COURSE
MATHEMATICAL METHODS
OF PHYSICS.
R.J.Kooman
University of Leiden
spring 2007.
TABLE OF CONTENTS.
Chapter I. Ordinary linear differential equations.
1.1. Linear first order differential equations. 1.2. Linear differential equations with constant coefficients. 1.3. The determinant of Wronski. 1.4. Frobenius’ method: power series solutions.
Reduction of the order. 1.5. Singular points. 1.6. The hypergeometric differential equation. 1.7.
The confluent hypergeometric differential equation. 1.8. The adjoint differential operator. 1.9.
Integral solutions of differential equations. 1.10. Asymptotic expansions; the method of steepest descent.
Chapter II. Hilbert spaces, Fourier series and operators.
2.1. Banach spaces and Hilbert spaces. 2.2. Orthogonal sets and Fourier series. 2.3. Classical Fourier series. 2.4. Bounded operators. 2.5. Compact operators. 2.6. The spectral theorem for compact normal operators. 2.7. Distributions.
Chapter III. Integral equations.
3.1 Volterra integral equations of the second kind. 3.2 Fredholm integral equations of the second kind. Integral equations with separated kernel. 3.3. Solution by an integral transform.
Chapter IV. Sturm-Liouville theory.
4.1 Unbounded operators. 4.2 Sturm-Liouville systems. 4.3 Green’s functions for Sturm-Liouville operators. 4.4 Asymptotic behaviour of the solutions of Sturm-Liouville problems. 4.5 Application:
separation of variables and PDE’s
Chapter V. Partial differential equations.
5.1 General concepts. 5.2 Quasilinear PDE’s of first order. Characteristics. 5.3 Linear PDE’s of second order. Classification. 5.4 The diffusion equation. 5.5 The elliptic case: the equation of Laplace; harmonic functions. 5.6 The equation of Helmholtz. 5.7. The hyperbolic case: the wave equation in one and several dimensions.
Chapter VI. Tensor algebra.
6.1. The dual of a vector space. 6.2. Tensors and tensor products. Tensor product of vector spaces. 6.3. Symmetric and antisymmetric tensors; the wedge product. 6.4. Cartesian tensors.
6.5. Application: isotropic elastic bodies. 6.6. The Hodge star operator.
Chapter VII. Tensor analysis and differential geometry.
7.1. Tensor analysis in Euclidian space. Tangent and cotangent space. The metric tensor. 7.2.
The covariant derivative. Parallel displacement. 7.3. Divergence, curl and Laplacian in arbitrary coordinates. 7.4. Differentiable manifolds. 7.5. Integration of p-forms. 7.6. The exterior derivative;
Poincar´e’s lemma and Stokes’ theorem. 7.7. The Lie derivative. Interior product of a p-form and a vector. Time derivative of integrals. Divergence and flux of a vector field. 7.8. Riemannian and pseudo-Riemannian manifolds. Isometries and Killing vector fields. 7.9. Connections and geodesics. 7.10. The Riemann curvature tensor. Geodesic deviation. 7.11. General relativity.
7.12. Lorentz vectors and tensors. 7.13. The Hodge star operator; the Maxwell equations.
Chapter VIII. Groups and representations.
8.1. Groups: general concepts and definitions (subgroups, homomorphisms, quotient groups, direct product). 8.2. Representations of finite groups. Characters. Tensor product of representations.
Representation of a subgroup. 8.3. Physical applications: dipole moments, degeneracy of energy states, normal modes, vibrational modes of a water molecule.
Chapter IX. Lie groups and Lie algebras.
9.1. Matrix groups. Infinitesimal transformations. 9.2. Lie groups. The Lie algebra of a Lie group.
The exponential map. 9.3. The structure of Lie algebras. The adjoint representation. The Killing form. Compact Lie algebras. 9.4. Representation of compact Lie groups. 9.5. Representation of Lie algebras. Casimir operators. SO(3) and SO(3, 1).
Chapter X. Calculus of variations.
10.1 The functional derivative. 10.2 The Euler Lagrange equation. 10.3. Lagrange multiplicators.
10.4. The case of free boundary conditions. 10.5. Geodesics. 10.6. Eigenvalue problems. 10.7.
Noether’s theorem.
MATHEMATICAL METHODS OF PHYSICS.
