MATH 3423 - Physical Mathematics II, Section 001 - Fall 2013
MWF 11:30-12:20 p.m., 160 Gould Hall

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

Office Hours: Mon 2:30-3:30 p.m., Wed 10:15-11:15 a.m., or by appointment, in 802 PHSC.

Please take a couple of minutes to fill out your evaluation of the course!
Here is a link to the evaluation web-site: http://eval.ou.edu; it closes on December 8th (Sunday).

Prerequisite: 2443 (Calculus and Analytic Geometry IV), 3413 (Physical Mathematics I).

Course catalog description: Prerequisite: 2443, 3413. 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 4-10, 17-20.
A list of errata in the book (collected by Prof. Daniel Sober).

Check out the OU Math Blog! It is REALLY interesting!

Check out the Problem of the Month!

Homework:

• Homework 1, due August 30 (Friday).
• Homework 2, due September 6 (Friday).
• Homework 3, due September 16 (Monday).
• Homework 4, due September 27 (Friday).
• Homework 5, due October 4 (Friday).
• Homework 6, due October 16 (Wednesday).
• Homework 7, due November 4 (Monday). (Note the new due date!)
• Homework 8, due November 8 (Friday).
• Homework 9, due November 15 (Friday).
• Homework 10, due November 25 (Monday).
• Homework 11, due December 6 (Friday).

Content of the lectures:

