{"id":381,"date":"2023-09-05T22:01:13","date_gmt":"2023-09-05T22:01:13","guid":{"rendered":"https:\/\/physics.uri.edu\/muller\/?page_id=381"},"modified":"2023-10-10T14:45:23","modified_gmt":"2023-10-10T14:45:23","slug":"phy510n","status":"publish","type":"page","link":"https:\/\/physics.uri.edu\/muller\/phy510n\/","title":{"rendered":"PHY510N"},"content":{"rendered":"\n<h1 class=\"wp-block-heading\">Teaching<\/h1>\n\n\n\n<div class=\"sub_nav\"><a title=\"Electricity and Magnetism\" href=\"https:\/\/physics.uri.edu\/muller\/phy331n\/\">PHY331<\/a>|<a title=\"Mathematical Methods of Physics I\" href=\"https:\/\/physics.uri.edu\/muller\/phy510n\/\">PHY510<\/a> |<a title=\"Equilibrium Statistical Physics\" href=\"https:\/\/physics.uri.edu\/muller\/phy525n\/\">PHY525<\/a><\/div>\n\n\n\n<div>&nbsp;<\/div>\n\n\n\n<p><a rel=\"license\" href=\"http:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\"><img decoding=\"async\" style=\"border-width: 0\" src=\"https:\/\/i.creativecommons.org\/l\/by-nc-sa\/4.0\/88x31.png\" alt=\"Creative Commons License\"><\/a>This work is licensed under a <a rel=\"license\" href=\"http:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License<\/a>.<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">Mathematical Methods of Physics<\/h1>\n\n\n\n<p>These lecture notes for the courses PHY510\/610 are a work in progress at this time. Amendments, improvements, and corrections wil be made continually. The topics listed are interrelated in intricate ways. They do not reflect a sequence of course materials. The lectures visit each topic multiple times.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Complex Analysis&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Complex numbers&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd7-A.pdf\">[gmd7-A]<\/a><\/li>\n\n\n\n<li>Stereographic projection<\/li>\n\n\n\n<li>Complex functions<\/li>\n\n\n\n<li>Complex derivatives<\/li>\n\n\n\n<li>Differential operators<\/li>\n\n\n\n<li>Orthogonal families of curves<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Line integrals&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd7-B.pdf\">[gmd7-B]<\/a><\/li>\n\n\n\n<li>Cauchy&#8217;s theorem<\/li>\n\n\n\n<li>Cauchy integrals<\/li>\n\n\n\n<li>Taylor series<\/li>\n\n\n\n<li>Laurent series<\/li>\n\n\n\n<li>Residues<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Trigonometric relations made transparent by complex variables&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex16.pdf\">[gex16]<\/a><\/li>\n\n\n\n<li>Search for zeros of functions in the complex plane I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex18.pdf\">[gex18]<\/a><\/li>\n\n\n\n<li>Constructing inverse trigonometric functions I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex19.pdf\">[gex19]<\/a><\/li>\n\n\n\n<li>Constructing analytic function from harmonic function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex21.pdf\">[gex21]<\/a><\/li>\n\n\n\n<li>Derivatives of inverse trigonometric functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex22.pdf\">[gex22]<\/a><\/li>\n\n\n\n<li>Harmonic and analytic functions I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex54.pdf\">[gex54]<\/a><\/li>\n\n\n\n<li>Applications of L&#8217;Hospital&#8217;s rule&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex55.pdf\">[gex55]<\/a><\/li>\n\n\n\n<li>Orthogonal family of functions I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex56.pdf\">[gex56]<\/a><\/li>\n\n\n\n<li>Conducting plates intersecting at right angle&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex57.pdf\">[gex57]<\/a><\/li>\n\n\n\n<li>Electric potential and field at edge of large conducting plate&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex58.pdf\">[gex58]<\/a><\/li>\n\n\n\n<li>Fringe electric potential and fringe field of parallel plates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex59.pdf\">[gex59]<\/a><\/li>\n\n\n\n<li>Differential operators acting on complex functions I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex60.pdf\">[gex60]<\/a><\/li>\n\n\n\n<li>Line integral in complex plane I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex70.pdf\">[gex77]<\/a><\/li>\n\n\n\n<li>Green&#8217;s theorem adapted to complex functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex78.pdf\">[gex78]<\/a><\/li>\n\n\n\n<li>Application of Cauchy&#8217;s theorem