MATH 3423 - Physical Mathematics II, Section 001 - Fall 2016
MWF 2:30-3:20 p.m., 222 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Tue 12:00-1:00 p.m., Fri 1:30-2:30 p.m., or by appointment, in 802 PHSC.

First day handout

Prerequisite: 2443 (Calc and Analytic Geom IV) or MATH 2934 (Diff & Int Calc III), 3413 (Physical Math I).

Course catalog description: The Fourier transform and applications, a survey of complex variable theory, linear and nonlinear coordinate transformations, tensors, elements of the calculus of variations. (F, Sp)

Text: D. A. McQuarrie, Mathematical Methods for Scientists and Engineers, University Science Books, Sausalito, CA, 2003.
The course will cover (parts of) chapters 4, 8-10, 15, 17-20.
A list of errata in the book (collected by Prof. Daniel Sober from CUA).

Homework (after the due date solutions are deposited in Bizzell Library and can be checked out from the Main Desk)

• Homework 1, due September 2 (Friday).
• Homework 2, due September 9 (Friday).
• Homework 3, due September 19 (Monday).
• Homework 4, due September 30 (Friday).
• Homework 5, due October 10 (Monday).
• Homework 6, due October 17 (Monday).
• Homework 7, due October 31 (Monday).
• Homework 8, due November 7 (Monday).
• Homework 9, due November 14 (Monday).
• Homework 10, due December 2 (Friday).
• Homework 11, due December 9 (Friday).

Content of the lectures:

• Lecture 1 (Mon, Aug 22): Complex numbers and the complex plane: a brief tour of the history of the concept of a number:
  • counting objects - introducing the natural numbers N;
  • what is 3−5? - introducing the integers Z;
  • what is 3 divided by 5? - introducing the rational numbers Q;
  • the roots of the quadratic equation x²−2=0 are ±2^1/2 which are not rational numbers (with a proof) - introducing the real numbers R;
  • what are the roots of the quadratic equation x²+1=0? - defining the number i:=(−1)^1/2;
  • introducing the complex numbers C - all numbers of the form x+iy with x,y∈R.
  Every quadratic equation has exactly two roots in C (if counted with their multiplicities, i.e., a double real root is counted as two roots).
• Lecture 2 (Wed, Aug 24): Complex numbers and the complex plane: complex numbers z=x+iy; the set of complex numbers C; real x=Re(z) and imaginary y=Im(z) parts of a complex number z; complex conjugate y^*=x−iy of a complex number z=x+iy; addition, subtraction, multiplication and division of complex numbers; modulus |z|=(x²+y²)^1/2 of a complex number z=x+iy; remark: |z₁z₂|=|z₁||z₂|, |z₁/z₂|=|z₁|/|z₂|, but |z₁+z₂| is NOT equal to |z₁|+|z₂|; the complex plane - representing a complex number z=x+iy as a point (x,y) in the plane; domains in the complex plane (circles, ellipses); geometric interpretation of addition of complex numbers and of complex conjugation; polar form of complex numbers [Sec. 4.1]
  Functions of a complex variable: a complex-valued function ƒ(z) of a complex variable z=x+iy ƒ(z)=u(x,y)+iv(x,y), where u(x,y) and v(x,y) are real-valued functions of two variables; examples [pages 165-166 of Sec. 4.2]
  Euler's formula and the polar form of complex numbers: defining the exponent of a complex number by a power series (the same series as the Taylor series of an exponent of a real number); derivation of Euler's formula e^y=cos(y)+isin(y) for y∈R; the "magic" relation e^iπ+1=0; exponent of a complex number z=x+iy: e^z=e^x+iy=e^xe^iy=e^x[cos(y)+isin(y)]; Cartesian and polar coordinates in the complex plane - writing the complex number z=x+iy as z=re^iy [pages 169-171 of Sec. 4.3]
• Lecture 3 (Fri, Aug 26): Euler's formula and the polar form of complex numbers (cont.): arg and Arg of a complex number: arg(θ)=Arg(Θ)+2πn, n∈Z; multiple-valued functions (arg is a function with infinitely many values); elementary rules for working with complex numbers written in polar form; de Moivre's formula and its use to derive trigonometric identities; integer powers and roots of complex numbers, examples [pages 171-175 of Sec. 4.3]
  Trigonometric and hyperbolic functions: definitions and examples [pages 176-178 of Sec. 4.4]
  The logarithms of complex numbers: definition [page 181 of Sec. 4.5]
  Reading assignments: (1) Branch cuts in the complex plane [page 171 of Sec. 4.3]; (2) compuing a definite integral of e^−αtsin(t) over t from 0 to ∞ by using Euler's formula [pages 172-173 of Sec. 4.3]
  
• Lecture 4 (Mon, Aug 29): The logarithms of complex numbers (cont.): elementary properties of ln(z); Ln(z) versus ln(z); examples [Sec. 4.5]
  Powers of complex numbers: definition: z^c:=e^cln(z); examples: integer real powers, rational real powers, irrational real powers, complex powers [Sec. 4.6]
  Functions, limits, and continuity of complex-valued functions of a complex variable: a brief discussion of branch cuts; limits and continuity; analogies with functions of two real variables [skim through pages 870-871 of Sec. 18.1]
  Differentiation: The Cauchy-Riemann equations: definition of a derivative of a complex-valued function ƒ(z)=u(x,y)+iv(x,y) of a complex variable z=x+iy; computing derivatives of the complex-valued function ƒ(z)=3x−iy by taking the limit Δz→0 for different choices of Δz: when Δz=Δx∈R, the limit is 3, and when Δz=iΔy with Δy∈R, the limit is −1; derivation of Cauchy-Riemann equations [pages 875-876 of Sec. 18.2; please read the handout]
  Thinking assignment: Think of two complex numbers, z₁ and z₂, such that Ln(z₁z₂) is not equal to Ln(z₁)+Ln(z₂); recall that ln(z₁z₂)=ln(z₁)+ln(z₂) always holds!
