By Snieder R.
Unlike conventional textbooks for college students and execs within the actual sciences, this booklet offers its fabric within the type of difficulties. the second one variation includes new chapters on dimensional research, variational calculus, and the asymptotic overview of integrals. The booklet can be utilized via undergraduates and lower-level graduate scholars. it will probably function a stand-alone textual content, or as a resource of difficulties and examples to enrich different textbooks. First version Hb (2001): 0-521-78241-4 First variation Pb (2001): 0-521-78751-3
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Extra info for A Guided Tour of Mathematical Methods: For the Physical Sciences
10). This series hides an intriguing result. 10) suggest that a function f (x) is speciﬁed for all values of its argument x when all the derivatives are known at a single point x = 0. This means that the global behavior of a function is completely contained in the properties of the function at a single point. In fact, this is not always true. 9) is an inﬁnite series, and the sum of inﬁnitely many terms does not necessarily lead to a ﬁnite answer. 15). A series can only converge when the terms go to zero as n → ∞, because otherwise every additional term changes the sum.
2. For such a change δr, the corresponding change δ f is by deﬁnition equal to zero, hence (∇ f · δr) = 0. Problem f Use the properties of the inner product of two vectors to show that this identity implies that the vectors ∇ f and δr are for this special perturbation perpendicular: ∇ f ⊥ δr . This last identity of course only holds when the step δr is taken along the surface where the function f is constant. 1 Properties of the gradient vector f = const. 49 r Fig. 2 Contour lines deﬁned by the condition f = const.
A series can only converge when the terms go to zero as n → ∞, because otherwise every additional term changes the sum. 15) are given by x n ; these terms only go to zero as n → ∞ when |x| < 1. 9) only converges when x is smaller than a certain critical value called the radius of convergence. Details on the criteria for the convergence of series can be found in for example Boas  or Butkov . The second reason why the derivatives at one point do not necessarily constrain the function everywhere is that a function may change its character over the range of parameter values that is of interest.
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