PROBLEM SET.
spring 2007.
R.J.Kooman, University of Leiden.
Chapter I: Ordinary linear differential equations.
1a. Use Frobenius’ method to find two linearly independent solutions of the differential equation y00(z)+
y(z) = 0.
b. Show that the power series converge for all z ∈ C.
c. Why are the two solutions you found linearly independent?
2a. Find two linearly independent solutions of the Airy equation y00(z) + zy(z) = 0.
b. Show that the power series converge for all z ∈ C.
3. Show that y(z) is a solution of the Hermite equation y00(z) − 2zy0(z) + λy(z) = 0 if and only if w(z) = y(z)e−z2/2 is a solution of the equation
w00(z) + (λ + 1 − z2)w(z) = 0.
4a. Solve the differential equation using the substitution z = es:
4y00(z) +1
zy0(z) − 1
z2y(z) = 0.
b. Give the singular points in C ∪ {∞}. Which singular points are regular?
5a. Give two linearly independent solutions of Laguerre’s equation about z = 0:
zy00(z) + (1 − z)y0(z) + λy(z) = 0.
b. For which values of λ is there a polynomial solution?
c. Show that the power series converge for all z ∈ C.
d. Is z = ∞ an ordinary or a singular point of the DE? If singular, is it regular or irregular?
6a. Show that substitution of x = cos θ transforms Chebyshev’s equation (1 − x2)y00(x) − xy0(x) + λy(x) = 0 into the constant coefficient equation w00+ λw = 0.
b. Show that for λ = N2 the DE has a polynomial solution of degree N .
c. Show that the general solution of the Chebyshev equation for λ = N2is y(x) = A cos(N arccos x)+
B sin(N arccos x). For which A, B is the solution a polynomial?
7a. Give two linearly independent power series solutions about z = 0 of the DE y00(z) − 2z
1 − z2 y0(z) − n(n + 1)
(1 − z2)z2 y(z) = 0 where n ∈ Z≥0.
b. For what z ∈ C do the power series converge?
c. Give the singular points of the DE in C ∪ {∞}. Are they regular or not?
8. Give the singular points of the Bessel equation x2y00(x) + xy0(x) + (x2− ν2)y(x) = 0 and find out if they are regular or not.
9. Consider the DE
x2y00(x) − 3xy0(x) + 4y(x) = 0.
a. Use Frobenius’ method to find a solution of the DE. Call it y1(x)
b. Give a solution y2 that is linearly independent from y1. Use reduction of the order.
10. Prove the following properties of the Bessel functions Jν (using the power series representation):
a. d
dx(xνJν(x)) = xνJν−1(x).
b. d
dx(x−νJν(x)) = −x−νJν+1(x).
c. J−n(x) = (−1)nJn(x) for n ∈ Z.
11. Let y00(z) + P (z)y0(z) + Q(z)y(z) = 0 be a differential equation with three regular singular points in z = 0, 1 en z = ∞ (and no other singular points).
a. Show that P (z) = p(z)
z(z − 1), Q(z) = q(z)
z2(z − 1)2 where p(z), q(z) are polynomials of degree 1 and 2, respectively.
b. Show that the are numbers α, β ∈ C such that u(z) = zα(z − 1)βy(z) is a solution of the hyperge- ometric differential equation
z(1 − z)u00(z) + (b − (a + c + 1)z)u0(z) − acu(z) = 0. (†) c. Show that (†) has, besides F (a, c; b; z), also a solution z1−bF (a − b + 1, c − b + 1; 2 − b; z). Why are
these solutions linearly independent if b 6= 1? What happens if b = 1?
d. Show that the elliptic function of the first kind K(z) = Z π/2
0
p dθ
1 − z2sin2θ is equal toπ 2F (1
2,1
2; 1; z2).
12a. Show that J1/2(x) = r2
π sin x
√x en J−1/2(x) = r2
π cos x
√x .
b. Write J3/2(x) and J−3/2(x) in terms of x, sin x and cos x. (Use problem 10).
c. Show that for n = 0, 1, 2, . . . there exist polynomials Pn and Qn or degrees n and n − 1 resp. such that
Jn+1/2(x) = x−n−1/2(Pn(x) cos x+Qn(x) sin x), J−n−1/2(x) = x−n−1/2(Pn(x) sin x−Qn(x) cos x).