• Lecture 5 (Wed, Aug 31): Differentiation: The Cauchy-Riemann equations (cont.): functions ƒ:C→C analytic at a point z₀; functions analytic in an open susset D of C; entire functions (analytic in C); practical "rule" for recognizing whether a function ƒ(z) is analytic - it is analytic if it can be written as a differentiable (in the sense of Caluculus I) function of z=x+iy (but not of x and y separately); rules for differentiation of complex-valued functions of complex variables; a function u(x,y) is said to be harmonic if it satisfies Laplace's equation Δu(x,y)=0, where Δ:=∂_xx+∂_yy is the Laplace's operator (or Laplacian); the real, u(x,y), and the imaginary, v(x,y), parts of an analytic functions ƒ(z)=u(x,y)+iv(x,y) are harmonic functions; regular and singular points; isolated and non-isolated singularities; poles of order n; examples; Cauchy-Riemann equations if z is written in polar form; branch points as singularities [pages 875-878, 880, 881 of Sec. 18.2]
  Reading/thinking assignment: The harmonic function v(x,y) is said to be a harmonic conjugate of the harmonic function u(x,y) if the function ƒ(z)=u(x,y)+iv(x,y) is differentiable, i.e., if u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations; think how you would find v(x,y) if u(x,y) is known (say, u(x,y)=x²−y²+8x) [page 878 of Sec. 18.2]
• Lecture 6 (Fri, Sep 2): Complex integration: Cauchy's theorem: definition of integration along a path in the complex plane; examples; properties of complex integration; terminology: closed curve, simple curve, connected domain, simply-connected domain, multiply-connected domain; Cauchy-Goursat Theorem; proof Cauchy-Goursat Theorem in the case when ƒ'(z) is continuous (so that we can apply Green's Theorem from Calculus); proof of the independence of an integral of an analytic function of the path but only on the endpoints (in a simply-connected domain) [pages 882-888 of Sec. 18.3]
• Lecture 7 (Wed, Sep 7): Complex integration: Cauchy's theorem (cont.): theorem that an integral of a continuous function ƒ over a path C is equal to the antiderivative F of ƒ at the end of C minus F at the start of C (but the antiderivative F must be a single-valued function!); examples; proof of the Principle of Path Deformation [pages 888-891 of Sec. 18.3]
  Cauchy's integral formula: statement of the theorem; derivation of the formula [pages 894-895 of Sec. 18.4]
  Reading assignment: Read Example 2 on pages 895-896.
• Lecture 8 (Fri, Sep 9): Cauchy's integral formula (cont.): applications of Cauchy's integral formula to compute integrals of functions with simple poles; generalization to the case of non-simple poles; applications of the generalized Cauchy's integral formula to compute integrals of functions with multiple poles; Cauchy's inequality: if ƒ is analytic on and inside the circle C of radius R centered at a∈C, and |ƒ(z)|≤M for any z∈C, then |ƒ⁽ⁿ⁾(a)|≤Mn!/Rⁿ; Liouville theorem: a bounded entire function must be a constant [pages 895-898 of Sec. 18.4; Problems 21 and 23 of Sec 18.4]
• Lecture 9 (Mon, Sep 12): Taylor series and Laurent series: geometric series; power series; radius of convergence R, circle of convergence |z−R|=a, and disk of convergence |z−R|<a of a power series; the sum of a convergent Taylor series is analytic inside the disk of convergence; every analytic function can be represented by a power series in a disk; Taylor series of an analytic function; explanation of phenomena occurring in functions of real variables in the light of what is happening in the complex plane - the interval of convergence of a power series Σ_nc_n(z−a)ⁿ with real a∈R and real coefficients c_n∈R converges on an interval (a−r,a+r) symmetric around the point a (at the ends of the interval the power series may converge or diverge, so that the power series may converge on (a−r,a+r), [a−r,a+r), (a−r,a+r], [a−r,a+r], or (−∞,∞)) [pages 901-904 of Sec. 18.5]
• Lecture 10 (Wed, Sep 14): Taylor series and Laurent series (cont.): explanation of phenomena occurring in functions of real variables in the light of what is happening in the complex plane - the Taylor series of ƒ(x)=1/(1+x²) (about 0) is divergent for |x|>1 because the function ƒ(z)=1/(1+z²) has singularities at ±i, hence the disk of convergence is only |z|<1 (the Taylor series of 1/(1+x²) about 0 is divergent also at the boundary points x=1 and x=−1 but the convergence/divergence at the boundary does not follow from the general theory and must be studied separately); derivation of the Laurent series; principal part of a Laurent series (the terms with n≤−1); residue of ƒ at a (the coefficient c₋₁ in the Laurent expansion of ƒ about a) [pages 905-907 of Sec. 18.5]
  Reading assignment: Study the text on pages 907, 908 where the Laurent expansions of the function 1/[(1−z)(2−z)] are found in different regions of C, and read Example 5 on pages 908, 909.
