Some Mathematical Methods of PhysicsMcGraw-Hill, 1960 - 300 páginas |
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Página 53
... dual space : To each vector x in the space determined by the basis u , there corresponds a vector x + in the dual space determined by the basis ut such that = ( 4.7 ) where x is the complex conjugate of x1 , and u , * u , = 8 ;; ( 4.8 ) ...
... dual space : To each vector x in the space determined by the basis u , there corresponds a vector x + in the dual space determined by the basis ut such that = ( 4.7 ) where x is the complex conjugate of x1 , and u , * u , = 8 ;; ( 4.8 ) ...
Página 58
... Dual Space If x = Σπ.x. then the corresponding vector in the dual space is With the definition one has or x * = Σx , u + y = Lx y = Lx Y1 = Σ Lisxs Hence Ÿ1 = Xx‚L ;; or y + = x + L + ( 4.21 ) This result suggests the notation y * = x + ...
... Dual Space If x = Σπ.x. then the corresponding vector in the dual space is With the definition one has or x * = Σx , u + y = Lx y = Lx Y1 = Σ Lisxs Hence Ÿ1 = Xx‚L ;; or y + = x + L + ( 4.21 ) This result suggests the notation y * = x + ...
Página 59
... dual space , just as the n - column y and the matrix L are , respectively , the representatives of the vector y and the linear operator in the n - dimen- sional vector space . Note that the above development is equivalent to the ...
... dual space , just as the n - column y and the matrix L are , respectively , the representatives of the vector y and the linear operator in the n - dimen- sional vector space . Note that the above development is equivalent to the ...
Contenido
34 | 12 |
Solution for Diagonalizable Matrices | 21 |
The Evaluation of a Function of a Matrix for an Arbitrary Matrix | 38 |
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approximation arbitrary ax² basis Bessel functions boundary conditions Chap coefficients column consider constant continuous systems contour coordinates corresponding cylindrical functions d²/dx² defined definition denoted determinant diagonal differential equation Dirac notation domain eigencolumns eigenfunctions eigenvectors elements evaluate expansion F₁ finite number follows formula Fourier given Green's function Hence Hermitian Hermitian matrix Hermitian operator infinite integral inverse Laplacian linear operator linearly independent lowest eigenvalue Mathematical matrix McGraw-Hill Book Company method multiplication nonsingular normal number of degrees obtained orthonormality conditions Physics problem relations representation result Ritz method scattering sinh solution solve spherical spherical harmonics string Substitution theorem transform trial functions vanish variable vector space Verify w₁ wave write written x₁ Y₁ yields York zero ηπχ ди ду дх