The spherical Bessel functions m = 0, 1, 2, . . . are defined as jm(x) = r π
2xJm+1/2(x) and nm(x) =
r π
2xJ−m−1/2(x).
d. Show that jm(x) and nm(x) are solutions of the DE
x2y00(x) + 2xy0(x) + (x2− m(m + 1))y(x) = 0.
13a. Show that 1 2π
Z π
−π
cosnθdθ =
½ 1 2n ·¡ n
n/2
¢ for n ∈ Z, n even
0 for n odd.
b. Prove that 1 2π
Z π
−π
eiz cos θdθ = J0(z).
(Hint: give a power series for the integrand and use a.) 14. (zeroes of the Bessel function.) Let ym(x) =√
xJm(x) for m ∈ R.
a. Show that
ym00(x) + µ
1 +1/4 − m2 x2
¶ ym(x).
b. Show that y1/2(x) = a sin x for some a > 0.
c. Use (b) and theorem 1.5 to prove that Jm(x) has for |m| ≤ 1/2 infinitely many positive (and also infinitely many negative) zeroes.
d. Use problem 10 to show that Jm(x) has infinitely many zeroes for all real values of m. (use induction to [m].)
15. (multipole expansion.) Consider the function F (z, x) = 1
√1 − 2zx + z2.
a. Fix x ∈ R, |x| ≤ 1. Show that F (z, x) is a (complex) analytic function is for |z| < 1. Let the power series be
X∞ n=0
An(x)zn.
b. Show that (1 − x2)Fxx− 2xFx+ z(zF )zz = 0 for x ∈ R, |x|, |z| < 1.
c. Prove that it follows from (b) that An(x) is a solution of the n-th order Legendre equation (1 − x2)y00− 2xy0+ n(n + 1)y = 0.
d. Prove that An(x) = Pn(x).
Let x, y be vectors in RN such that kyk < kxk. Let θ be the angle between x and y.
e. Show that
1
kx + yk = 1 kxk
X∞ n=0
µkyk kxk
¶n
Pn(cos θ).
16. The modified Bessel function of the first kind is defined as Iν(z) = e−iπν/2Jν(iz) =
X∞ m=0
(z/2)2m+ν m!Γ(m + ν + 1).
An integral expression for Iν(z) is
Iν(x) = 1 2πi
Z
C
e(x/2)(z+1/z) dz zν+1
where the curve C ⊂ C starts in −∞, approaches the origin z = 0, circles it counterclockwise and goes back to −∞.
a. Give (the first term of) an asymptotic expression for Iν(x) as x ∈ R, x → ∞.
17. The Bessel equation is given by
z2y00(z) + zy0(z) + (z2− ν2)y(z) = 0. (∗∗) a. Give the Bessel equation in self-adjoint form.
b. What is the adjoint equation of (**)?
c. Solve the adjoint equation. Express the solution in terms of the solutions of the Bessel equation.
(Hint: use the Lagrange identity.)
18. Express the zeroth order Bessel function as an integral Z
C
f (t)eixtdt. Use the method described in
§1.9 (or see Chapter 15 of Hassani).
19. Give integral expressions of two linearly independent solutions of Airy’s equation y00(z)+zy(z) = 0.
20. Show that Z ∞
x
ex2−t2dt ∼ 1 2x− 1
22x3 + 1 · 3
23x5 − 1 · 3 · 5 24x7 + . . .
Chapter II. Hilbert spaces.
1. The functions e2πnix/` (n ∈ Z) form an orthogonal basis of the Hilbert space H = L2(−`, `).
a. Explain that the functions 1 and sin 2πnx/`, cos 2πnx/` for n ∈ Z, n > 0 also form an orthogonal basis of H.
b. Give the Fourier series of the function x with respect to the latter basis.
2. Let H = L2(−∞, ∞)w with weight function e−x2. An orthogonal basis is given by the Hermite polynomials {H0(x), H1(x), . . .} where the degree of Hn is n. Give the Fourier series of x2 with respect to this basis. (Do not look up the form of the Hermite polynomials; the information given here should be sufficient to give the answer.)