• Lecture 11 (Fri, Sep 16) Taylor series and Laurent series (cont.): more examples of expansions in Laurent series by using the formula for the sum of a geometric series (making sure that we use the expansion of 1/(1−w) only when |w|<1) [pages 907-909 of Sectoin 18.5]
  Residues and the Residue Theorem: definition of residue, statement and discussion of the Residue Theorem; an example (computing integral of z⁵e^1/z over the unit circle in positive direction) [pages 911-912 of Sec. 18.6]
• Lecture 12 (Mon, Sep 19) Residues and the Residue Theorem (cont.): proof of the Residue Theorem; examples of using the Residue Theorem; generalizations - contour in negative direction, contour enclosing more than one singularity, non-simple contours; essential singularities; a criterion useful in recognizing poles of order N: a point a is a pole of order N of ƒ(x) if and only if ƒ(x)=g(x)/(x−a)^N for some analytic function g(x); formula for computing the residue of a function at a pole of order N: Res ƒ(a)=lim_z→a(d^N−1/dz^N−1)[(x−a)^Nƒ(z)]/(N−1)!; examples [pages 913-917 of Sec. 18.6]
• Lecture 13 (Wed, Sep 21): Exam 1 [on the material from Sec. 4.1-4.6, 18.1-18.5, covered in Lectures 1-10 and Homework assignments 1-3]
• Lecture 14 (Fri, Sep 23): Evaluation of real definite integrals: integrals of F(sinθ,cosθ) from 0 to 2π; example of computing the value of integral of 1/(5+cosθ) for θ from 0 to 2π; using symmetries of the integrand to compute integrals over other intervals (i.e., for the integrand in the example on can see that the value of the integral over [0,π] will be half of the value of the integral over [0,2π]) [pages 929-930 of Sec. 19.2]
• Lecture 15 (Mon, Sep 26): Evaluation of real definite integrals (cont.): integrals of a rational function F(x)=P(x)/Q(x) (where P(x) and Q(x) are polynomials) over the real line for F(x) decaying fast enough as x→∞; example: computing integral over the real axis of 1/(1+x²); remark: choosing different contours; using symmetry: if the real integral is over x from 0 to ∞ and the integrand F(x) is even, then the integral over [0,∞) is half of the integral over (−∞,∞); integrals of functions with a branch cut; example: computing integral of x^p−1/(1+x) over [0,∞); remarks on integrals with singularities on the contours (as in Problem 18 on page 936) [pages 930-932, 934-936 of Sec. 19.2]
  Reading assignment: Evaluating integrals of F(x)cos(mx) or F(x)sin(mx) over the real line, where F(x) is a rational function of x [pages 933-934 of Sec. 19.2]
• Lecture 16 (Wed, Sep 28 ): Conformal mapping: conformal mapping - a mapping that preserves the angles between curves at the points of intersection; examples: the stereographic projection and the Mercator projection are conformal mappings;
  proof that every analytic function defines a conformal mapping; discussion: if w=ƒ(z) where ƒ(z) is analytic, and we consider the complex numbers z=x+iy∈C and w=u+iv∈C as points (x,y)∈R² (u,v)∈R², then a curve z=Z(t) gets mapped by ƒ to a curve w=W(t):=ƒ(Z(t)), and the tangent vector to the curve Z(t) at the point Z(0) is Z'(0), while the tangent vector to the curve W(t) at the point W(0)=ƒ(Z(0)) is W'(0)=ƒ'(Z(0))Z'(0), which implies that the mapping ƒ rotates the tangent vector Z'(0) by Arg(ƒ'(Z(0)) (in addition to "stretching it");
  Laplacian operator in R²=C: Δ=∂_xx+∂_yy; changes of variables in R² defined by a conformal mapping preserve the Laplace's equation, i.e., if φ(x,y) satisfies Laplace's equation φ_xx+φ_yy=0 (in other words, if φ(x,y) is a harmonic function), then after a conformal change of variables x=X(u,v), y=Y(u,v), the new function ψ(u,v):=φ(X(u,v),Y(u,v)) satisfies Laplace's equation ψ_uu+ψ_vv=0 [pages 959, 960, 967, 968 of Sec. 19.5]
  Reading/thinking assignment (optional): vocabulary used to describe elementary geometric transformations in the complex plane and the corresponding functions describing them:
  (1) translation: adding a complex number, ƒ(z)=z+a for some a∈C;
  (2) rotation: multiplication by a complex number of modulus 1, ƒ(z)=e^iβz for some β∈R;
  (3) dilation: "stretching" the complex plane by a real factor, ƒ(z)=βz for some β∈R;
  (4) inversion: ƒ(z)=1/z ("swapping" the inside and the outside of a unit circle);
  (5) linear fractional transformation: ƒ(z)=(a+bz)/(c+dz) where a, b, c, and d are complex numbers with ad−bc≠0; the linear fractional transformation generalizes the mappings from (1)-(4) [see the figures on pages 955, 956, 959 of Sec. 19.5]
• Lecture 17 (Fri, 30): Conformal mapping and boundary value problems: set-up of the boundary-value problem (BVP) for the Laplace's equation: given a domain D in C with boundary ∂D, find a function φ:D→R satisfying the partial differential equation (PDE) φ(x,y)=0 for all (x,y)∈D and the boundary conditions (BCs) φ|_∂D=ƒ, where ƒ:∂D→R is a given function and φ|_∂D:∂D→R stands for the restriction of the function φ to the boundary ∂D of the domain D;
