Physics 501 Math Models of Physics Problems Luis Anchordoqui Wednesday, September 4, 13 1 Bibliography L. A. Anchordoqui and T. C. Paul, ``Mathematical Models of Physics Problems’‘ (Nova Publishers, 2013) G. F. D. Duff and D. Naylor, ``Differential Equations of Applied Mathematics’‘ (John Wiley & Sons, 1966) G. B. Arfken, H. J. Weber, and F. E. Harris ``Mathematical Methods for Physicists’’ (7th Edition) (Academic Press, 2012) Wednesday, September 4, 13 2 Provisional Course Outline (Please note this may be revised during the course to match coverage of material during lectures, etc.) 1st week - Elements of Linear Algebra 2nd week - Analytic Functions 3rd week - Integration in the Complex Plane 4th week - Isolated Singularities and Residues 5th week - Initial Value Problem (Picard’s Theorem) 6th week - Initial Value Problem (Green Matrix) 7th week - Boundary Value Problem (Sturm-Liouville Operator) 8th week - Midterm-exam (October 23) 9th week - Boundary Value Problem (Special Functions) 10th week - Fourier Series and Fourier Transform 11th week - Hyperbolic Partial Differential Equation (Wave equation) 12th week - Parabolic Partial Differential Equation (Diffusion equation) 13th week - Elliptic Partial Differential Equation (Laplace equation) 14th week - Midterm-exam (December 11) Wednesday, September 4, 13 3 Elements of Linear Algebra 1.1 Linear Spaces 1.2 Matrices and Linear Transformations Wednesday, September 4, 13 4 Linear Spaces Definition 1.1.1. F + field A is a set together with two operations and · �, µ, ⌫ F : for which all axioms below hold 8 2 (i) closure sum � + µ and product � µ again belong to F � � · (ii) associative law � + (µ + ⌫) = (� + µ) + ⌫ & � (µ ⌫) = (� µ) ⌫ � � · · · · (iii) commutative law � + ⌫ = ⌫ + � & � µ = µ � � � · · (iv) distributive laws � (µ + ⌫) = � µ + � ⌫ � � · · · (� + µ) ⌫ = � ⌫ + µ ⌫ and · · · (v) existence of an additive identity there exists an element � � 0 F for which � + 0 = � 2 (vi) existence of a multiplicative identity there exists an element � � 1 F with 1 = 0 for which 1 � = � 2 6 · (vii) existence of additive inverse to every � F there corresponds � � 2 an additive inverse � such that � + � = 0 � � (viii) existence of multiplicative inverse to every � F � � 2 1 1 � � � = 1 there corresponds a multiplicative inverse � such that � · Wednesday, September 4, 13 5 Example 1.1.1. F Underlying every linear space is a field examples are R and C Definition 1.1.2. A linear space V is a collection of objects with a (vector) addition and scalar multiplication defined which is closed under both operations Such a vector space satisfies following axioms: ➢ commutative law of vector addition x + y = y + x, x, y V 8 2 ➢ associative law of vector addition x + (y + w) = (x + y) + w, x, y, w V 8 2 0 x + 0 = x, x V ➢ There exists a zero vector such that 8 2 Wednesday, September 4, 13 6 x V ➢ To every element 2 x there corresponds an inverse element � x + ( x) = 0 such that � ➢ associative law of scalar multiplication (�µ ) x = � (µ x), x V and �, µ F 8 2 2 ➢ distributive laws of scalar multiplication (� + µ) x = � x + µ x, x V and �, µ F 8 2 2 � (x + y) = � x + � y, x, y V and � F 8 2 2 1 x = x, x V ➢ · 8 2 Wednesday, September 4, 13 7 Example 1.1.2. n Cartesian space R is prototypical example n of real -dimensional vector space x = (x , . . . , x ) n x Let 1 n be an ordered tuple of real numbers i x to which there corresponds a point with these Cartesian x coordinates and a vector with these components We define addition of vectors by component addition x + y = (x + y , . . . , x + y ) 1 1 n n (1.1.1.) and scalar multiplication by component multiplication � x = (�x , . . . ,�x ) (1.1.2.) 1 n Wednesday, September 4, 13 8 Definition 1.1.3. Given a vector space V over a field F a subset W of V is called subspace if W is vector space over F under operations already defined on V Corollary 1.1.1 W V V A subset of a vector space is a subspace of , (i)W is nonempty (ii) if x, y W ☛ then x + y W 2 2 (iii) x W and � F ☛ then � x W 2 2 · 2 After defining notions of vector spaces and subspaces next step is to identify functions that can be used to relate one vector space to another Functions should respect algebraic structure of vector spaces so we require they preserve addition and scalar multiplication Wednesday, September 4, 13 9 Definition 1.1.4. V W F Let and be vector spaces over field A linear transformation from V to W is a function T : V W ! T (�x + µy) = �T (x) + µT (y) such that (1.1.3.) for all vectors x, y V �, µ F and all scalars 2 2 If a linear transformation is one-to-one and onto it is called vector space isomorphism ☛ or simply isomorphism Definition 1.1.5. S = x , , x V F Let be a set of vectors in vector space over field 1 n · · · n � F y = � x Any vector of form i i for i 2 i=1 X S is called linear combination of vectors in S V V Set is said to span if each element of S can be expressed a s linear combination of vectors in Wednesday, September 4, 13 10