I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex79.pdf\">[gex79]<\/a><\/li>\n\n\n\n<li>Argument theorem in complex analysis&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex80.pdf\">[gex80]<\/a><\/li>\n\n\n\n<li>Poisson integrals for circle in complex plane&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex81.pdf\">[gex81]<\/a><\/li>\n\n\n\n<li>Application of Cauchy&#8217;s theorem II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex82.pdf\">[gex82]<\/a><\/li>\n\n\n\n<li>Poisson integrals for half plane in complex analysis&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex83.pdf\">[gex83]<\/a><\/li>\n\n\n\n<li>Applications of Poisson integrals in complex analysis I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex84.pdf\">[gex84]<\/a><\/li>\n\n\n\n<li>Laurent series of analytic functions I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex85.pdf\">[gex85]<\/a><\/li>\n\n\n\n<li>Laurent series of analytic functions II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex86.pdf\">[gex86]<\/a><\/li>\n\n\n\n<li>Laurent series of analytic functions III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex87.pdf\">[gex87]<\/a><\/li>\n\n\n\n<li>Residues of isolated singularities of complex functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex88.pdf\">[gex88]<\/a><\/li>\n\n\n\n<li>Definite integral via residue theorem I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex89.pdf\">[gex89]<\/a><\/li>\n\n\n\n<li>Definite integral via residue theorem II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex90.pdf\">[gex90]<\/a><\/li>\n\n\n\n<li>Definite integral via residue theorem III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex91.pdf\">[gex91]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Complex functions in electroststics&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gam1.pdf\">[gam1]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Special functions&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Elementary functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4.pdf\">[gmd4]<\/a><\/li>\n\n\n\n<li>Special functions<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Gamma function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4A.pdf\">[gmd4A]<\/a><\/li>\n\n\n\n<li>Beta function<\/li>\n\n\n\n<li>Incomplete Gamma function<\/li>\n\n\n\n<li>Binomial series<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Error function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4B.pdf\">[gmd4B]<\/a><\/li>\n\n\n\n<li>Fresnel integrals<\/li>\n\n\n\n<li>Elliptic integrals and elliptic functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4C.pdf\">[gmd4C]<\/a><\/li>\n\n\n\n<li>Legendre polynomials and functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4D.pdf\">[gmd4D]<\/a><\/li>\n\n\n\n<li>Spherical harmonics&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4E.pdf\">[gmd4E]<\/a><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Bessel functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd4F.pdf\">[gmd4F]<\/a><\/li>\n\n\n\n<li>Modified Bessel functions<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Recurrence relation for Gamma function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex2.pdf\">[gex2]<\/a><\/li>\n\n\n\n<li>Relation between Gamma and Beta function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex3.pdf\">[gex3]<\/a><\/li>\n\n\n\n<li>Complete elliptic integrals: series expansion&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex8.pdf\">[gex8]<\/a><\/li>\n\n\n\n<li>Area and circumference of an ellipse&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex9.pdf\">[gex9]<\/a><\/li>\n\n\n\n<li>Plane-pendulum oscillations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex10.pdf\">[gex10]<\/a><\/li>\n\n\n\n<li>Plane-pendulum rotations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex11.pdf\">[gex11]<\/a><\/li>\n\n\n\n<li>Electric field of charged ring and elliptic integrals&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex34.pdf\">[gex34]<\/a><\/li>\n\n\n\n<li>Magnetic field of circular current and elliptic integrals&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex35.pdf\">[gex35]<\/a><\/li>\n\n\n\n<li>Duplication relation between Gamma for Gamma function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex39.pdf\">[gex39]<\/a><\/li>\n\n\n\n<li>Euler&#8217;s product representation of Gamma