  idea of the method - use a conformal mapping to simlpify the domain and/or the boundary conditions, then solve the simplified problem, and finally go back to the original coordinates;
  example: solving Laplace's equation Δψ(u,v)=0 in the upper half plane v>0 with boundary conditions ψ(u,0)=β for u<0 and ψ(u,0)=α for u>0 by using the conformal (i.e., analytic) function u+iv=w=ƒ(z)=exp(z)=exp(x+iy) and the fact that the solution of Laplace's equation Δφ(x,y)=0 in the infinite strip 0<y<π with boundary conditions φ(x,0)=α, φ(x,π)=β is φ(x,y)=α+(β−α)y; the concrete expressions for the mapping are obtained by separating the real and imaginary parts in u+iv=w=ƒ(z)=exp(z)=exp(x+iy)=e^x[cos(y)+isin(y)]=e^xcos(y)+ie^xsin(y), i.e., u=U(x,y)=e^xcos(y), v=V(x,y)=e^xsin(y), the inverse transformations are obtained by inverting these relations: x=X(u,v)=(1/2)ln(u²+v²), y=Y(u,v)=arctan(v/u), so that the solution of the BVP for ψ is ψ(u,v)=φ(X(u,v),Y(u,v))=α+(β−α)Y(u,v)=α+(β−α)arctan(v/u) [pages 970-973 of Sec.19.6]
  Reading/thinking assignment: Look at Table 19.1 in the book illustrating the use of conformal transformations for mapping one domain in C to another [pages 962-966 of Sec. 19.5]
• Lecture 18 (Mon, Oct 3): Fourier transform: integral transforms; integral kernel K(s,t) of an integral transform; example: Laplace transform, K(s,t)=H(s)e^−st; definition of Fourier transform (FT), K(ω,t)=e^−iωt; inverse FT; definition of the Dirac δ-function δ(x) and the "shifted" δ-function δ_a(x)=δ(x−a); derivatives of δ(x) defined by integral of δ_a⁽ⁿ⁾(x) times ƒ(x) over R equals (−1)ⁿƒ⁽ⁿ⁾(a); FT of δ_a(x); using the inverse FT of the FT of δ(x) to prove the integral representation of δ(x) as (1/2π) multiplied by integral of e^iωt with respect to ω over the whole real line; the FT of e^iω₀t (where k₀ is a real constant) is δ_ω0(ω)=δ(ω−ω₀); shifting properties of the FT; FT of derivatives ƒ⁽ⁿ⁾; remark: the value of (the inverse FT of the FT of ƒ) at the point a is equal to ƒ(a) if ƒ is continuous at a, and is equal to the average value of the left limit (i.e., as x→a⁻) and the right limit (i.e., as x→a⁺) of ƒ(x) [pages 845-851 of Sec. 17.5; skip Example 2 on page 849]
• Lecture 19 (Wed, Oct 5): Fourier transform (cont.): definition of a convolution ƒ∗g of two functions ƒ and g; commutativity, ƒ∗g=g∗ƒ, and associativity,(ƒ∗g)∗h=ƒ∗(g∗h), of the convolution; the FT of ƒ∗g is equal to (2π)^1/2 times the product of the FT of ƒ and the FT of g; Parseval's (Plancherel's) theorem [pages 851-854 of Sec. 17.5; skip the 3-dimensional FT on page 852 and Example 2 on pages 852, 853]
  Fourier transforms and partial differential equations: solving the heat equation, u_t=α²u_xx on the whole real line (i.e., for x∈R), with initial condition u(x,0)=g(x) for a given function g (the initial temperature) by performing Fourier transform with respect to the spatial variable [pages 856, 857 of Sec. 17.6]
• Lecture 20 (Mon, Oct 10): Fourier transforms and partial differential equations (cont.): fundamental solution of the heat equation on R (with initial condition δ(x−a)); the solution of the heat equation on R with an arbitrary boundary condition u(x,0)=g(x) is equal to the convolution of g(x) and the fundamental solution;
  wave equation on R: u_xx(x,t)=(1/v²)u_tt(x,t); general solution of the wave equation on R - superposition of waves propagating to the left and waves propagating to the right with velocity v: u(x,t)=φ(x+vt)+ψ(x−vt); checking that this expression satisfies the wave equation (by using the chain rule); solving the wave equation by peforming a Fourier transform with respect to the spatial variable x for a general initial position and zero initial velocity [pages 859-860 of Sec. 17.6]
• Lecture 21 (Wed, Oct 12): Fourier transforms and partial differential equations (cont.): finishing the derivation of the solution of the wave equation from Lecture 20 (using the representation of the delta function as an integral); D'Alembert's formula for the solution of a general initial-value problem for the wave equation on R; formula for differentiating with respect to a parameter of an integral whose limits are functions of the parameter and whose integral is a function of the integration variable and of the parameter, using this formula to prove that the D'Alembert's formula indeed gives the solution of the initial-value problem for the wave equation; Laplace equation in the upper half-plane - expressing the FT of u(x,y) as a product and using the formula for the FT of a convolution of two functions to derive the Poisson's integral formula for the solution of Laplace's equation in the upper half-plane [pages 860-861 of Sec. 17.6]
  Reading assignment: Read the derivation of the fact that the derivative of the Heaviside function H_a is the Dirac delta-function δ_a from page 14 of the latest version of the Notes on the Fourier transform handout.
