MATH 3423 - Physical Mathematics II, Section 001 - Fall 2019
TR 10:30-11:45 a.m., 809 PHSC

Instructor: Nikola Petrov, 1101 PHSC, npetrov AT math.ou.edu

Office Hours: Tue 9:30-10:30 a.m., Wed 12:00-1:00 p.m., or by appointment, in 1101 PHSC.

First day handout

Prerequisites: MATH 2443 (Calc and Analytic Geom IV) or MATH 2934 (Diff & Int Calc III), MATH 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:

• Homework 1, due August 29 (Thursday).
• Homework 2, due September 5 (Thursday).
• Homework 3, due September 12 (Thursday).
• Homework 4, due September 19 (Thursday).
• Homework 5, due October 3 (Thursday).
• Homework 6, due October 10 (Thursday).
• Homework 7, due October 22 (Tuesday).
• Homework 8, due November 7 (Thursday).
• Homework 9, due November 14 (Thursday).
• Homework 10, due November 21 (Thursday).
• Homework 11, due December 5 (Thursday).

Content of the lectures:

• Lecture 1 (Tue, Aug 20)
  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);
  • minimizing the potential energy of a membrane hanging in the Earth's gravity field;
  • Queen Dido's isoperimetric problem;
  • minimizing the area of a soap film.
  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 interval for given values of q(t) at the initial and the final moments
  [pages 985-986 of Sec. 20.1]
• Lecture 2 (Thu, Aug 22)
  The Euler-Lagrange equation (cont.):
  • 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;
  • 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;
  • the Lagrangian function is the difference between the kinetic and the potential energy: L=T−U;
  • deriving 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;
  • example: derivation of the equations of motion of a point mass attracted to a (much heavier) center through a central force; observing that the angular momentum and the energy are conserved and using this to integrate the equations of motion;
  • a discussion of the connection between the symmetries and conservation laws:
    • homogeneity of time implies law of conservation of energy;
    • homogeneity of space implies law of conservation of momentum;
    • isotropy of space implies law of conservation of angular momentum
  [pages 986-989 of Sec. 20.1, 995-998 of Sec. 20.2]
• Lecture 3 (Tue, Aug 27)
  Multidimensional variational problems:
  • Euler-Lagrange equation when the unknown function depends on several variables, i.e., has the form u(t,x);
  • setting up the problem for the planar motion of a string of length l, linear density λ, tension τ in the Earth's gravity field g=−gk, assuming that the ends of the string are firmly attached; the shape of the string at time t is described by z=u(x,t), where x∈[0,l] and t∈[0,∞);
  • derivation of the expression for the density of the kinetic energy, λu_t²/2,
  • derivation of the expression for the density of the gravitational potential energy, λgu;
  • derivation of the expression for the density of the elastic potential energy, τ[(1+u_x²)^1/2−1]≈τu_x²/2.
• Lecture 4 (Thu, Aug 29)
  Multidimensional variational problems (cont.):
  • setting up the variational problem for a vibrating string;
  • derivation of the Euler-Lagrange equation for the planar motion of a string hanging in the gravity field;
  • discussion of the physical meaning of the solution u(x,t)=ƒ(x−ct)+g(x+ct) of the wave equation as waves propagating to the right and to the left with speed c=(τ/λ)^1/2;
  • setting up and solving the boundary value problem for the shape of a hanging string attached to two points (at the same height);
  • finding the shape of the steady state of the string and its maximumum hanging
  [pages 1015-1017 of Sec. 20.5]
• Lecture 5 (Tue, Sep 3)
  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; more generally, every polynomial equation of degree k has exactly k roots in C (if counted with their multiplicities);
  • 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;
  • circles in the complex plane;
  • geometric interpretation of addition of complex numbers and of complex conjugation;
  • polar form of complex numbers
  [Sec. 4.1]
• Lecture 6 (Thu, Sep 5)
  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;
  • exampes
  [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
  • 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
  [pages 169-173 of Sec. 4.3]
  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 7 (Tue, Sep 10)
  Euler's formula and the polar form of complex numbers:
  • integer powers and roots of complex numbers, examples
  [pages 174, 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 of log(z) and Log(z): log(z)=ln|z|+iarg(z), Log(z)=ln|z|+iArg(z);
  • elementary properties of log(z); examples;
  • Log(z) versus log(z)
  [Sec. 4.5]
  Powers of complex numbers:
  • definition: z^c:=e^clog(z);
  • examples: integer real powers, rational real powers, irrational real powers, complex powers
  [Sec. 4.6]
• Lecture 8 (Thu, Sep 12)
  Powers of complex numbers (cont.):
  • more examples: z^(21/2), (1+i)^i.
  Functions, limits, and continuity of complex-valued functions of a complex variable:
  • a brief discussion of multi-valued functions;
  • a brief discussion of branch cuts;
  • limits and continuity;
  • analogies with functions of two real variables
  [skim through pages 870-872 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,
    • when Δz=iΔy with Δy∈R, the limit is −1,
    • conclusion: the function ƒ(z)=3x−iy is not differentiable;
  • derivation of the Cauchy-Riemann (CR) equations;
  • functions ƒ:C→C analytic at a point z₀;
  • functions analytic in an open subset 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 Calculus I) function of z=x+iy (but not of x and y separately);
  • rules for differentiation of complex-valued functions of complex variables (the same as the rules from Calculus I);
  • 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, examples
  [pages 875-879 of Sec. 18.2]
  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!
  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 9 (Tue, Sep 17)
  Differentiation: The Cauchy-Riemann equations (cont.):
  • poles of order n, examples;
  • essential singularities (which are not poles of any finite order);
  • (optional) Cauchy-Riemann equations in polar form
  [pages 878, 880 of Sec. 18.2]
  Complex integration: Cauchy's Theorem:
  • recalling the definition of line integrals from Calculus;
  • definintion of an integral over a curve C in C: parameterize the curve C and compute the integral by using the integration rules from Calculus;
  • an illustrative example;
  • an important example: the integral of zⁿ over a (positively oriented) unit circle centered at the origin is 0 if n≠−1, and 1πi if n=−1;
  • properties of integrals over a curve:
    • linearity,
    • additivity of paths (integral over the "sum" of two paths is the sum of the integrals over each path),
    • integral over the "opposite" path is equal to minus the integral over the original path;
    • an upper bound in the modulus of an integral;
  • terminology:
    • closed curve,
    • simple curve,
    • closed simple curve,
    • connected domain,
    • simply connected domain,
    • multiply-connected domain;
  • Cauchy-Goursat Theorem;
  • proof of Cauchy-Goursat Theorem in the case when ƒ'(z) is continuous (so that we can apply Green's Theorem from Calculus)
  [pages 882-887 of Sec. 18.3]
• Lecture 10 (Thu, Sep 19)
  Complex integration: Cauchy's Theorem (cont.):
  • Corollary of Cauchy-Goursat's Theorem: if C₁ and C₂ are two simple piecewise smooth curves having the same initial and final points, and the function ƒ is analytic in a simply connected domain that contains both C₁ and C₂, the integrals of ƒ along C₁ and along C₂ are the same (i.e., the integral depends only on the endpoints);
  • examples;
  • Remark: the fact that the integral of a function ƒ over a closed curve C is zero does not imply that the function is analytic inside C;
  • Theorem: 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;
  • Principle of Path Deformation (with a proof)
  [pages 888-891 of Sec. 18.3]
• Lecture 11 (Tue, Sep 24)
  Exam 1 [on the material from Sections 20.1, 20.2, 20.5, 4.1-4.6, 18.1-3 covered in Lectures 1-9; the tricks about complex integrals learned in Lecture 10 will also be useful]