function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex53.pdf\">[gex53]<\/a><\/li>\n\n\n\n<li>Polygamma function: series representation, recurrence relation&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex92.pdf\">[gex92]<\/a><\/li>\n\n\n\n<li>Polygamma function: integral representation, asymptotic series&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex93.pdf\">[gex93]<\/a><\/li>\n\n\n\n<li>Complementary error function: asymptotic expansion&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex94.pdf\">[gex94]<\/a><\/li>\n\n\n\n<li>Relations between error function and Fresnel integrals&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex95.pdf\">[gex95]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Matrix Operations&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fundamental properties, operations, and relations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd6-A.pdf\">[gmd6-A]<\/a><\/li>\n\n\n\n<li>Determinants&nbsp;<\/li>\n\n\n\n<li>Inverse matrix<\/li>\n\n\n\n<li>Orthogonality and unitarity<\/li>\n\n\n\n<li>Systems of linear equations<\/li>\n\n\n\n<li>Eigenvalues and eigenvectors<\/li>\n\n\n\n<li>Mathematica notebook&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd6-A.nb\">[gmd6-A.nb]<\/a><\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Matrix operations I: matrix multiplication&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex40.pdf\">[gex40]<\/a><\/li>\n\n\n\n<li>Matrix operations II: Laplace expansion of determinant&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex41.pdf\">[gex41]<\/a><\/li>\n\n\n\n<li>Matrix operations III: inverse square matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex42.pdf\">[gex42]<\/a><\/li>\n\n\n\n<li>Matrix operations IV: orthogonal matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex43.pdf\">[gex43]<\/a><\/li>\n\n\n\n<li>Matrix operations V: unitary matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex44.pdf\">[gex44]<\/a><\/li>\n\n\n\n<li>Matrix operationss VI: determinant of matrix product&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex45.pdf\">[gex45]<\/a><\/li>\n\n\n\n<li>Matrix operations VII: system of linear equations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex46.pdf\">[gex46]<\/a><\/li>\n\n\n\n<li>Matrix operations VIII: eigenvectors of symmetric matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex47.pdf\">[gex47]<\/a><\/li>\n\n\n\n<li>Matrix operations IX: eigenvectors of Hermitian matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex48.pdf\">[gex48]<\/a><\/li>\n\n\n\n<li>Matrix operations X: eigenvectors of orthogonal matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex49.pdf\">[gex49]<\/a><\/li>\n\n\n\n<li>Matrix operations XI: eigenvectors of unitary matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex50.pdf\">[gex50]<\/a><\/li>\n\n\n\n<li>Matrix operations XII: eigenvectors of asymmetric matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex51.pdf\">[gex51]<\/a><\/li>\n\n\n\n<li>Matrix operations XIII: eigenvectors of transition matrix&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex52.pdf\">[gex52]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Vector Analysis&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Vector addition&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd1-A.pdf\">[gmd1-A]<\/a><\/li>\n\n\n\n<li>Dot product of vectors<\/li>\n\n\n\n<li>Cross product of vectors<\/li>\n\n\n\n<li>Triple product of vectors<\/li>\n\n\n\n<li>Reciprocal vectors<\/li>\n\n\n\n<li>Vector functions<\/li>\n\n\n\n<li>Scalar and vector fields<\/li>\n\n\n\n<li>Differential operators<\/li>\n\n\n\n<li>Identities involving differential operators<\/li>\n\n\n\n<li>Differentials of scalars<\/li>\n\n\n\n<li>Differentials of vectors<\/li>\n\n\n\n<li>Mathematica notebook <a href=\"http:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd1-A.nb\">[gmd1-A.nb]<\/a> <\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Vector integrations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd1-B.pdf\">[gmd1-B]<\/a><\/li>\n\n\n\n<li>Line integrals<\/li>\n\n\n\n<li>Surface integrals<\/li>\n\n\n\n<li>Differential operators from integrals<\/li>\n\n\n\n<li>Integral theorems<\/li>\n\n\n\n<li>Green&#8217;s identities<\/li>\n\n\n\n<li>Integration by parts generalized<\/li>\n\n\n\n<li>Helmholtz theorem<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Electric dipole