• Lecture 22 (Fri, Oct 14): Vector spaces and linear transformations: vector space (linear space) V - a set of elements (called vectors) with an operation addition of two vectors with properties
  • (A1) associativity,
  • (A2) existence of a zero vector 0,
  • (A3) existence of an opposite vector for any vector u∈V,
  • (A4) commutativity,
  and an operation multiplication of a vector by a number with properties
  • (M1) α(βu)=(αβ)u,
  • (M2) distributivity with respect to addition of numbers,
  • (M3) distributivity with respect to addition of vectors,
  • (M4) normalization (1u=u);
  simple properties:
  • uniqueness of 0,
  • uniqueness of the opposite vector to a vector u,
  • 0u=0,
  • the opposite to u is (−1)u,
  • u+u=2u;
  subspace of a linear space; linearly independent and linearly dependent vectors; span of a set of vectors - the set of all linear combinations of these vectors; basis of a vector space - a set of linearly independent vectors {v₁,...,v_k} such that span{v₁,...,v_k}=V; dimension of a linear space V - the number of vectors in a basis of V [pages 436-439 of Sec. 9.5]
• Lecture 23 (Mon, Oct 17): Vector spaces and linear transformations (cont.): expansion of an arbitrary vector u∈V in a basis v₁,...,v_n; components of a vector u∈V;
  matrices, addition of matrices of the same size, multiplication of a matrix by a number; the set Mat_m,n(R) of m×n matrices with real entries forms a (real) vector space of dimension mn; a possible basis B^(k,l) (with k=1,...,m and l=1,...,n) of Mat_m,n(R) defined by (B^(k,l))_ij=δ_ikδ_jl (i.e., having 1 at the (k,l)th position and all other entries equal to 0);
  transposed matrix; matrix multiplication; matrix multiplication is associative (Exercise: prove this!), but generally non-commutative; zero matrix (O)_ij=0 for any indices i and j; diagonal matrix (only for square matrices) (A)_ij=0 if i≠j; unit matrix (only for square matrices) (I)_ij=δ_ij;
  Levi-Civita symbol ε_i1...in in Rⁿ; determinant of an n×n matrix; det(AB)=det(A)det(B); a matrix is said to be singular (non-singular) if its determinant is zero (non-zero); a matrix is invertible exactly when it is non-singular; trace of a matrix; tr(A+B)=tr(A)+tr(B);
  linear operators (linear transformations) from V to V;
  matrix elements of a linear operator in a basis [pages 439-440 of Sec. 9.5, 418-425 of Sec. 9.3, 400-403 of Sec. 9.1]
• Lecture 24 (Wed, Oct 19): Exam 2 [on the material from Sec. 18.6, 19.2, 19.5, 19.6, 17.5, 17.6, covered in Lectures 11, 12, 14-21 and Homework assignments 4-6]
• Lecture 25 (Fri, Oct 21): Vector spaces and linear transformations (cont.): components of the vector Au (where A is a linear operator with a matrix A) written as a column vector are equal to the product of the matrix A and the column vector u; composition of operators; the matrix of a composition of two operators is equal to the product of the matrices of the two operators (in the same order);
  normed vector space; examples of norms: ||u||_p for p∈[0,∞] (including p=∞) [pages 457, 458 of Sec. 10.1, 419-423 of Sec. 9.3]
• Lecture 26 (Mon, Oct 24): Vector spaces and linear transformations (cont.): inner product vector space; norm in an inner product vector space ||u||:=(⟨u,u⟩)^1/2; Cauchy-Schwarz inequality (with proof); example of an inner product in Rⁿ corresponding to a symmetric positive-definite n×n matrix Q;
  space of functions from [a,b] to R; endowing this space with a structure of a vector space by defining addition of functions (ƒ+g)(x):=ƒ(x)+g(x) and multipication of a function by a number (αƒ)(x):=αƒ(x); examples of norms in function spaces; L_p([a,b]) spaces; inner product in a function space, weight function; angle between two vectors, orthogonal vectors [pages 444-446 of Sec. 9.6]
  Reading assignment: A Change of basis handout.
• Lecture 27 (Wed, Oct 26): Vector spaces and linear transformations (cont.): vector spaces of polynomials; inner products in vector spaces of polynomials; examples of sets of orthogonal polynomials (Legendre, Laguerre, Hermite, Chebyshev polynomials);
  quantum mechanics notations: ket vectors |u⟩ (ordinary vectors), bra vectors ⟨u| (linear functionals on the space of ket vectors);
  definition of a linear functional on a vector space V; endowing the set of linear functionals on V with operations addition of two functionals and multilpying a functional by a number: (l₁+l₂)(u):=l₁(u)+l₂(u), (αl)(u):=αl(u) ∀u∈V, ∀α∈R; dual space V^* of the linear space V - the linear space of linear functionals on V;
  Riesz Theorem - in an inner product vector space V, every linear functional can be represented as an inner product with an appropriately chosen vector from V; ⟨v| is the linear functional of taking inner product with the vector |v⟩∈V; thanks to the Riesz Theorem we can denote by ⟨v+αw| the linear functional ⟨v|+α⟨w|;
  |u⟩⟨v| is an operator on V; Π_u:=||u||⁻²|u⟩⟨u| is the orthogonal projection operator onto the one-dimensional subspace spanned by the vector u [pages 447-448 of Sec. 9.6]
  Reading/thinking assignment: Question: if |u₁⟩, ..., |u_n⟩ is an orthogonal basis of V, what is the meaning of the linear operator equal to the sum of Π_ui:=||u_i||⁻²|u_i⟩⟨u_i| over i from 1 to n=dim(V)?