3. Let H be the Hilbert space `2(C). The map T : H → C is given by T (x) =P∞
n=1xn/n.
a. Show that T is a well-defined linear operator.
b. Is T bounded? If so, give its norm kT k.
c. Illustrate the Riesz representation theorem for the case of T .
4. The evaluation operator E : D ⊂ L2(−π, π) → C is given by E(f ) = f (0). Its domain D = D(E) is the linear subspace of continuous functions in L2(−π, π) (more precisely, functions having a continuous representant).
Show that E is not a bounded operator.
5a. Show that the spectrum of the differentiation operator D = dxd on L2(−π, π) is C.
b. Use the functions cos nx to show that D is not bounded.
6. Let H = `2(C) and let L, R be the left- and right-shift operators
L(x1, x2, . . .) = (x2, x3, . . .), R(x1, x2, . . .) = (0, x1, x2, . . .).
Give the adjoint operators L† and R†.
7. Give an example of a Hilbert space and a linear operator T : H → H such that im(T†) is not equal to (ker(T ))⊥.
8. Let T : H → H be a hermitian operator with domain H. Suppose that hx, T (x)i = 0 for all x ∈ H.
Show that hx, T (y)i = 0 for all x, y ∈ H and hence, that T = 0.
9. Let H be the Hilbert space L2(−1, 1) and let C be the subset of continuous functions in H.
a. Give an example to show that C is not a closed subset.
b. Show that the closure of C is H. (The closure of C is the smallest closed subspace that contains U.) Hint: show that the orthogonal complement of C is the zero set {0}. Use that the Legendre polynomials form an orthogonal basis of H.
10. Let H = `2(C) and let the operator C : H → H be given by
C(x1, x2, x3, . . .) = (x1, x2/2, x3/3, . . .).
a. Show that C is a bounded hermitian operator. What is kCk?
b. Show that C is a compact operator.
The operator C0 is given by
C0(x1, x2, . . .) = (x2/2, x3/3, . . .).
c. Why is C0 compact? You may use (b).
d. Give the spectra of C and C0. Is zero an eigenvalue?
11. Let H be the Hilbert space L2(0, 1). Let X(x) = x. The operator T : H → H is given by T (f ) = Xf .
a. Show that T is bounded and give the value of kT k.
b. Show that T is hermitian.
c. Give the eigenvalues of T . d. Show that σ(T ) = [0, 1].
e. Is T compact?
12. Let H = `2(C). The operator T : H → H is given by
T (x1, x2, x3, . . .) = (x2− x1, x3− x2, x4− x3, . . .).
a. Show that λ ∈ C is an eigenvalue of T if and only if |λ + 1| < 1.
b. Show that T is bounded and give the value of kT k.
c. Show that σ(T ) = {λ ∈ C : |λ + 1| ≤ 1}.
d. Is T compact?
13. Let R : `2(C) → `2(C) be the right-shift operator. Prove that σ(R) = {λ ∈ C : |λ| ≤ 1}.
14. Calculate the distribution derivative d dx|x|.
15. Prove that xδ(x) = −δ0(x).
16. Show that the following sequences {δn}∞n=1 are delta-sequences:
a. δn = n
√πe−n2x2. b. δn = sin nx
πx .
17a. Let f : R → R be a differentiable function with zeroes x1, x2, . . . such that f0(xi) 6= 0 in i = 1, 2, . . ..
Show that
δ(f (x)) = X∞ n=1
1
|f0(xn)|δ(x − xn).
b. Give the value of the integralR∞
−∞δ(x2− π2) cos xdx.
Chapter III. Integral equations.
1. For what values of λ has the equation
f (x) = x + λ Z π
0
f (t) sin(x + t)dt
a solution?
2. Consider the integral equation
f (x) = x2+ λ Z 1
0
(1 + xt)f (t)dt.
Give the characteristic values and the eigenfunctions. Solve the equation. For what values of λ does the series converge?
3. Solve f (x) = xa+ λ Z ∞
0
e−(x+t)f (t)dt. where a ≥ 0. Are there any values of λ for which there is no solution?
4. Solve the following equations:
a. f (x) = x +1 2
Z 1
−1
(x + t)f (t)dt.
b. f (x) = x + Z x
0
f (t)dt.
c f (x) = λ Z π
0
f (t) sin(x − t)dt.