• Lecture 12 (Thu, Sep 26):
  Cauchy's integral formula:
  • statement and proof of Cauchy's integral formula;
  • 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
  [Sec. 18.4, Problems 21 and 23 of Sec 18.4]
  Reading assignment: Read Example 2 on pages 895-896 and Example 4 on pages 897-898.
• Lecture 13 (Tue, Oct 1):
  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 (−∞,∞));
    • 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)
  [pages 901-906 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.
• Lecture 14 (Thu, Oct 3):
  Taylor series and Laurent series (cont.):
  • Laurent series of e^z/(z−1) about 1.
  [Example 5 on pages 908, 909]
  Residues and the Residue Theorem:
  • definition of residue;
  • statement and discussion of the Residue Theorem;
  • proof of the Residue Theorem;
  • an example of using the Residue Theorem (computing integral of z⁵e^1/z over the unit circle in positive direction);
  • an example of using the Residue Theorem (computing integral of 1/[(z−1)(z−3+i)] over a circle of radius 5 centered at 0 in positive direction);
  • 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) such that g(a)≠0;
  • 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)!;
  • more examples.
  [pages Sec. 18.6; please read all the examples in this section]
  Evaluation of real definite integrals:
  • integrals of F(sinθ,cosθ) from 0 to 2π;
  • example: computing the value of integral of 1/(5+4cosθ) for θ from 0 to 2π.
  [pages 929-930 of Sec. 19.2]
  Reading assignment: Study the text on pages 930 about using the symmetries of the integrand to compute integrals in which the integration over θ is not from 0 to 2π.
• Lecture 15 (Tue, Oct 8):
  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 the 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]
  Conformal mapping:
  • conformal mapping - a mapping that preserves the angles between curves at the points of intersection;
  • examples of simple conformal transformations in C (we will prove later that they are conformal):
    • translation: adding a complex number, ƒ(z)=z+a for some a∈C;
    • rotation: multiplication by a complex number of modulus 1, ƒ(z)=e^iβz for some β∈R;
    • dilation: "stretching" the complex plane by a real factor, ƒ(z)=βz for some β∈R;
    • inversion: ƒ(z)=1/z ("swapping" the inside and the outside of a unit circle);
    • 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 described above;
  • examples: the stereographic projection and the Mercator projection are conformal mappings
  [see the figures on pages 955-958 of Sec. 19.5]
  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 (Thu, Oct 10):
  Conformal mapping:
  • proof that every analytic function defines a conformal mapping: 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]
  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)=arccot(u/v), 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)arccot(u/v)
  [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 17 (Tue, Oct 15):
  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);
  • 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
  [pages 845-851 of Sec. 17.5; you may skip Example 2 on page 849]
• Lecture 18 (Thu, Oct 17)
  Fourier transform (cont.):
  • Parseval's (Plancherel's) theorem
  [page 853 of Sec. 17.5]
  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
  • fundamental solution of the heat equation on R (with initial condition δ(x−a));
  • wave equation on R: u_xx(x,t)=(1/v²)u_tt(x,t); solving the wave equation by peforming a Fourier transform with respect to the spatial variable x for a general initial position u(x,0)=ƒ(x) and zero initial velocity u_t(x,0) (using the representation of the delta function as an integral)
  [pages 856, 857, 859, 860 of Sec. 17.6]
• Lecture 19 (Mon, Oct 22):
  Fourier transforms and partial differential equations (cont.):
  • 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;
  • 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);
  • 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]