  Answer: the sum of the operators Π_ui over i from 1 to n=dim(V) is the identity operator on V, I:V→V defined by Iv=v ∀v∈V; orthonormal basis: if {|u_i⟩}_i=1,...,n is an orthogonal basis of V, then the vectors {|v_i⟩}_i=1,...,n defined by |v_i⟩:=|u_i⟩/||u_i|| form an orthonormal basis: ⟨v_i,v_k⟩=δ_ik; completeness property of an orthonormal system of vectors: sum of |i⟩⟨i| over i equals the identity operator I [pages 10, 11 of the notes from Lecture 27]
• Lecture 28 (Fri, Oct 28): Vector spaces and linear transformations (cont.): discussion of the completeness property of an orthonormal system of vectors; examples of incomplete and complete orthonormal sets of vectors in R³; the components a_ij of a linear operator A in an orthonormal basis |i⟩ are equal to a_ij=⟨i|A|i⟩; writing a linear operator in an orthonormal basis in as a double sum of |i⟩a_ij⟨j| over i and j (derivation: write A=IAI, then write each identity operator I as a sum of projection operators onto span(|i⟩) by the completeness relatoin, and use that a_ij=⟨i|A|i⟩); examples of linear operators in R²: rotation R_α by angle α, dilation ("stretching") D_μ by a factor of μ>0, reflection P with respect to a line.
• Lecture 29 (Mon, Oct 31): Orthogonal transformations: definition of an orthogonal transformation in an inner product vector space (a transformation that preserves the inner product); an orthogonal transformation also preserves the angles; orthogonality condition on the matrix of the operator: A^TQA=Q; for the Euclidean inner product Q=I=(δ_ij), the orthogonality condition reads A^TA=I; simple consequences: for A orthogonal, A⁻¹=A^T, det(A)=±1 [pages 457-460 of Sec. 10.1]
  Eigenvalues and eigenvectors: definition of an eigenvector and the corresponding eigenvalue of a linear operator and of a matrix; the eigenvalues λ_i satisfy the polynomial equation det(A−λI)=0; an example of computing the eigenvalues and eigenvectors of a matrix; if all eigenvalues of a matrix areal and distinct, then the corresponding eigenvectors are linearly independent and, therefore, form a basis [pages 462-466 of Sec. 10.2]
• Lecture 30 (Wed, Nov 2): Symmetric and Hermitean matrices: definition of a symmetric matrix (S^T=S), antisymmetric matrix (A^T=−A), and Hermitean matrix (H^T=H^*); a symmetric operator (⟨u|S|v⟩=⟨v|S|u⟩), an antisymmetric operator (⟨u|A|v⟩=−⟨v|A|u⟩), and a Hermitean operator (⟨u|H|v⟩=⟨v|H|u⟩^*); physical importance of symmetric/Hermitean matrices: the moment of inertia of a solid body and the kinetic energy of a point in generalized coordinates are given by symmetric matrices, and the Hamilton's operator in quantum mechanics is Hermitean; decomposing an arbitrary square matrix into its symmetric and antisymmetric parts: C=C^(s)+C^(a), where C^(s)=(C+C^T)/2, C^(a)=(C−C^T)/2; proof that if S=(a_ij) is symmetric and A=(b_ij) is antisymmetric, then the double sum of s_ija_ij over all values of i and j equals zero; properties of symmetric/Hermitean matrices: all their eigenvalues are real, and the eigenvectors corresponding to distinct eigenvalues are orthogonal to one another; if all eigenvalues λ_i of a symmetric/Hermitean matrix S are distinct (they are automatically real), then its eigenvectors form an orthogonal basis which can be normalized to obtain an orthonormal basis |i⟩:=|f_i⟩; this orthonormal basis satisfies ⟨i|j⟩=δ_ij and sum over all i of |i⟩⟨j| is the identity operator I with matrix I=(δ_ij); the operator S in this basis of eigenvectors of S is equal to sum over all values of i of |i⟩λ_i⟨i| (i.e., S is a sum of projections onto the subspaces spanned by the basis vectors multiplied by the corresponding eigenvalue; equivalently, the matrix S in this basis is diagonal with the eigenvalues λ_i on the main diagonal [pages 466-469 of Sec. 10.2]
  Ordinary differential equations and linear algebra: definition of the k-fold composition A^k of the operator A:V→V; definition of the k-fold product A^k of the matrix A; definition of exp(A)=e^A; application: the solution of the initial value problem x'(t)=Ax(t), x(0)=x⁽⁰⁾ for a constant matrix A=(a_ij) is x(t)=exp(tA)x⁽⁰⁾; Question: What is exp(tA) if A is diagonal with entries λ₁,...,λ_n, or a 2×2 matrix with entries λ on the main diagonal, and 1 in the upper right corner? [pages 494, 495 of Sec. 10.5]