5. Transform the differential equation
y00(x) + xy0(x) + y(x) = 0, y(0) = 1, y0(0) = 0
into a Volterra integral equation of the second type. Use partial integration to remove derivatives from within the integral. Solve the integral equation.
Chapter IV. Sturm-Liouville systems.
1. Consider the inhomogeneous Sturm-Liouville system
y00(x) + λy(x) = 0, y(0) = 0, y0(π) = 0.
a. Give the eigenvalues and eigenfunctions and state the orthogonality relation for the eigenfunctions.
b. Give the Fourier series of the function f (x) = 1 (with respect to the eigenfunctions).
c. Apply Parseval’s theorem to the function f (x) = 1.
d. Give the Green’s function G(x, t) for the operator Ly = y00 on [0, π] with boundary values y(0) = y0(π) = 0. Give both an explicit form and the Fourier series.
e. Solve the inhomogeneous boundary value problem
y00(x) = f (x), y(0) = y0(π) = 0.
Give the solution in the form of an integral.
f. Consider the inhomogeneous boundary value problem
y00(x) + y(x) = f (x), y(0) = y0(π) = 0.
For what f (x) is there a solution? Give the solution in the case that it exists, in whatever form you like.
2. Consider the inhomogeneous S.L. problem
y00(x) + n2y(x) = sin mx, y(0) = y(π) = 0, where m, n are positive integers.
a. Fix n. For what values of m is there a solution? (Use the Fredholm alternative.)
b. Use the theory of Fredholm integral equations to solve the system, in the case that a solution exists.
3. Consider the Sturm-Liouville system y00+ λy = 0, y0(0) = 0, y0(1) = 0 op [0, 1].
a. Give the eigenvalues and the corresponding eigenfunctions. State the orthogonality relation for the eigenfunctions.
b. Give the Fourier series of the function f (x) = x (with respect to the eigenfunctions).
c. Apply Parseval’s theorem to the function f (x) = x.
d. Give the Green’s function G(x, t) for the operator Ly = y00+ (π2/4)y on [0, 1] with the boundary values y0(0) = y0(1) = 0. What is the Fourier series of G(x, t)?
e. Solve the inhomogeneous boundary value problem
y00(x) + (π2/4)y(x) = f (x), y0(0) = y0(1) = 0.
Give the solution in the form of an integral.
f. Consider the inhomogeneous boundary value problem
y00(x) = f (x), y0(0) = y0(1) = 0.
Give a condition on f (x) such that there is a solution.
g. Give an integral form of the solution in the case that it exists. (Express the solution as a single integral.)
4. Consider the singular Sturm-Liouville system on the interval [−1, 1] given by the Legendre equation (1 − x2)y00(x) − 2xy0(x) + λy(x) = 0, where y(x), (1 − x2)y0(x) are bounded in (−1, 1).
a. Show that for all n = 0, 1, . . . there is a polynomial eigenfunction of degree n.
b. Show that Pn(x) = dn
dxn(x2− 1)2n is a solution of the Legendre equation. Show that it is a polynomial of degree n and that Pn(1) = 1. (Pn(x) is called the n-th Legendre polynomial).
c. Argue that Z 1
−1
Pn(x)Pm(x)dx = 0 if m 6= n so that the Legendre polynomials form a system of orthogonal polynomials.
Remark: By the Stone-Weierstrasz theorem mentioned in chapter 2, the Legendre polynomials form an orthogonal basis of L2(−1, 1).
5. Consider the boundary value problem
r2R00(r) + rR(r) + (λr2− n2)R(r) for n = 0, 1, . . ., with R(1) = 0, and R(r) continu in r = 0.
a. Write the differential equation in self-adjoint form and show that we obtain a singular Sturm- Liouville problem.
b. Show that the eigenvalues are αnj2 (j = 1, 2, . . .) where 0 < αn1< αn2. . . are the positive zeroes of the Bessel function Jn and that the eigenfunctions are yn(r) = Jn(αnjr).
c. Give the orthogonality relation for the eigenfunctions.
6. Consider the Sturm-Liouville system y00+λy = 0 with boundary conditions y(0) = 0, y0(1)−2y(1) = 0.
a. Find the eigenvalues and the eigenfunctions. Show explicitly that there are infinitely many eigen- values λ1< λ2< . . . and that λn/n2→ C as n → ∞ (with C 6= 0 a constant.
b. Give the Green’s function for the system.
