 1 one thus has sn = 1 − z n+1 . 15); so then the series converges with sum equal to 1/(1 − z). We write ∞ zn = n=0 1 1−z if | z | < 1.
12 π 0 1 π 1 2it π te t (e2it ) dt = − 2i 0 2i 0 1 2πi 1 π 2it π = πe + − e dt = 2i 2i 0 2i 1 π π + (e2πi − 1) = . = 2i 4 2i te2it dt = 1 π 2i 0 1 2it e 4 e2it (t) dt π 0 The following inequality is often applied to estimate integrals: b a f (t) dt ≤ b a | f (t) | dt for b ≥ a. 22) For real-valued functions this is a well-known inequality. Here we omit the proof for complex-valued functions. 1). 22) is the following inequality. If | f (t) | ≤ M on the integration interval [a, b], then b a f (t) dt ≤ b a | f (t) | dt ≤ b a M dt = M(b − a).
A history of vector analysis : the evolution of the idea of a vectorial system by Michael J. Crowe