• Lecture 31 (Fri, Nov 4): Ordinary differential equations and linear algebra (cont.): computing exp(At) when A is diagonal; computing exp(At) when A is a 2×2 matrix with entries λ on the main diagonal and 1 in the upper right corner by solving the initial value problem x'(t)=Ax(t), x(0)=x⁽⁰⁾, in which case the answer is x(t)=exp(tA)x⁽⁰⁾; computing exp(At) in the case when A has only real eigenvalues by first choosing a basis in which A is in normal form [pages 471-475 of Sec. 10.3]
• Lecture 32 (Mon, Nov 7): Ordinary differential equations and linear algebra (cont.): proof that e^SAS−1=Se^AS⁻¹ and using this to solve the initial value problem x'(t)=Ax(t), x(0)=x⁽⁰⁾; constructing the matrix S of change of basis as the eigenvectors stacked together as row vectors (in the case when all eigenvalues are real and distinct); constructing the matrix S if the matrix A is symmetric (then the eigenvalues are real, the eigenvectors are orthogonal, and the matrix S is orthogonal, so that S⁻¹=S^T); solving systems of constant coefficient ODEs by writing the solution x(t) as a superposition of e^λ_jtu_j where λ_j are the eigenvalues (assumed all real and distinct) and u_j are the corresponding eigenvectors [pages 493-495 of Sec. 10.5]
• Lecture 33 (Wed, Nov 9): Vectors in plane polar coordinates: Cartesian coordinates in R²; position, velocity and acceleration of a particle moving in R²; polar coordinates in R²; unit vectors e_r and e_θ in direction of positive change of r, resp. θ (while keeping all other coordinates fixed); components u_r and u_θ of a vector u in the basis (e_r, e_θ): u=u_re_r+u_θe_θ; explicit expressions for e_r and e_θ from geometry: e_r=cos(θ)i+sin(θ)j, e_θ=−sin(θ)i+cos(θ)j; derivatives of e_r and e_θ: de_r/dθ=e_θ, de_θ/dθ=−e_r; expressions for the velocity and the acceleration in the basis (e_r, e_θ): v(t)=r'(t)=r'(t)e_r+r(t)e_θ, a(t)=v'(t)=(r''−rθ'²)e_r+(2r'θ'+rθ'')e_θ; central forces and angular momentum conservation (Example 1); obtaining the components in an orthonormal basis by using dot product; scale factors (metric coefficients) h_r and h_θ defined by ∂r/∂r=:h_re_r, ∂r/∂θ=:h_θe_θ; fundamental relations: dr=h_re_rdr+h_θe_θdθ, ds²=h_r²dr²+h_θ²dθ²; geometric derivation of the expression for ds² [pages 349, 350 of Sec. 8.1, pages 355-358 of Sec. 8.2]
• Lecture 34 (Fri, Nov 11): Vectors in plane polar coordinates (cont.): operations in orthogonal (in particular, polar) coordinates: differential operators, derivation of the expression for the gradient, ∇=h_r⁻¹e_r∂_r+h_θ⁻¹e_θ∂_θ, divergence in polar coordinates, Laplacian in polar coordinates [pages 358-361 of Sec. 8.2]
  Curvilinear coordinates: converting from Cartesian (x,y,z) to general curvilinear coordinates (u₁,u₂,u₃) [page 378 of Sec. 8.5]
• Lecture 35 (Mon, Nov 14): Curvilinear coordinates (cont.): tangent vectors ∂r/∂u_j to the curves r(u₁,u₂,u₃) when u_j increases and all u_i with i≠j are kept constant; volume element in R³: |(∂r/∂u₁×∂r/∂u₂)⋅∂r/∂u₃|;
  in the rest of this lecture we consider only orthogonal curvilinear coordinates: metric coefficients h_j:=|∂r/∂u_j|, so that ∂r/∂u₁=h₁e₁, ∂r/∂u₂=h₂e₂, ∂r/∂u₃=h₃e₃; infinitesimal displacement element dr=(∂r/∂u₁)du₁+(∂r/∂u₂)du₂+(∂r/∂u₃)du₃=h₁e₁du₁+h₂e₂du₂+h₃e₃du₃; volume element in orthogonal curviliner coordinates in R³: dV=dxdydz=h₁h₂h₃du₁du₂du₃; area element in orthogonal curviliner coordinates in R²: dA=dxdy=h₁h₂du₁du₂; the gradient in orthogonal curvilinear coordinates in R³: ∇=h₁⁻¹e₁(∂/∂u₁)+h₂⁻¹e₂(∂/∂u₂)+h₃⁻¹e₃(∂/∂u₃); example: geometric determination of the metric coefficients and the volume element in spherical coordinates in R³ [pages 378-379 of Sec. 8.5]
  Thinking assignment: draw/imagine the polar, cylindrical, and spherical coordinates, think about the geometric meaning of the vectors ∂r/∂u_j, and figure out without doing any calculations the expressions for the metric coefficients h_j in each case.