field&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex26.pdf\">[gex26]<\/a><\/li>\n\n\n\n<li>Magnetic dipole field&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex27.pdf\">[gex27]<\/a><\/li>\n\n\n\n<li>Reciprocal vectors&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex28.pdf\">[gex28]<\/a><\/li>\n\n\n\n<li>Expansion of vector in non-orthonormal basis&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex29.pdf\">[gex29]<\/a><\/li>\n\n\n\n<li>Diagonals of a parallelogram&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex30.pdf\">[gex30]<\/a><\/li>\n\n\n\n<li>Law of sines for spherical triangle&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex32.pdf\">[gex32]<\/a><\/li>\n\n\n\n<li>Common identities used in electrostatics and elsewhere&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex33.pdf\">[gex33]<\/a><\/li>\n\n\n\n<li>Vector divisions?&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex36.pdf\">[gex36]<\/a><\/li>\n\n\n\n<li>Gradients of related scalar functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex37.pdf\">[gex37]<\/a><\/li>\n\n\n\n<li>Work done by conservative force I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex63.pdf\">[gex63]<\/a><\/li>\n\n\n\n<li>Work done by conservative force II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex64.pdf\">[gex64]<\/a><\/li>\n\n\n\n<li>Electrostatic field of two point charges&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex65.pdf\">[gex65]<\/a><\/li>\n\n\n\n<li>Electrostatic field of three point charges&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex66.pdf\">[gex66]<\/a><\/li>\n\n\n\n<li>Vector functions I <a href=\"http:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex124.pdf\">[gex124]<\/a> <\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Coordinate Systems&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>General coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd2-A.pdf\">[gmd2-A]<\/a><\/li>\n\n\n\n<li>Orthogonal coordinates<\/li>\n\n\n\n<li>Gradient, divergence, curl, and Laplacian<\/li>\n\n\n\n<li>Determination of scale factors<\/li>\n\n\n\n<li>Cartesian coordinates<\/li>\n\n\n\n<li>Cylindrical coordinates<\/li>\n\n\n\n<li>Spherical coordinates<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Parabolic cylindrical coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd2-B.pdf\">[gmd2-B]<\/a><\/li>\n\n\n\n<li>Paraboloidal coordinates<\/li>\n\n\n\n<li>Elliptic cylindrical coordinates<\/li>\n\n\n\n<li>Prolate spheroidal coordinates<\/li>\n\n\n\n<li>Oblate spheroidal coordinates<\/li>\n\n\n\n<li>Ellipsoidal coordinates<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Vector fields in curvilinear coordinates I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex67.pdf\">[gex67]<\/a><\/li>\n\n\n\n<li>Vector fields in curvilinear coordinates II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex68.pdf\">[gex68]<\/a><\/li>\n\n\n\n<li>Vector fields in curvilinear coordinates III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex69.pdf\">[gex69<\/a>]<\/li>\n\n\n\n<li>Laplacian operating on vector field&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex70.pdf\">[gex70]<\/a><\/li>\n\n\n\n<li>Parabolic cylindrical coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex71.pdf\">[gex71]<\/a><\/li>\n\n\n\n<li>Paraboloidal coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex72.pdf\">[gex72]<\/a><\/li>\n\n\n\n<li>Elliptic cylindrical coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex73.pdf\">[gex73]<\/a><\/li>\n\n\n\n<li>Prolate spheroidal coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex74.pdf\">[gex74]<\/a><\/li>\n\n\n\n<li>Oblate spheroidal coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex75.pdf\">[gex75]<\/a><\/li>\n\n\n\n<li>Ellipsoidal coordinates&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex76.pdf\">[gex76<\/a>]<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Tensor Analysis&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd5-A.pdf\">[gmd5-A]<\/a><\/li>\n\n\n\n<li>Inertia tensor<\/li>\n\n\n\n<li>From matrices to tensors<\/li>\n\n\n\n<li>Tensors in real coordinate space<\/li>\n\n\n\n<li>Contravariance versus covariance<\/li>\n\n\n\n<li>Invariance<\/li>\n\n\n\n<li>Mixed variance<\/li>\n\n\n\n<li>Affine tensors<\/li>\n\n\n\n<li>Cartesian tensors<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Tensor operations&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd5-B.pdf\">[gmd5-B]<\/a><\/li>\n\n\n\n<li>Quotient