7. Apply a Liouville substitution to Bessel’s equation (xy0)0(x) +
µ x − ν2
x
¶
y(x) = 0
to bring in into the form
v00(t) + µ
1 −ν2− 1/4 t2
¶
v(t) = 0.
Let A = A(t), φ = φ(t) be functions such that
v(t) = A sin φ, v0(t) = A√ S cos φ
where S = S(t) = 1 −ν2− 1/4
t2 . (See also §4.4 of the lecture notes.) a. Show that
φ0(t) = 1 − ν2− 1/4
2t2 + O(1
t3), A0(t)
A(t) = O(1 t3).
b. Integrate the above equations the show that
φ(t) = t − φ∞+ ν2− 1/4
2t + O(1
t2), A(t) = A∞+ O(1 t2) where A∞6= 0.
c. Conclude that v(t) = A∞sin(t − φ∞+ν2− 1/4
2t ) + O(1
t2) as t → ∞.
d. Give the asymptotic behaviour of the solutions of the Bessel equation as x → ∞. (Do not bother which values of φ∞, A∞ belong to Jν and J−ν (or Yν).)
8. Consider the wave equation utt= ∆u for t > 0 on the square G = {(x, y) ∈ R2 : 0 < x, y < 1} in R2 with homogeneous boundary conditions u(0, y, t) = u(1, y, t) = u(x, 0, t) = u(x, 1, t) = 0. Use separation of variables to find de frequencies of the eigenmodes.
9. Consider the one-dimensional heat equation ut = kuxx where u(x, t) is the temperature of a bar 0 ≤ x ≤ 1. At time t = 0 the temperature is given by u(x, 0) = f (x), the left end of the bar is kept at a constant temperature u(0, t) = 0 and the right end is isolated, so ux(1, t) = 0 (there is no heat current).
Solve this initial- and boundary values problem by separation of variables.
10. Solve Laplace’s equation ∆u = 0 on the unit disk {x2+ y2 < 1} in R2 with boundary condi- tion u(x, y) =
½1 if x2+ y2= 1, y > 0
−1 if x2+ y2= 1, y < 0 by separation of variables. Use polar coordinates (the Laplacian in polar coordinates is ∆u = urr+1
rur+ 1 r2uφφ).
Chapter V. Partial differential equations.
1. Let f : R2 → R be a differentiable function that is invariant under the dilatation group, i.e.
f (x, y) = f (ax, ay) for x, y, a ∈ R and a 6= 0.
a. Show, by considering an infinitesimal transformation, that f satisfies the first order PDE xfx+ yfy= 0.
b. Solve the PDE and show that f is a function of y/x (or x/y) only.
2. Consider the PDE xux+ yuy= u where u = u(x, y) is a real-valued function on R2. a. Give the characteristics.
b. Impose on u the condition u(x, 0) = φ(x) for some function φ. Is this boundary value problem well-posed?
c. Solve the boundary value problem
xux+ yuy = u, u(x, 1) = φ(x) where ψ is some differentiable function on R.
3. Consider the PDE ux+ 2xuy = C where u = u(x, y) is a real-valued function on R2 and C is a real constant.
a. Give the characteristics.
b. Solve the boundary value problem with boundary condition u(0, y) = ψ(y) where ψ is some differ- entiable function on R.
c. Now impose instead of (b) the boundary condition u(x, 0) = ψ(x). What condition must be imposed on ψ in order that there is a solution?
4. Consider the second-order PDE
uxx+ 4uxy + uyy+ 2ux+ 4uy+ 2u = 0.
Transform it into a PDE in standard form (5.13’) for some function w and express w as a function of u.
5. Consider the diffusion equation on the half line x > 0
ut− kuxx= 0 for x > 0, t > 0 u(x, 0) = φ(x) for x > 0 u(0, t) = 0 voor t > 0
. (†)
We can use the solution formula for the diffusion equation on R by defining φ(x) properly on the negative x-axis: let φ(−x) = −φ(x).
a. Why is this a good choice? In what way would you extend ψ if the boundary condition on t = 0 were ux(0, t) = 0?
b. Give a formula for the solution of † as an integral from x = 0 to ∞. What is the fundamental solution for the half-line?