  Reading/thinking assignment: skim through Sec. 8.4, "Spherical coordinates," identify there the expressions for ds², dV, h_ρ, h_θ, h_φ, ∇, and look at the derivations of the expressions for div, curl, and the Laplacian Δ=∇² (Warning: instead of (ρ,θ,φ), the book uses the notations (r,φ,θ) for the spherical coordinates!) [Sec. 8.4]
• Lecture 36 (Wed, Nov 16): Exam 3 [on the material from Sec. 9.1, 9.3, 9.5, 9.6, 10.1-10.3, 10.5, covered in Lectures 22, 23, 25-32 (see also the lecture notes from these lectures and Homework assignments 7-9)]
• Lecture 37 (Fri, Nov 18): General (not necessarily orthogonal) curvilinear coordinates: metric tensor g=(g_ij), where g_ij:=(∂r/∂u_i)⋅(∂r/∂u_j); for orthogonal curvilinear coordinates (g_ij) is diagonal, with g_ii=h_i²; the square, ds², of the line element ds is equal to the double sum over i and j of g_ijdu_idu_j; length of a parametrically defined curve Γ in R³_x,y,z: if the curve is given by a function U:[a,b]→R³_u1,u2,u3 which afterwards is mapped to R³_x,y,z, then the length of Γ is given integral over t∈[a,b] of [Σ_i,kg_ik(dU_i/dt)(dU_k/dt]^1/2; integral of a function ƒ:R³_x,y,z→R over a curve Γ in R³_x,y,z; proof that the area element in R² is given by dA=[det(g_ij)]^1/2du₁du₂; integral of a function ƒ:R²_x,y→R over a 2-dimensional region in R²_x,y.
• Lecture 38 (Mon, Nov 21): General (not necessarily orthogonal) curvilinear coordinates (cont.): the volume element dV in R³ in curvilinear coordinates is equal to [det(g_ij)]^1/2du₁du₂du₃.
  Calculus of variations - introduction: Fermat's principle in optics: paths of light rays in geometric optics are such that the "optical length" of a path, i.e., the time light needs to get from a point A to a point B, is minimal; Snell's law in optics as a consequence of Fermat's minimal time principle; Johann Bernoulli's brachistochrone challenge (1696).
  The Euler-Lagrange equation: functionals; the action I[q] as a functional of a function q(t); idea of obtaining the function q(t) from the condition δI[q]=0 of extremizing the action over a given time inteval for given values of q(t) at the initial and the final moments [pages 985-987 of Sec. 20.1]
• Lecture 39 (Mon, Nov 28): The Euler-Lagrange equation (cont.): functionals; the action I[q] as a functional of a function q(t); derivation of the Euler-Lagrange equation from the condition of extremizing the action: δI[q]=0; generalization for the case of N degrees-of-freedom system: q(t)=(q₁(t),q₂(t),...,q_N(t)), δI[q]=0, Euler-Lagrange equations; the Lagrangian function is the difference between the kinetic and the potential energy: L=T−U; deriving the Newton's second law from a variational principle; generalized coordinates q_i(t), generalized momenta p_i=∂L/∂q_i', generalized forces ∂L/∂q_i [pages 986-988, 993 of Sec. 20.1]
• Lecture 40 (Wed, Nov 30): The Euler-Lagrange equation (cont.): deriving the Lagrangian function and Euler-Lagrange equations for the Bernoulli's brachistochrone problem; Euler-Lagrange equations when the Lagrangian depends on several functions: L=L(q(t),q'(t),t), where q(t)=(q₁(t),q₂(t),...q_n(t)); energy conservation: if the physical system is autonomous, i.e., its Lagrangian does not depend explicitly on time, L=L(q(t),q'(t)), then E=q_i'∂L/∂q_i'−L=const [pages 988-991 of Sec. 20.1]
• Lecture 41 (Fri, Dec 2): Two laws of physics in variational form: generalization when the "Lagrangian" depends on higher derivatives of q(t): L=L(q(t),q'(t),q''(t),...q⁽ⁿ⁾(t),t); Euler-Lagrange equation when the unknown function depends on several variables, i.e., has the form u(t,x,y).
  Multidimensional variational problems: setting up the problem for the motion of the membrane ("soap film") whose shape at time t is described by z=u(x,y,t), (x,y)∈D⊂R² of a membrane attached to a horizontal rim ∂D in a gravity field g=−gk; derivation of the expression for the potential energy, which is equal to the sum of the potential energy of the membrane in the gravity field (equal to the double integral over D of ρgu(x,y), where ρ is the area density of the mass of the membrane) and the tension energy (equal to the surface tension τ times the difference between the stretched and the unstretched areas of the membrane); obtaining the expression for the stretching energy of the membrane in lowest approximation: (τ/2)|∇u|²=(τ/2)(u_x²+u_y²).
• Lecture 42 (Mon, Dec 5): Multidimensional variational problems (cont.): setting up the variational problem for a vibrating membrane; derivation of the Euler-Lagrange equation for the motion of a membrane hanging in the gravity field; discussion of the physical meaning of the equation, the speed of the waves is c=(τ/ρ)^1/2; incorporating an air-resistance term in the equation; setting up and solving the boundary value problem for the shape of a hanging circular membrane attached at its boundary: deriving the expression for the Laplacian of a function depending only on r but not on θ by using two methods (direct differentiation and using the scale factors h_r=1 and h_θ=r); solving the boundary-value problem for the radial part [pages 1015-1017 of Sec. 20.5]

Good to know: the greek_alphabet, some useful notations.