• Lecture 1 (Mon, Aug 21): Reminder: Fourier series: Fourier series of a periodic function (using complex exponents or sines and cosines), even and odd functions, sine and cosine series, convergence of Fourier series, Gibbs phenomenon [Sec. 15.1-15.3].
  Reading assignment: Parseval's theorem, physical interpretation of Parseval's theorem [in Sec. 15.3]
• Lecture 2 (Wed, Aug 21): Fourier transform: integral transforms, integral kernel of a transform, example: Laplace transform; definition of Fourier transform (FT), 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); FT of δ(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; FT of e^ik₀t (where k₀ is a real constant); FT of a wave train and relation with the Heisenberg uncertainly relation; shifting properties of the FT; FT of derivatives f⁽ⁿ⁾; definition of a convolution f∗g of two functions f and g, commutativity and associativity of the convolution [pages 845-851 of Sec. 17.5].
• Lecture 3 (Fri, Aug 23): Fourier transform (cont.): convolution properties of the FT, Parseval's theorem [pages 851-854 of Sec. 17.5].
  Fourier transforms and partial differential equations: solving the heat equation on the whole real line by using Fourier transform (for a general initial condition and for an initial condition δ(x−a)), fundamental solution of the wave equation on R [pages 856-857 of Sec. 17.6]
• Lecture 4 (Mon, Aug 26): Fourier transforms and partial differential equations (cont.): solving the wave equation on the whole real line by using Fourier transform (for a general initial condition and zero initial velocity), physical interpretation of the solution; solving Laplace's equation in the upper half-plane by using Fourier transform, Poisson's integral formula [pages 859-861 of Sec. 17.6].
• Lecture 5 (Wed, Aug 28): Fourier transforms and partial differential equations (cont.): multidimensional Fourier transform; solving the heat equation in two spatial dimensions for the case of a rotationally symmetric initial condition - derivation of the fundamental solution of the equation (which turns out to be a product of two fundamental solutions of the heat equation in one spatial dimension) [pages 861, 862 of Sec. 17.6].
  Complex numbers and the complex plane: natural numbers N, integer numbers Z, rational numbers Q, proof that 2^1/2 is not in Q, real numbers R, need for square root of negative numbers in solution of quadratic equations; complex numbers C, real and imaginary parts of a complex number, complex numbers as solutions of quadratic equations, complex conjugate, addition, subtraction, multiplication and division of complex numbers, modulus of a complex number [pages 160-162 of Sec. 4.1]
• Lecture 6 (Fri, Aug 30): Complex numbers and the complex plane (cont.): modulus and argument, the complex plane [pages 162-164 of Sec. 4.1]
  Functions of a complex variable: definition and examples [Sec. 4.2]
  Euler's formula and the polar form of complex numbers: derivation of Euler's formula, Cartesian and polar coordinates in the complex plane, arg and Arg, de Moivre's formula and its use to derive trigonometric identities [pages 169-173 of Sec. 4.3]
• Lecture 7 (Wed, Sep 4): Euler's formula and the polar form of complex numbers (cont.): arg and Arg, branch cuts in the complex plane, multiple-valued functions, integer powers and roots of complex numbers, examples [pages 171-175 of Sec. 4.3]
  Trigonometric and hyperbolic functions: definitions and examples [Sec. 4.4]
• Lecture 8 (Fri, Sep 6): The logarithms of complex numbers: definition and elementary properties, examples [Sec. 4.5]
  Powers of complex numbers: definition, examples: integer real powers, rational 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 [Sec. 18.1]
  Differentiation: The Cauchy-Riemann equations: computing derivatives of a complex-valued function of a complex variable along different lines in the complex plane; derivation of Cauchy-Riemann equations [pages 875-876 of Sec. 18.2]
• Lecture 9 (Mon, Sep 9): Differentiation: The Cauchy-Riemann equations (cont.): analytic functions, entire functions, rules for differentiation of complex-valued functions of complex variables, regular points, isolated and non-isolated singularities, poles of order n [pages 876-878 of Sec. 18.2].
• Lecture 10 (Wed, Sep 11): Differentiation: The Cauchy-Riemann equations (cont.): conjugate pair of harmonic functions (the real and imaginary parts of an analytic function), problem for finding one of them if the other is known; Cauchy-Riemann equations if z is written in polar form; branch points as singularities [pages 878-881 of Sec. 18.2]
  Complex integration: Cauchy's theorem: definition and properties of integration along a path in the complex plane, examples; simple closed curves, simply-connected regions, multiply-connected regions; Cauchy-Goursat Theorem [pages 882-887 of Sec. 18.3]
• Lecture 11 (Fri, Sep 13): Complex integration: Cauchy's theorem (cont.): definition of a more restricted version of the Cauchy-Goursat Theorem (called the Cauchy Theorem) in which f/(z) is required to be 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); theorem that an integral of a continuous function f over a path C is equal to the antiderivative F of f 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 887-891 of Sec. 18.3]