rule<\/li>\n\n\n\n<li>Arrays of elements &#8212; vectors, matrices, tensors<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inertia tensor from momentum of rigid body&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex96.pdf\">[gex96]<\/a><\/li>\n\n\n\n<li>Inertia tensor from rotational kinetic energy of rigid body&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex97.pdf\">[gex97]<\/a><\/li>\n\n\n\n<li>Tensor summation convention I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex98.pdf\">[gex98]<\/a><\/li>\n\n\n\n<li>Tensor summation convention II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex99.pdf\">[gex99]<\/a><\/li>\n\n\n\n<li>Tensor summation convention III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex100.pdf\">[gex100]<\/a><\/li>\n\n\n\n<li>From polar to rectangular coordinates and back: Jacobians&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex112.pdf\">[gex112]<\/a><\/li>\n\n\n\n<li>Uniform vector field in plane made into a tensor&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex113.pdf\">[gex113]<\/a><\/li>\n\n\n\n<li>Radial and azimuthal fields in a plane&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex117.pdf\">[gex117]<\/a>&nbsp;<\/li>\n\n\n\n<li>Application of quotient rule I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex118.pdf\">[gex118]<\/a><\/li>\n\n\n\n<li>Application of quotient rule II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex119.pdf\">[gex119]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Summation convention&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gam2.pdf\">[gam2]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Ordinary Differential Equations<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First-order ODEs&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd10-A.pdf\">[gmd10-A]<\/a><\/li>\n\n\n\n<li>Second-order ODEs<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First-order ODE: separation of variables I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex4.pdf\">[gex4]<\/a><\/li>\n\n\n\n<li>First-order ODE: separation of variables II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex5.pdf\">[gex5]<\/a><\/li>\n\n\n\n<li>First-order ODE: separation of variables III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex6.pdf\">[gex6]<\/a><\/li>\n\n\n\n<li>First-order ODE: exact differential I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex7.pdf\">[gex7]<\/a><\/li>\n\n\n\n<li>First-order ODE: integrating factor I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex12.pdf\">[gex12]<\/a><\/li>\n\n\n\n<li>First-order ODE: linearity&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex13.pdf\">[gex13]<\/a><\/li>\n\n\n\n<li>First-order ODE: homogeneity&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex14.pdf\">[gex14]<\/a><\/li>\n\n\n\n<li>First-order ODE: Bernoulli type&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex15.pdf\">[gex15]<\/a><\/li>\n\n\n\n<li>First-order ODE: convertibility&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex17.pdf\">[gex17]<\/a><\/li>\n\n\n\n<li>Second-order ODEs reducible to first order&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex20.pdf\">[gex20]<\/a><\/li>\n\n\n\n<li>First-order ODE: general, particular, and singular solutions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex23.pdf\">[gex23]<\/a><\/li>\n\n\n\n<li>ODE for two-parameter family of conics&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex24.pdf\">[gex24]<\/a><\/li>\n\n\n\n<li>First-order ODE: Clairaut type&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex25.pdf\">[gex25]<\/a><\/li>\n\n\n\n<li>Second-order ODE: fixed points and isoclines I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex101.pdf\">[gex101]<\/a><\/li>\n\n\n\n<li>Second-order ODEs: fixed points and isoclines II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex102.pdf\">[gex102]<\/a><\/li>\n\n\n\n<li>Plane pendulum with attenuation: fixed points and phase flow&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex103.pdf\">[gex103]<\/a>&nbsp;<\/li>\n\n\n\n<li>Coupled first-order ODEs: fixed points and flow dynamics I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex104.pdf\">[gex104]<\/a><\/li>\n\n\n\n<li>Coupled first-order ODEs: fixed points and flow dynamics II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex105.pdf\">[gex105]<\/a><\/li>\n\n\n\n<li>Coupled