Some Mathematical Methods of PhysicsThis well-rounded, thorough treatment for advanced undergraduates and graduate students introduces basic concepts of mathematical physics involved in the study of linear systems. The text emphasizes eigenvalues, eigenfunctions, and Green's functions. Prerequisites include differential equations and a first course in theoretical physics. The three-part presentation begins with an exploration of systems with a finite number of degrees of freedom (described by matrices). In part two, the concepts developed for discrete systems in previous chapters are extended to continuous systems. New concepts useful in the treatment of continuous systems are also introduced. The final part examines approximation methods — including perturbation theory, variational methods, and numerical methods — relevant to addressing most of the problems of nature that confront applied physicists. Two Appendixes include background and supplementary material. 1960 edition. |
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Página vii
16 1.9 Linear Independence 17 Chapter 2 Solution for Diagonalizable Matrices . . . 21 2.1 Solution by Taylor Series . . . . . . . . 21 2.2 Eigenvalues and Eigencolumns . . . . . . . 22 2.3 Superposition .
16 1.9 Linear Independence 17 Chapter 2 Solution for Diagonalizable Matrices . . . 21 2.1 Solution by Taylor Series . . . . . . . . 21 2.2 Eigenvalues and Eigencolumns . . . . . . . 22 2.3 Superposition .
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1.9 Linear Independence Up to now, square matrices have been separated into two categories, singular matrices and nonsingular matrices. It is further possible to define an index, the rank of a matrix, which indicates the degree of ...
1.9 Linear Independence Up to now, square matrices have been separated into two categories, singular matrices and nonsingular matrices. It is further possible to define an index, the rank of a matrix, which indicates the degree of ...
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A nonsingular matrix may be characterized by the statement that any column y can be written as a linear superposition of the columns of the matrix. To proceed further, it is useful to introduce the concept of linear independence.
A nonsingular matrix may be characterized by the statement that any column y can be written as a linear superposition of the columns of the matrix. To proceed further, it is useful to introduce the concept of linear independence.
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Thus, if A has less than n linearly independent eigencolumns, the resultant linear superposition will depend on less than n arbitrary constants, whereas an arbitrary column is specified by all n elements. If A does have n linearly ...
Thus, if A has less than n linearly independent eigencolumns, the resultant linear superposition will depend on less than n arbitrary constants, whereas an arbitrary column is specified by all n elements. If A does have n linearly ...
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A further result, proven in this section, is that A has at least as many linearly independent eigencolumns as it has distinct eigenvalues, so that if A is nondegenerate it is certain that A may be diagonalized.
A further result, proven in this section, is that A has at least as many linearly independent eigencolumns as it has distinct eigenvalues, so that if A is nondegenerate it is certain that A may be diagonalized.
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applied approximate arbitrary base vectors basis Bessel function boundary conditions Chap chapter coefficients column commute complete consider constant continuous systems contour corresponding cylindrical functions defined definition denoted determinant diagonal diagonalizable differential equation Dirac notation domain eigen eigencolumns eigenfunctions eigenvalue equation eigenvector elements evaluate expansion find finite number first follows formula Fourier given Green’s function Hence Hermitian matrix Hermitian operator infinite integral Introduction inverse Laplacian linear operator linearly independent lowest eigenvalue matrix McGraw-Hill Book Company membrane method multiplication nonsingular normal normal matrix Note number of degrees obtained orthonormality conditions perturbation plane procedure QUANTUM MECHANICS relations representation result Ritz method satisfies satisfy scattering solve specified spherical spherical harmonics string Substitution theorem theory tion trial functions vanish variable vector space verified wave write written yields York zero
Información bibliográfica
| Título | Some Mathematical Methods of Physics Dover Books on Physics |
| Autores | Gerald Goertzel, Nunzio Tralli |
| Editor | Courier Corporation, 2014 |
| ISBN | 048679332X, 9780486793320 |
| Largo | 320 páginas |
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