• Lecture 12 (Mon, Sep 16): Cauchy's integral formula: derivation of the formula, applications, generalization to the case of non-simple poles, examples [Sec. 18.4]
• Lecture 13 (Wed, Sep 18): Exam 1 [on the material from Sec. (15.1-15.3), 17.5, 17.6, 4.1-4.6, 18.1, 18.2, covered in Lectures 1-9 and half of Lecture 10]
• Lecture 14 (Fri, Sep 20): Cauchy's integral formula (cont.): Cauchy's inequality: |f⁽ⁿ⁾(a)|≤Mn!/Rⁿ; Liouville theorem: a bounded entire function must be a constant [Problems 21 and 23 of Sec 18.4]
  Taylor series and Laurent series: geometric series; power series; disk of convergence, circle of convergence, and radius of convergence 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 [pages 901-904 of Sec. 18.5]
• Lecture 15 (Mon, Sep 23): 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 f(x)=1/(1+x²) (about 0) is divergent for |x|>1 because the function f(z)=1/(1+z²) has singularities at ±i, hence the disk of convergence is only |z|<1; derivation of the Laurent series [pages 904-907 of Sec. 18.5]
• Lecture 16 (Wed, Sep 25): Taylor series and Laurent series (cont.): principal part of a Laurent series, residue, examples of Laurent series [pages 907-909 of Sec. 18.5]
  Residues and the Residue Theorem: definition of residue, statement and discussion of the Residue Theorem [pages 911-912 of Sec. 18.6]
• Lecture 17 (Fri, Sep 27): 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 f(x) if and only if f(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(f,a)=lim_z→a[(x−a)^Nf(z)]^(N−1)/(N−1)!, examples [pages 913-917 of Sec. 18.6]
• Lecture 18 (Mon, Sep 30): Evaluation of real definite integrals: integrals of F(sinθ,cosθ) from 0 to 2π, example, using symmetries of the integrand to compute integrals over [0,π], etc.; 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, choosing different contours, 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 (−∞,∞) [pages 929-933 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 19 (Wed, Oct 2): Evaluation of real definite integrals (cont.): integrals of functions with a branch cut; integrals with singularities on the contours (Problem 18 on page 936) [pages 934-936 of Section 19.2]
  Conformal mapping: examples of elementary maps of the complex plane: translation, rotation, dilation, inversion, linear fractional transformation; conformal transformations - transformations preserving the angles [pages 954-956 958, 959 of Sec. 19.5]
• Lecture 20 (Fri, Oct 4): Conformal mapping (cont.): conformal mapping (preserving the angles between curves at the points of intersection); the stereographic projection and the Mercator projection are conformal mappings; proof that every analytic function defines a conformal mapping; examples of conformal mappings; 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 959-968 of Sec. 19.5]
• Lecture 21 (Mon, Oct 7): Conformal mapping and boundary value problems: idea of the method, example: solving Laplace's equation ΔΦ(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=f(z)=exp(πz/a)=exp[π(x+iy)/a] and the fact that the solution of Laplace's equation ΔΦ(x,y)=0 in the infinite strip 0<y<a with boundary conditions Φ(x,0)=α, Φ(x,a)=β is Φ(x,y)=α+π(β-α)y/a [pages 970-973 of Sec.19.6]
• Lecture 22 (Wed, Oct 9): Conformal mapping and fluid flow: problem to be solved: steady irrotational flow of inviscid incompressible fluid in a two-dimensional domain; derivation of the continuity equation ∂ρ/∂t+div(ρV)=0; incompressibility is equivalent to div(V)=0; irrotational motion: curl(V)=0, which is locally equivalent to V==∇φ for some function φ called the velocity potential; for irrotational motion of an incompressible fluid the velocity potential φ is harmonic: 0=div(V)=div(∇φ)=Δφ; FROM THIS POINT ON IT IS FOR R² ONLY: if Ω(z)=φ(x,y)+iψ(x,y); is an analytic function, then its real part φ(x,y) defines a velocity field of irrotational motion of incompressible fluid in R² by V=∇φ; fact: for an analytic function Ω(z)=φ(x,y)+iψ(x,y) ("complex potential"), the level curves φ(x,y)=C₁ and ψ(x,y)=C₂ are orthogonal at each intersection point (which follows from the Cauchy-Riemann equations); corollary: the level curves ψ(x,y)=C₂ of the "stream function" ψ(x,y) are integral lines of the vector field V=∇φ; boundary condition: V is tangent to the boundary ∂D of the domain D in which we are solving the problem, hence the boundary must be a level curve of the stream function ψ; idea: to find the velocity V(x,y) of an irrotational motion of incompressible fluid in a domain D in R², try to find an analytic function Ω(z)=φ+iψ (complex potential) such that the boundary ∂D is a level curve of the imaginary part ψ (stream function), then the velocity V is the gradient of the real part φ (velocity potential), and the level curves of ψ are the stream lines of the flow [pages 977-981 of Sec. 19.7]
• Lecture 23 (Mon, 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 24 (Wed, Oct 16): 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; 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; 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; linear operators (linear transformations) from V to V; matrix elements of a linear operator in a basis; 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 439-440 of Sec. 9.5, (418-425 of Sec. 9.3), 400-403 of Sec. 9.1]
