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Dr. Abdelhay Salah

Assistant Professor

Faculty Staff - Theoretical Physics

كلية العلوم
2A64
course

501 phys

This course offers advanced knowledge in Vector Analysis, Vector Analysis in Curved Coordinates and Tensors, Functions of Complex variable I, Functions of Complex variable II, Differential equations, Sturm- Liouville Theory-Orthogonal Functions.
 
Title
1 Vector Analysis 1
1.1 Definitions, Elementary Approach . . . . . . . . . . . . . . . . . . . . . 1
1.2 Rotation of the Coordinate Axes . . . . . . . . . . . . . . . . . . . . . . 7
 
1.3 Scalar or Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Vector or Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 
1.5 Triple Scalar Product, Triple Vector Product . . . . . . . . . . . . . . . 25
1.6 Gradient, ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Divergence, ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.8 Curl, ∇× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 
1.9 Successive Applications of ∇ . . . . . . . . . . . . . . . . . . . . . . . 49
1.10 Vector Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.11 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.12 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.13 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
 
1.14 Gauss’ Law, Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . 79
1.15 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.16 Helmholtz’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
 
2 Vector Analysis in Curved Coordinates and Tensors 103
2.1 Orthogonal Coordinates in R3 . . . . . . . . . . . . . . . . . . . . . . . 103
2.2 Differential Vector Operators . . . . . . . . . . . . . . . . . . . . . . . 110
2.3 Special Coordinate Systems: Introduction . . . . . . . . . . . . . . . . 114
2.4 Circular Cylinder Coordinates . . . . . . . . . . . . . . . . . . . . . . . 115
2.5 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 123
 
2.6 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.7 Contraction, Direct Product . . . . . . . . . . . . . . . . . . . . . . . . 139
2.8 Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.9 Pseudotensors, Dual Tensors . . . . . . . . . . . . . . . . . . . . . . . 142
2.10 General Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.11 Tensor Derivative Operators . . . . . . . . . . . . . . . . . . . . . . . . 160
 
6 Functions of a Complex Variable I Analytic Properties, Mapping 403
6.1 Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
6.2 Cauchy–Riemann Conditions . . . . . . . . . . . . . . . . . . . . . . . 413
6.3 Cauchy’s Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 418
6.4 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . 425
6.5 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
 
6.6 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
6.7 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
6.8 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
7 Functions of a Complex Variable II 455
7.1 Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
7.2 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
7.3 Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . . 489
 
9 Differential Equations 535
9.1 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 535
9.2 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . 543
9.3 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
9.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
9.5 Series Solutions—Frobenius’ Method . . . . . . . . . . . . . . . . . . . 565
 
9.6 A Second Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
9.7 Nonhomogeneous Equation—Green’s Function . . . . . . . . . . . . . 592
9.8 Heat Flow, or Diffusion, PDE . . . . . . . . . . . . . . . . . . . . . . . 611
 
10 Sturm–Liouville Theory—Orthogonal Functions 621
10.1 Self-Adjoint ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
10.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
10.3 Gram–Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . 642
10.4 Completeness of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 649
10.5 Green’s Function—Eigenfunction Expansion . . . . . . . . . . . . . . . 662
 
List Required Textbooks
1-G.Arfken,Mathematical Methods for Physicist ;Academic Press ;New York ;1970
2. List Essential References Materials (Journals, Reports, etc.)
1.R.P.Kanwal;Linear Integral Equations “Theory and Techniques”;Burkhauser,Boston ;1997
2.G.stephenson ;Mathematical Methods for sciences Students 2nd edition ;Longman,U.K.;1992
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