first-order ODEs: fixed point and limit cycle&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex106.pdf\">[gex106]<\/a><\/li>\n\n\n\n<li>Coupled first-order ODEs: R\u00f6ssler band strange attractor&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex107.pdf\">[gex107]<\/a><\/li>\n\n\n\n<li>Second-order ODE: reduction to first-order ODE I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex108.pdf\">[gex108]<\/a><\/li>\n\n\n\n<li>Second-order ODE: reduction to first-order ODE II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex109.pdf\">[gex109]<\/a><\/li>\n\n\n\n<li>Linear second-order ODE with degenerate roots&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex110.pdf\">[gex110]<\/a><\/li>\n\n\n\n<li>Linearly damped harmonic oscillator: general solution&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex111.pdf\">[gex111]<\/a><\/li>\n\n\n\n<li>Linear inhomogeneous ODE: undetermined constant parameters I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex114.pdf\">[gex114]<\/a><\/li>\n\n\n\n<li>Linear inhomogeneous ODE: undetermined constant parameters II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex115.pdf\">[gex115]<\/a><\/li>\n\n\n\n<li>Second-order ODE: reduction to first order III&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex116.pdf\">[gex116]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Linear ODEs&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gam8.pdf\">[gam8]<\/a><\/li>\n\n\n\n<li>General structure of ODE and solution<\/li>\n\n\n\n<li>Homogeneous ODE with constant coefficients<\/li>\n\n\n\n<li>Linearly damped harmonic oscillator<\/li>\n\n\n\n<li>Particular solution of inhomogeneous ODE<\/li>\n\n\n\n<li>Method of undetermined constant parameters<\/li>\n\n\n\n<li>Method of variation of parameters<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>ODEs of classical dynamical systems&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gam3.pdf\">[gam3]<\/a><\/li>\n\n\n\n<li>Dynamical systems of two variables<\/li>\n\n\n\n<li>Isoclines<\/li>\n\n\n\n<li>Fixed points<\/li>\n\n\n\n<li>Conservative forces<\/li>\n\n\n\n<li>Limit cycle<\/li>\n\n\n\n<li>Dynamical systems of three variables<\/li>\n\n\n\n<li>Is classical mechanics a deterministic theory?<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Integral Transforms&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>General form and types&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd8.pdf\">[gmd8]<\/a><\/li>\n\n\n\n<li>Laplace transform&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd8A.pdf\">[gmd8A]<\/a><\/li>\n\n\n\n<li>Fourier transform [gmd8B]<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Linerar ODE solved via Laplace transform&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex61.pdf\">[gex61]<\/a><\/li>\n\n\n\n<li>Laplace transform of derivatives of functions&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex62.pdf\">[gex62]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Partial Differential Equations<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Classification&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd11-A.pdf\">[gmd11-A]<\/a><\/li>\n\n\n\n<li>Structure og general solution according to type<\/li>\n\n\n\n<li>Subsidiary conditions<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>PDEs solved and solutions visualized&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex120.pdf\">[gex120]<\/a><\/li>\n\n\n\n<li>PDE solved via reduction to ODE I <a href=\"http:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex121.pdf\" data-type=\"URL\" data-id=\"penrose.uri.edu\/Gerhard\/PHY510N\/wgex121.pdf\">[gex121]<\/a> <\/li>\n\n\n\n<li>PDE solved via reduction to ODE II <a href=\"http:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex122.pdf\">[gex122]<\/a> <\/li>\n\n\n\n<li>Laplace equation for conducting hyperbolic trough <a href=\"http:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex123.pdf\">[gex123]<\/a> <\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Laplace equation [gam4] <\/li>\n\n\n\n<li>Diffusion equation [gam5]<\/li>\n\n\n\n<li>Wave equation [gam6]<\/li>\n\n\n\n<li>Schr\u00f6dinger equation [gam7]<\/li>\n\n\n\n<li>Fokker-Planck equation [gam9]<\/li>\n\n\n\n<li>Navier-Stokes equation [gam10]<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Probabilities&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Elementary