6. Let A and B be two points in R2 and let ` be the (closed) segment between A and B. Let H = R2\`. For X ∈ H let u(X) be the directed angle between the half-lines XA and XB,
−π < u(X) < π. Show that u is a harmonic function on H.
7. Show that if u(r, φ) is harmonic on the disk {r < R} in R2, then the function v(r, φ) := u(R2/r, φ) is harmonic on the exterior {r > R}.
b. Prove (5.23).
8a. Show that for 0 ≤ r < 1:
1 + 2 X∞ n=1
rncos nθ = 1 − r2 1 + r2− 2r cos θ.
b. Use Poisson’s formula to solve the following Dirichlet problem on the unit disk {r < 1} in R2:
∆u(x, y) = 0 voor x2+ y2< 1 u(x, y) = 1 als x2+ y2= 1, y > 0 u(x, y) = 0 als x2+ y2= 1, y < 0.
Use (a) and write the solution in the form of a Fourier cosine series.
c. Derive the following closed form for the solution:
u(x, y) = 1 2 + 1
πArg
µ1 + x + iy 1 − x − iy
¶ .
9. The mean value theorem for harmonic functions in Rn. Let a ∈ Rn (n > 2) and 0 < ² < R and let u be a harmonic function in B(a, R) = {x ∈ Rn : kx − ak = R}.
Apply Green’s second identity (5.15) for G = B(a, ²) and v = uf + c where uf is the fundamental solution of the Laplace equation and c is some real constant. Conclude that
I
kx−ak=²
∂u
∂ndn−1A = 0, u(a) = 1 Ωn²n−1
I
kx−ak=²
u(x)dn−1A,
where Ωn is the surface area of the unit ball B(0, 1) in Rn.
10. Show that (∆ + k2)eikr
r = −4πδ(x) for x ∈ R3, r = kxk and k2∈ R.
11. Suppose that the function u = u(r) satisfies urr+ n − 1
r ur+ k2u = 0. Let w = r−1ur. Prove that wrr+ n + 1
r wr+ k2w = 0.
12. A movie problem. Let u(x, t) be a solution of the one-dimensional wave equation utt− c2uxx= 0 for t > 0, x ∈ R with initial conditions u(x, 0) = φ(x), ut(x, 0) = ψ(x).
a. Let ψ(x) = 0, φ(x) =
ncos x if |x|, π/2
0 otherwise. Draw the graph of u(x, t) for both small and large values of t.
b. Let φ(x) = 0, ψ(x) =
ncos x if |x|, π/2
0 otherwise. Draw the graph of u(x, t) for both small and large values of t.
13. The Doppler effect. We consider a source that moves with speed 0 < v < c along the x-axis and which sends a signal that is observed by some stationary observer on the x-axis. This is modelled by the boundary value problem
utt− c2uxx= 0 voor x ∈ R, x 6= vt, t > 0 u(vt, t) = sin ωt t > 0
u(x, 0) = ut(x, 0) = 0 x 6= 0
a. Give the solution u(x, t). Distinguish between the cases x < vt, vt < x < ct and x > ct. (Hint: the (x, t)-plane is divided into two parts by the straight line x = vt along which the source is moving.
On x = vt the solution is continuous but not differentiable. On each of the parts x < vt, x > vt the solution is differentiable and satisfies the wave equation so that d’Alembert’s formula holds for suitable functions φ, ψ. We must extend these functions to the whole of R in order to find a solution. We can use the value u(vt, t) together with continuity of the solution. Compare the boundary value problem (5.28) of the lecture notes.
14. Refraction of a one dimensional wave at the boundary of two media with different propagation speeds.
Consider the initial value problem
½ utt(x, t) = c(x)2uxx(x, t) u(x, 0) = f (x), ut(x, 0) = 0
¾
(x ∈ R, t > 0)
met c(x) =
½c1 for x > 0
c2 for x < 0 for certain c1, c2> 0, and f (x) =
nsin x for −2π ≤ x ≤ −π
0 otherwise .
a. Solve the initial value problem. Express the solution u(x, t) in terms of the function f . Assume that u and ux are continuous at the boundary x = 0.
b. Draw the region in the (x, t)-plane where u(x, t) 6= 0 in the case that c1> c2.
c. Discuss reflection and transmission/refraction of the wave at the boundary x = 0. Does the the sign of the solution change?
d. What happens if c1< c2?
