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 crucial calculus of vector and tensor services of house and time. Worked-out difficulties and strategies. 1968 version
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Extra info for A history of vector analysis : the evolution of the idea of a vectorial system
48). 49). 6). Exercises 1. The complementary error function is defined by Show that where Also show that for Exercises 43 2. The logarithmic integral is defined by where the integral is taken along any path belonging to the plane with two cuts along the segments (—oo,0] and [1, oo) of the real axis. Show that where for |z| < 1 and |arg z| < 3. The complementary incomplete gamma function F(a, x) is defined by Show that when a and x are positive, as x -> +00. Furthermore, show that the nth error term is bounded in absolute value by the (n + l)th term of the series and has the same sign, when n > a — 1.
46) at v = \ and let It is easily shown that By writing with 28 I Fundamental Concepts of Asymptotics we also have where and F(a, x) being the complementary incomplete Gamma function defined by On making the substitution t = x(l + T), the last integral can be written as Since it follows that if a > 1 and x > a — 1. Therefore if x > 2N - 1. 53). 51) then yields 5. 57). 45), there still remains the question of how good are these bounds. 59) shows immediately that the latter result is definitely superior to the former, even without numerical computation.
Two functions f(z) and #(2) defined in some neighborhood of z0 are said to be asymptotically equal, written if for every n > 0. Two functions having the same asymptotic expansion are asymptotically equal, and the converse is also true. Even this degree of generality is not sufficient to describe the asymptotic behavior of many of the known functions of mathematics. The form as z -> 20, with the meaning as z->z0, where N^ N2,... are arbitrary fixed integers, must often be used to give asymptotic information for many of the higher transcendental functions.
A history of vector analysis : the evolution of the idea of a vectorial system by Michael J. Crowe