• Lecture 20 (Thu, Oct 24):
  Exam 2 [on the material from Sections 18.3-18.6, 19.2, 19.5, 19.6, 17.5, 17.6 covered in Lectures 10, 12-18]

• Lecture 21 (Tue, Oct 29):
  Vector spaces:
  • vector space (linear space) V - a set of elements (called vectors) with an operation addition of two vectors with properties
    • (A1) associativity: (u+v)+w=u+(v+w) ∀ u,v),w∈V,
    • (A2) existence of a zero vector 0: ∃0∈V s.t. u+0=u ∀u∈V,
    • (A3) existence of an opposite vector: ∀u∈V ∃u'∈V s.t. u+u'=0,
    • (A4) commutativity: u+v=v+u,
    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: (α+β)u=αu+βu ∀α,β∈R ∀u∈V,
    • (M3) distributivity with respect to addition of vectors: α(u+v)=αu+αv ∈V,
    • (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;
  • Rⁿ as a linear space;
  • example: linear (in)dependence of vectors in Rⁿ;
  • 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
  • expansion of an arbitrary vector u∈V in a basis v₁,...,v_n;
  • components of a vector u∈V
  [pages 436-440 of Sec. 9.5]
  Matrices:
  • definition of a matrix,
  • 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
  [pages 418-425 of Sec. 9.3]
  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 handout Notes on the Fourier transform.
• Lecture 22 (Thu, Oct 31):
  Matrices (cont.):
  • Levi-Civita symbol ε_i1...in in Rⁿ;
  • writing the cross product u×v of two vectors in R³ by using the Levi-Civita symbol ε_ijk;
  • 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;
  • property of trace: tr(A+B)=tr(A)+tr(B); tr(αA)=α tr(A), tr(A₁A₂...A_k)=tr(A_kA₁A₂...A_k−1)=... (cyclic permutations of the product A₁A₂...A_k);
  • trace and determinant in the characteristic polynomial of a 2×2 matrix A: det(A−λI)=λ²−tr(A)λ+det(A)
  [pages 400-402 of Sec. 9.1]
  Linear operators:
  • linear operators (linear transformations) from V to V;
  • matrix elements of a linear operator in a basis
  • 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)
  [pages 455-458 of Sec. 10.1]
  Normed and inner product vector spaces:
  • normed vector space;
  • inner product vector space;
  • norm in an inner product vector space ||u||:=(⟨u,u⟩)^1/2;
  • Cauchy-Schwarz inequality (with proof)
  [pages 444-446 of Sec. 9.6]
• Lecture 23 (Tue, Nov 5):
  Normed and inner product vector spaces (cont.):
  • examples of norms: ||u||_p for p∈[0,∞] (including p=∞);
  • 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;
  • vector spaces of polynomials;
  • inner products in vector spaces of polynomials;
  • examples of sets of orthogonal polynomials (Legendre, Laguerre, Hermite, Chebyshev polynomials)
  [pages 444-448 of Sec. 9.6]
  Vector spaces, operators, functionals, etc.:
  • 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 a linear space structure, by defining the operations
    • addition of two functionals: (l₁+l₂)(u):=l₁(u)+l₂(u), and
    • multilpying a functional by a number: (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
  [pages 444-448 of Sec. 9.6]
  Reading assignment: A Change of basis handout.
• Lecture 24 (Thu, Nov 7): Vector spaces, operators, functionals, etc. (cont.):
  • let ⟨v| be the linear functional of taking inner product with the vector |v⟩∈V (recall Riesz Theorem);
  • 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;
  • a set of vectors {|u_i⟩}_i=1,...,n, denoted in an abbreviated way as {|i⟩}_i=1,...,n, is said to be orthogonal if ⟨i|j⟩=0 whenever i≠j, and orthonormal if ⟨i|j⟩=δ_ij;
  • an orthonormal system of vectors {|i⟩}_i=1,...,n in the linear space V is said to be complete if Σ_i=1,...,n|i⟩⟨i| is the identity operator I in V;
  • 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 relation, and use that a_ij=⟨i|A|i⟩).
