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 + X n - Σxu == i = 1 + where x , is the complex conjugate of x , and u , * u ...
... 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 + X n - Σxu == i = 1 + where x , is the complex conjugate of x , and u , * u ...
Página 57
... ( 4.17 ) the single relation ( ax + y ) = xLx + Ly . The converse is also true , a knowledge of the VECTOR SPACES AND LINEAR OPERATORS 57 The Representation of Linear Operators by Matrices The Operator in the Dual Space.
... ( 4.17 ) the single relation ( ax + y ) = xLx + Ly . The converse is also true , a knowledge of the VECTOR SPACES AND LINEAR OPERATORS 57 The Representation of Linear Operators by Matrices The Operator in the Dual Space.
Página 58
... Dual Space If X x = Σux then the corresponding vector in the dual space is With the definition one has or Hence or + X = = Σxu + y = Lx y = Lx Yi = ΣLisx ; ÿ1 = Σx , Lis y + = x + L + ( 4.21 ) This result suggests the notation y + = x + ...
... Dual Space If X x = Σux then the corresponding vector in the dual space is With the definition one has or Hence or + X = = Σxu + y = Lx y = Lx Yi = ΣLisx ; ÿ1 = Σx , Lis y + = x + L + ( 4.21 ) This result suggests the notation y + = x + ...
Contenido
Perturbation of Eigenvalues | 14 |
The Laplacian v2 in One Dimension | 18 |
Solution for Diagonalizable Matrices | 21 |
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approximate arbitrary ax² basis Bessel function boundary conditions chap coefficients column consider constant continuous systems contour coordinates corresponding cylindrical functions d²/dx² defined denoted determinant diagonal differential equation Dirac notation domain eigen eigencolumns eigenfunctions eigenvalue equation eigenvector eikr evaluate expansion finite number follows Fourier given Green's function Hence Hermitian Hermitian matrix Hermitian operator infinite integral inverse Laplace transform Laplacian linear operator linearly independent lowest eigenvalue matrix membrane method multiplication nonsingular normal obtained orthonormality conditions plane problem procedure relations representation result satisfies the boundary scattering sinh solve spherical spherical harmonics string Substitution theorem trial functions vanish variable vector space Verify wave write written y₁ yields York zero ηπχ πο ποχ ди ду дх