Studies in theoretical physics. Volume 1, Fundamental mathematical methods /

Studies in Theoretical Physics, Volume 1: Fundamental mathematical methods is the first of the six-volume series in theoretical physics. It provides the mathematical methods that any physical sciences and engineering undergraduate might need in upper-division courses in classical mechanics, quantum...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Erenso, Daniel (Autor), Montemayor, Victor (Victor J.) (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2022]
Colección:IOP (Series). Release 22.
IOP ebooks. 2022 collection.
Temas:
Mathematical physics.
Mathematics and computation.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Series and convergence
  • 1.1. Sequence and series
  • 1.2. Testing series for convergence
  • 1.3. Series representations of real functions
  • 1.4. Sequence, series and Mathematica
  • 1.5. Homework assignment
  • 2. Complex numbers, functions, and series
  • 2.1. Complex numbers
  • 2.2. Complex infinite series
  • 2.3. Powers and roots of complex numbers
  • 2.4. Algebraic versus transcendental functions
  • 2.5. Complex numbers, functions and Mathematica
  • 2.6. Homework assignment
  • 3. Vectors
  • 3.1. Vector fundamentals
  • 3.2. Vector addition
  • 3.3. Vector multiplication
  • 3.4. Vectors and equations of a line and a plane
  • 3.5. Vectors and Mathematica
  • 3.6. Homework assignment
  • 4. Matrices and determinants
  • 4.1. Important terminologies
  • 4.2. Matrix arithmetic and manipulation
  • 4.3. Matrix representation of a set of linear equations
  • 4.4. Solving a set of linear equations using matrices
  • 4.5. Determinant of a square matrix
  • 4.6. Cramer's rule
  • 4.7. The adjoint and inverse of a matrix
  • 4.8. Orthogonal matrices and the rotation matrix
  • 4.9. Linear dependence and independence
  • 4.10. Gram-Schmidt orthogonalization
  • 4.11. Matrices and Mathematica
  • 4.12. Homework assignment
  • 5. Introduction to differential calculus I
  • 5.1. Partial differentiation
  • 5.2. Total differential
  • 5.3. The multivariable form of the chain rule
  • 5.4. Extremum (max/min) problems
  • 5.5. The method of Lagrangian multipliers
  • 5.6. Change of variables
  • 5.7. Partial differentiation and Mathematica
  • 5.8. Homework assignments
  • 6. Introduction to differential calculus II
  • 6.1. First-order ordinary DE
  • 6.2. The first-order ODE and exact total differential
  • 6.3. First-order ODE and non-exact total differential
  • 6.4. Higher-order ODE
  • 6.5. The particular solution and the method of superposition
  • 6.6. The method of successive integration
  • 6.7. Introduction to partial differential equations
  • 6.8. Linear differential equations and Mathematica
  • 6.9. Homework assignment
  • 7. Integral calculus-scalar functions
  • 7.1. Integration in Cartesian coordinates
  • 7.2. Physical applications
  • 7.3. 1-D and 2-D curvilinear coordinates
  • 7.4. 3-D curvilinear coordinates : cylindrical
  • 7.5. 3-D curvilinear coordinate : spherical
  • 7.6. Scalar integrals and Mathematica
  • 7.7. Homework assignment
  • 8. Vector calculus
  • 8.1. Review of vector products
  • 8.2. Vectors product physical applications
  • 8.3. Vectors derivatives
  • 8.4. The gradient operator and directional derivative
  • 8.5. The divergence, the curl, and the Laplacian
  • 8.6. Line vector integrals
  • 8.7. Conservative vectors and exact differentials
  • 8.8. Double integral and Green's theorem
  • 8.9. The Stokes' theorem
  • 8.10. The divergence theorem
  • 8.11. Vector calculus and Mathematica
  • 8.12. Homework assignment
  • 9. Introduction to the calculus of variations
  • 9.1. Stationary points and geodesic
  • 9.2. The general problem of the calculus of variations
  • 9.3. The Brachistochrone problem
  • 9.4. The Euler-Lagrange equation in classical mechanics
  • 9.5. The calculus of variations and Mathematica
  • 9.6. Homework assignment
  • 10. Introduction to the eigenvalue problem
  • 10.1. Eigenvalue problem in physics
  • 10.2. Matrix review
  • 10.3. Orthogonal transformations and Dirac's notation
  • 10.4. Eigenvalues and eigenvectors
  • 10.5. Eigenvalue equation and Hermitian matrices
  • 10.6. The similarity transformation
  • 10.7. Eigenvalue equation and Mathematica
  • 10.8. Homework assignment
  • 11. Special functions
  • 11.1. The factorial, the gamma function, and Stirling's formula
  • 11.2. The beta function
  • 11.3. The error function
  • 11.4. Elliptic integrals
  • 11.5. The Dirac delta function
  • 11.6. Mathematica and special functions
  • 11.7. Homework assignments
  • 12. Power series and differential equations
  • 12.1. Power series substitution
  • 12.2. Orthonormal set of vectors and functions
  • 12.3. Complete set of functions
  • 12.4. The Legendre differential equation
  • 12.5. The Legendre polynomials
  • 12.6. The generating function for the Legendre polynomials
  • 12.7. Legendre series
  • 12.8. The associated Legendre differential equation
  • 12.9. Spherical harmonics and the addition theorem
  • 12.10. The method of Frobenius and the Bessel equation
  • 12.11. The orthogonality of the Bessel functions
  • 12.12. Fuch's theorem
  • 12.13. Mathematica and serious substitution method
  • 12.14. Homework assignments
  • 13. Partial differential equation
  • 13.1. PDE in physics
  • 13.2. Laplace's equation in Cartesian coordinates
  • 13.3. Laplace's equation in spherical coordinates
  • 13.4. Laplace's equation in cylindrical coordinates
  • 13.5. Poisson's equation
  • 13.6. Homework assignment
  • 14. Functions of complex variables
  • 14.1. Review of complex numbers
  • 14.2. Analytic functions
  • 14.3. Essential terminologies
  • 14.4. Contour integration and Cauchy's theorem
  • 14.5. Cauchy's integral formula
  • 14.6. Laurent's theorem
  • 14.7. The residue theorem
  • 14.8. Methods of finding residues
  • 14.9. Applications of the residue theorem
  • 14.10. The modified residue theorem
  • 14.11. Mathematica and complex functions
  • 14.12. Homework assignment
  • 15. Laplace transform
  • 15.1. Integral transform
  • 15.2. The Laplace transform
  • 15.3. Inverse Laplace transform
  • 15.4. Applications of Laplace transforms
  • 15.5. Mathematica and Laplace transform
  • 15.6. Homework assignment
  • 16. Fourier series and transform
  • 16.1. Average and root-mean-square values
  • 16.2. The Fourier series
  • 16.3. Dirichlet conditions
  • 16.4. Fourier series with spatial and temporal arguments
  • 16.5. The Fourier transform and inverse transform
  • 16.6. The Dirac delta function and the Fourier inverse transform
  • 16.7. Applications of the Fourier transform
  • 16.8. Fourier transform and convolution
  • 16.9. Mathematica, Fourier series, transform, and inverse transform.

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