• Lecture 25 (Tue, Nov 12): Vector spaces, operators, functionals, etc. (cont.):
  • writing the components of the vector Au as a column vector equal to the product of the matrix of the operator A and the column vector of the components of u;
  • 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.
  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, where Q=I=(q_ij) is the (symmetric positive definite) matrix defining the inner product: ⟨u,v⟩=Σ_i,j u_iq_ijv_j;
  • 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]
• Lecture 26 (Thu, Nov 14):
  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]
  Symmetric matrices:
  • definition of a symmetric matrix (S^T=S) and antisymmetric matrix (A^T=−A);
  • a symmetric operator (⟨u|S|v⟩=⟨v|S|u⟩) and an antisymmetric operator (⟨u|A|v⟩=−⟨v|A|u⟩);
  • physical importance of symmetric matrices: the moment of inertia of a solid body and the kinetic energy of a point in generalized coordinates are given by symmetric matrices;
  • 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=(s_ij) is symmetric and A=(a_ij) is antisymmetric, then the double sum of s_ija_ij over all values of i and j equals zero;
  • properties of symmetric matrices: all their eigenvalues are real, and the eigenvectors corresponding to distinct eigenvalues are orthogonal to one another
  [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;
  • computing exp(At) when A is diagonal;
  • 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, then exp(tA) is a matrix function B(t)=(b_ij(t)), where b₁₁(t)=e^λt, b₁₂(t)=te^λt, b₂₁(t)=0, b₂₂(t)=e^λt;
  • 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⁾.
• Lecture 27 (Tue, Nov 19): Ordinary differential equations and linear algebra (cont.):
  • Theorem: the eigenvalues of a symmetric matrix are real, and the eigenvectors corresponding to different eigenvectors are orthogonal;
  • diagonilizing a symmetric matrix S whose eigenvalues are distinct:
    • find the eigenvalues λ_j and the corresponding normalized eigenvectors v_j;
    • stack the normalized eigenvectors as columns to create a matrix O;
    • proof that this matrix is orthogonal, i.e., O^TO=I;
    • the matrix D=SAS⁻¹ is automatically diagonal, with its eigenvalues on the diagonal;
    • computing the exponential exp(At)=exp(S⁻¹DSt)=S⁻¹exp(Dt)S, using also that exp(Dt) is a diagonal matrix with entries e^λ_jt;
  • using this algorithm to solve an initial value problem for a linear system of ODEs x'(t)=Ax(t), x(0)=x⁽⁰⁾, where A is a constant symmetric matrix: the solution is x(t)=exp(At)x⁽⁰⁾;
  • another way to solve an initial value problem x'(t)=Ax(t), x(0)=x⁽⁰⁾ where A is a constant symmetric matrix: find the eigenvalues λ_j and the (not necessarily normalized) eigenvectors u_j, then the general solution of the system of ODEs is a sum over j of C_jexp(λ_jt)u_j, and finding the constants C_j from the initial conditions;
  • Example: solving x'(t)=(8/5)x(t)+(14/5)y(t), y'(t)=(14/5)x(t)−(13/5)y(t) with initial condition x(0)=(2,6).
  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
  • components u_r and u_θ of a vector u in the basis (e_r, e_θ): u=u_re_r+u_θe_θ;
  [pages 349, 350 of Sec. 8.1, page 355 of Sec. 8.2]
• Lecture 28 (Thu, Nov 21):
  Vectors in plane polar coordinates (cont.):
  • 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: ∂e_r/∂θ=e_θ, ∂e_θ/∂θ=−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;
  • 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.
  [pages 355-359 of Sec. 8.2]
• Lecture 29 (Tue, Nov 26):
  Exam 3 [on the material covered in Lectures 21-27]

Good to know: the greek_alphabet, some useful notations.