By Morris Kline

ISBN-10: 0486404536

ISBN-13: 9780486404530

Application-oriented creation relates the topic as heavily as attainable to technology. In-depth explorations of the spinoff, the differentiation and integration of the powers of *x*, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric capabilities, logarithmic and exponential services, ideas of integration, polar coordinates, even more. Examples. 1967 variation. resolution advisor on hand upon request.

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Application-oriented creation relates the topic as heavily as attainable to technology. In-depth explorations of the by-product, the differentiation and integration of the powers of x, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric features, logarithmic and exponential features, concepts of integration, polar coordinates, even more.

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**Extra info for Calculus: An Intuitive and Physical Approach (2nd Edition) (Dover Books on Mathematics)**

**Example text**

Proof. We have to prove that Φn (v) := Φ0 (Hn v), where Φ0 is defined in (19), possesses non-degenerate critical points at least for n large. 8 in [6]): for v(t, x) = η(t + x) − η(t − x) Φn (v) = 2πn2 η˙ 2 (t)dt − T Hence we can write √ 12n 48n4 1 Φn √ v = 2 a2 2 πa2 π 2 a22 12 η 2 (t) dt T Ψ(η) = + T η˙ 2 (s) ds − where 2 1 2 1 4 η 2 (s) ds T η˙ 2 (s) ds − T 1 4 a22 2n2 2 + v 2 L−1 v 2 + Ω π2 6 η 2 (t) dt 2 1 48n4 1 R(η) = 2 Ψ(η) + 2 R(η) 2 n a2 n η 2 (s) ds . T (111) 2 T and R : E → R is a smooth functional defined on E := {η ∈ H 1 (T) | η odd}.

V. I. Shulman, Periodic solutions of the equation utt − uxx + u3 = 0, Funct. Anal. Appl. 22, 332–333, 1988. [19] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21, no. 3, 22–37, 95, 1987. [20] S. B. Kuksin, Analysis of Hamiltonian PDEs Oxford Lecture Series in Mathematics and its Applications. 19. Oxford University Press, 2000. [21] S. Kuksin, J. P¨ oschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schr¨ odinger equation, Ann.

P¨ oschel, A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. , 23, 119-148, 1996. [24] J. P¨ oschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. , 71, no. 2, 269-296, 1996. W. Su, Persistence of periodic solutions for the nonlinear wave equation: a case of finite regularity, PhD Thesis, Brown University, 1998. [26] E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys. 3, 479-528, 1990. [27] A.

### Calculus: An Intuitive and Physical Approach (2nd Edition) (Dover Books on Mathematics) by Morris Kline

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