probabilities&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd9-A.pdf\">[gmd9-A]<\/a><\/li>\n\n\n\n<li>Elements of set theory<\/li>\n\n\n\n<li>Elements of probability theory<\/li>\n\n\n\n<li>Joint and conditional probabilities<\/li>\n\n\n\n<li>Statistical uncertainty and information<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Generalized Functions&nbsp;<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Dirac delta function&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/gmd3-A.pdf\">[gmd3-A]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Representations of the Dirac delta function I&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex1.pdf\">[gex1]<\/a><\/li>\n\n\n\n<li>Representation of the Dirac delta function II&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex38.pdf\">[gex38]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Functional Analysis [gmd12]<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Differential Geometry [gmd13]<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Calculus of Variation [gmd14]<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Searching for extremum with Lagrange multiplier&nbsp;<a href=\"https:\/\/penrose.uri.edu\/Gerhard\/PHY510N\/wgex31.pdf\">[gex31]<\/a><\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Group Theory and Symmetry Transformations [gmd15]<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Exercises:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Additional Materials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Some Relevant Textbooks and Monographs:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>L. C. Andrews:&nbsp;<em>Special Functions for Engineers and Applied Mathematicians<\/em>. Macmillan, 1985.<\/li>\n\n\n\n<li>L. C. Andrews:&nbsp;<em>Elementary Partial Differential Equations<\/em>. Academic Press, 1986.<\/li>\n\n\n\n<li>L. C. Andrews and B. K. Shivamoggi:&nbsp;<em>Integral Transforms for Engineers and Applied Mathematicians.<\/em>&nbsp;Macmillan, 1988.<\/li>\n\n\n\n<li>G. B. Arfken and H. J. Weber:&nbsp;<em>Mathematical Methods for Physicists.<\/em>&nbsp; Harcourt, 2001.<\/li>\n\n\n\n<li>R. Bronson and G. B. Costa:&nbsp;<em>Differential Equations<\/em>. McGraw-Hill, 2014.<\/li>\n\n\n\n<li>R. Courant and D. Hilbert:&nbsp;<em>Methoden der Mathematischen Physik I, II.<\/em>&nbsp;Springer, 1968.<\/li>\n\n\n\n<li>P. Dennery and A. Krzywicki:&nbsp;<em>Mathematics for Physicists.<\/em>&nbsp;Dover, 1995.<\/li>\n\n\n\n<li>G. H. Golub and C. F. van Loan:&nbsp;<em>Matrix Computations.<\/em>&nbsp;Johns Hopkins University Press, 1985.<\/li>\n\n\n\n<li>M. Karbach:&nbsp;<em>Mathematische Methoden der Physik.<\/em>&nbsp;De Gruyter, 2017.<\/li>\n\n\n\n<li>D. C. McKay:&nbsp;<em>Tensor Calculus.<\/em>&nbsp;McGraw Hill, 2011.<\/li>\n\n\n\n<li>D. E. Neuenschwander:&nbsp;<em>Tensor Calculus for Physics.<\/em>&nbsp;Johns Hopkins University Press, 2015<\/li>\n\n\n\n<li>M. R. Spiegel:&nbsp;<em>Advanced Mathematics for Engineers and Scientists.<\/em>&nbsp;McGraw-Hill, 1971.<\/li>\n\n\n\n<li>M. R. Spiegel, S. Lipschutz, and D. Spellman:&nbsp;<em>Vector Analysis<\/em>. McGraw-Hill, 2009.<\/li>\n\n\n\n<li>M. R. Spiegel, S. Lipschutz, J. J. Schiller,and D. Spellman:&nbsp;<em>Complex Variables<\/em>. McGraw-Hill, 2009.<\/li>\n\n\n\n<li>&#8230;<\/li>\n<\/ul>\n\n\n\n<p>Last updated: 09\/05,23<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Teaching PHY331|PHY510 |PHY525 &nbsp; This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Mathematical Methods of Physics These lecture notes for the courses PHY510\/610 are a work in progress at this time. Amendments, improvements, and corrections wil be made continually. The topics listed are interrelated in intricate ways. They do not reflect [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-twocol.php","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-381","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/pages\/381","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/comments?post=381"}],"version-history":[{"count":11,"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/pages\/381\/revisions"}],"predecessor-version":[{"id":582,"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/pages\/381\/revisions\/582"}],"wp:attachment":[{"href":"https:\/\/physics.uri.edu\/muller\/wp-json\/wp\/v2\/media?parent=381"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}