Mathematical Methods for Physics and Engineering

Abstract

The new edition of this highly acclaimed textbook contains several major additions, including more than four hundred new exercises (with hints and answers). To match the mathematical preparation of current senior college and university entrants, the authors have included a preliminary chapter covering areas such as polynomial equations, trigonometric identities, coordinate geometry, partial fractions, binomial expansions, induction, and the proof of necessary and sufficient conditions. Elsewhere, matrix decompositions, nearly-singular matrices and non-square sets of linear equations are treated in detail. The presentation of probability has been reorganised and greatly extended, and includes all physically important distributions. New topics covered in a separate statistics chapter include estimator efficiency, distributions of samples, t- and F-tests for comparing means and variances, applications of the chi-squared distribution, and maximum likelihood and least-squares fitting. In other chapters the following topics have been added: linear recurrence relations, curvature, envelopes, curve-sketching, and more refined numerical methods.

Multi-stable composite twisting structure for morphing applications

“…We expand U as a Taylor series and consider its variations dU , for small variations dq and da of q and a, respectively (Riley et al 2006),…”

Symmetry considerations on the stiffness and eigen deformations of plant fibers and structures made of them

“…Group elements may be divided into conjugacy classes, members of any class being connected by similarity transformations X= Y -1 ZY. The conjugacy classes do not overlap [3,5,6].…”

“…Otherwise the representation is Irreducible. A number of theorems apply to Irreducible Representations (Irreps); among others, orthogonality, completeness, and dimensionality theorems [3,5,6]. Using such theorems, a character table can be constructed for any discrete group.…”

“…In the particular case of normal modes, vibrations transforming with the same Irrep are degenerate, and the dimensionality of the Irrep indicates the degree of degeneracy. Abelian groups (all elements commuting) have only one-dimensional Irreps [3,5,6]. …”

“…In other words, properties are retained in symmetry transformations. Further, quite a few properties of structures may be deduced on the basis of the symmetries of the system, using group theory and group representation theory [1][2][3][4][5][6].…”