• Lecture 25 (Fri, Oct 18): Exam 2 [on the material from Sec. 18.3-18.6, 19.2, 19.5-19.7 covered the second half of Lecture 10 and in Lectures 11, 12, 14-22]
• Lecture 26 (Mon, Oct 21): Vector spaces and linear transformations (cont.): normed vector space; inner product vector space; norm in an inner product vector space ||u||:=(⟨u,u⟩)^1/2; Cauchy-Schwarz inequality (with proof); 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; space of functions from [a,b] to R, endowing this space with a structure of a vector space by defining addition of functions (f+g)(x):=f(x)+g(x) and multipication of a function by a number (αf)(x):=αf(x); examples of norms in function spaces; L_p([a,b]) spaces [pages 444-446 of Sec. 9.6]
• Lecture 27 (Wed, Oct 23): Vector spaces and linear transformations (cont.): inner product in a function space, weight function; 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); linear functional on a vector space V form a vector space called the dual space V^* of V; Riesz Theorem - in an inner product vector space, 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; ||u||⁻²|u⟩⟨u| is the orthogonal projection operator onto the one-dimensional subspace spanned by the vector v; Question: if |v₁⟩, ..., |v_n⟩ is an orthogonal basis of V, what is the meaning of the linear operator equal to the sum of ||v_i||⁻²|v_i⟩⟨v_i| over i from 1 to n? [pages 447-448 of Sec. 9.6]
• Lecture 28 (Fri, Oct 25): Vector spaces and linear transformations (cont.): Answer to the Question from Lecture 27: if |v₁⟩, ..., |v_n⟩ is an orthogonal basis of V, the sum of ||v_i||⁻²|v_i⟩⟨v_i| over i from 1 to n is the identity operator on V; completeness property of an orthonormal system of vectors; writing a linear operator in an orthonormal basis in as a double sum of |i⟩a_ij⟨j| over i and j; the components a_ij of a linear operator A in an orthonormal basis |i⟩ are equal to 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 with respect to a line; equivalence of norms, derivation of the equivalence of the norms ||u||₂ and ||u||_∞ in R² [pages 455-459 of Sec. 10.1]
• Lecture 29 (Mon, Oct 28): Vector spaces and linear transformations (cont.): action of a linear operator on the components of the vector: the ith component of Au is a_iju_j (sum over j) [pages 457-458 of Sec. 10.1]
  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, Oct 30): Symmetric and Hermitean matrices: definition of a symmetric matrix (A^T=A) and of a Hermitean matrix (A^T=A^*); a symmetric operator (⟨u|A|v⟩=⟨v|A|u⟩) and a Hermitean operator (⟨u|A|v⟩=⟨v|A|u⟩^*); physical importance of symmetric matrices moment of inertia of a solid body, kinetic energy of a point in generalized coordinates) and of Hermitean matrices (Hamilton's operator in quantum mechanics); 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 A=(a_ij) is symmetric and B=(b_ij) is antisymmetric, then the double sum of a_ijb_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 A 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 A in this basis of eigenvectors of A is equal to sum over all values of i of |i⟩λ_i⟨i| (i.e., A is a sum of projections onto the subspaces spanned by the basis vectors multiplied by the corresponding eigenvalue; equivalently, the matrix A 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 1): 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; 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⁾; solving systems of constant coefficient ODEs by using linear algebra [pages 471-475 of Sec. 10.3, 493-495 of Sec. 10.5]
• Lecture 32 (Mon, Nov 4): Ordinary differential equations and linear algebra (cont.): diagonalization of square matrices, computing exponents of square matrices by diagonalization; diagonalization of a symmetric (or Hermitean) matrix A, using that the matrix S whose columns are the normalized eigenvectors of A form an orthogonal matrix in order to invert S without doing any calculations (how?); other topics to study: simultaneous diagonalization, matrices with complex eigenvalues, spectral theory [pages 491-497 of Sec. 10.5]
  Vectors in plane polar coordinates: Cartesian coordinates in R², position, velocity and acceleration of a particle moving in R², polar coordinates [pages 349, 350 of Sec. 8.1, page 355 of Sec. 8.2]
• Lecture 33 (Wed, Nov 6): Vectors in plane polar coordinates (cont.): unit vectors e_r and e_θ in direction of positive change of r, resp. θ (while keeping all other coordinates constant); 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 355-358 of Sec. 8.2]
• Lecture 34 (Fri, Nov 8): Vectors in plane polar coordinates (cont.): differential operators in orthogonal (in particular, polar) coordinates: 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); area element in R², volume element in R³; orthogonal curvilinear coordinates: metric coefficients ∂r/∂u₁=:h₁e₁, ∂r/∂u₂=:h₂e₂, ∂r/∂u₃=:h₃e₃, infinitesiaml displacement element dr=(∂r/∂u₁)du₁+(∂r/∂u₂)du₂+(∂r/∂u₃)du₃=h₁e₁du₁+h₂e₂du₂+h₃e₃du₃ [pages 378-379 of Sec. 8.5]
• Lecture 35 (Mon, Nov 11): Curvilinear coordinates (cont.): 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 in spherical coordinates in R³.
  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; area element in R²: dA=[det(g_ij)]^1/2du₁du₂; volume element in R³: dV=[det(g_ij)]^1/2du₁du₂du₃.
• Lecture 36 (Wed, Nov 13): General (not necessarily orthogonal) curvilinear coordinates (cont.): semi-rigorous proof that the area element in R² is given by dA=(det(g_ij))^1/2du₁du₂; formula for the line element ds in general curvilinear coordinates: ds=(g_ijdu_idu_j)^1/2.
  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).
• Lecture 37 (Fri, Nov 15): The Euler-Lagrange equation: 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 when the "Lagrangian" depends on higher derivatives of q(t): L=L(q(t),q'(t),q''(t),...q⁽ⁿ⁾(t),t) [pages 986-988, 993 of Sec. 20.1]
• Lecture 38 (Mon, Nov 18): Exam 3 [on the material from Sec. 9.1, 9.3, 9.5, 9.6, 10.1, 10.2, 10.5, 8.2, 8.5 covered in Lectures 23, 24, 26-34]
• Lecture 39 (Wed, Nov 20): Two laws of physics in variational form: the Lagrangian function is the difference between the kinetic and the potential energy: L=T−U; derivation of the equation of the harmonic oscillator; 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)); generalized coordinates q_i(t), generalized momenta p_i=∂L/∂q_i', generalized forces ∂L/∂q_i [pages 996, 998-998 of Sec. 20.2]
• Lecture 40 (Fri, Nov 22): Two laws of physics in variational form (cont.): derivation of the Lagrangian and of the Euler-Lagrange equations of a planar double pendulum.
  Multidimensional variational problems: setting up the problem for the static shape z=u(x,y), (x,y)∈D⊂R² of a membrane attached to a rim horizontal ∂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).
• Lecture 41 (Mon, Nov 25): Multidimensional variational problems (cont.): obtaining the expression for the stretching energy of the membrane in lowest approximation: (τ/2)|∇u|, derivation of the Euler-Lagrange equation for the equilibrium shape of a membrane hanging in gravity field; 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; deriving the action functional of a vibrating membrane; obtaining the wave equation as the Euler-Lagrange equation corresponding to the action of a vibrating membrane, speed of propagation of disturbances of the membrane [pages 1015-1017 of Sec. 20.5]
• Lecture 42 (Mon, Dec 2): Multidimensional variational problems (cont.): Lagrangian density and Euler-Lagrange equations governing the motion of a membrane in the gravity field, physical interpretation of the terms in the Euler-Lagrange equation, incorporating an air-resistance term in the equation.
  The method of Lagrange multipliers: set-up - find the extrema of a function f:Rⁿ→R given that the argument must satisfy the contition ("constraint") g(x)=0; unconstrained optimization - deriving the condition ∇f(x^*)=0 or, equivalently, the conditions ∂f/∂x_i(x^*)=0, i=1,...,n from the fact that if f(x) has an (unconstrained) extremum at x^*, then df=∇f(x^*)⋅dx=0 and the components dx_i of the infinitesimal displacement vector dx=(dx₁,...,dx_n) are independent; if x is constrained by the condition g(x)=0, then the infinitesimal displacement vector dx must be tangent to the (n−1)-dimensional "surface" g(x)=0, so it must be perpendicular to ∇g, i.e., must satisfy (∇g)⋅dx=0.
• Lecture 43 (Wed, Dec 4): The method of Lagrange multipliers (cont.): derivation of the method of Lagrange multipliers, discussion of the physical meaning of the method - the term λ∇g is the "normal reactionforce" exerted by the surface g⋅dx=0 onto the "particle" constrained to move on this surface; an example; remark: the method of Lagrange multipliers is also useful in statistical mechanics [Sec. 6.9]
  Variational problems with constraints: set-up of the problem; an example: determining the shape of a uniform, flexible cable of length 2l and linear density ρ suspended at its ends from two points of equal height [pages 1001-1003 of Sec. 20.3]
• Lecture 44 (Fri, Dec 6): class cancelled due to weather.
• Final exam: Tuesday, Dec 10, 1:30-3:30 p.m.

Grading: Your grade will be determined by your performance on the following coursework:

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Your homework should have your name clearly written on it, and should be stapled. The problems should be written in the order they are given. No late homework will be accepted (unless you have a really compelling reason for turning it late)!

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or phone call from a doctor or a parent). You should come to class on time; if you miss a quiz because you came late, you won't be able to make up for it.

Useful links: the academic calendar, the class schedules.

Policy on W/I Grades : From Sept 3 (Tue) to Oct 25 (Fri), you can withdraw from the course with an automatic "W". Dropping after Oct 28 (Mon) requires a petition to the Dean. (Such petitions are not often granted. Even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the Academic Calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Conduct Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know: the greek_alphabet, some useful notations.