15. Does Huygens’ principle hold in 1 dimension? Explain your answer.
16a. Derive d’Alembert’s formula for the one-dimensional wave equation from Poisson’s formula for the three-dimensional wave equation (5.32) by the method of descent.
b. Use Duhamel’s principle to find a solution u = u(x, t) of the inhomogeneous one-dimensional wave equation (with source term) with homogeneous boundary conditions
∆u(x, t) = f (x, t), u(x, 0) = ut(x, 0) = 0.
c. Take f (x, t) = δ(x − x0)δ(t − t0) (for x0, t0 ∈ R fixed) and give the Green’s function for the one-dimensional wave equation on R.
Chapter VI. Tensor algebra.
1. Let V be a (real or complex) vector space with basis {e1, . . . , en}. Let A be an invertible (real or complex) n × n-matrix . Set fj = Aijeifor j = 1, . . . , n.
a. Why is {f1, . . . , fn} a basis of V ?
Let {e1, . . . , en} and {f1, . . . , fn} be the dual bases in V∗of {e1, . . . , en} and {f1, . . . , fn}, respec- tively.
b. Show that fj = (A−1)jiei for j = 1, . . . , n.
2. Let T be a tensor of rank (r, s) with components Tji11...j...irs with respect to some coordinate basis of the (finite-dimensional) vector space V and let T0 be the contraction of T with respect to the k-th upper (contravariant) index and the `-th lower (covariant) index
(T0)ij1...ˆp...ir
1...ˆp...js = Tji11...p...j...p...isr (†) (where the hat means that the corresponding index is omitted and where the Einstein summation convention has been used).
Show that after transformation to a different basis of V (and the corresponding dual basis of V∗) T0 transforms like a tensor of rank (r − 1, s − 1). In other words, contraction of a tensor yields indeed a tensor.
3. Let v1, . . . , vn be vectors in some vector space V . Prove that
v1∧ . . . ∧ vn= Xn i1=1
. . . Xn in=1
²i1...invi1⊗ . . . ⊗ vin.
(Note that this justifies in some sense the choice of coefficients in the definitions of the antisym- metriser and the wedge product of two tensors in §6.3).
4. Prove that the Levi-Civit`a-(pseudo)tensor ² with components ²ijk is the only Cartesian pseudoten- sor of rank 3 in R3 and that there are no other Cartesian (pseudo)tensors of rank 3 in any Rn for n > 1.
5. The tensor ² ⊗ ² (with components ²ijk²`mn) is a Cartesian tensor of rank 6 in R3 (why?). We know that all tensors of even rank are tensor products of the Kronecker-deltatensor. Show that in fact
²ijk²`mn =
¯¯
¯¯
¯¯
δi` δim δin δj` δjm δjn δk` δkm δkn
¯¯
¯¯
¯¯.
6. Let u, v be given Cartesian vector fields in Rn. Assume that there exists a linear connection between v and the tensor of second partial derivatives of u:
vi= Cijk` ∂2uj
∂xk∂x` (∗)
where x1, . . . , xn are Cartesian coordinates. Assume moreover that the tensor of coefficients C is isotropic, i.e.the values of the components Cijk` does not depend on the choice of the Cartesian
coordinates (it remains the same whenever the coordinate axes are translated or rotated). Show that (∗) can be written in the form
v = A∆u + B∇(∇ · u)
where ∆u = ∇ · ∇u is the Laplacian of u.
7. Consider in R3 the Cartesian rank-2-tensor I. With respect to a certain Cartesian coordinate system x1, x2, x3 the tensor I has components I11 = I1, I22 = I33 = I2 and Iij = 0 als i 6= j. (The matrix is then
I1 0 0 0 I2 0 0 0 I2
.) Determine how the components transform under a coordinate transformation (x1, x2, x3) → (x01, x02, x03) in the following cases:
a. The coordinate axes are rotated about the x1-axis about an angle θ.
b. The coordinate axes are rotated about the x3-axis about an angle θ.
8. Let T be a tensor of rank (r, s). Any component of T has r contravariant and s covariant indices.
Take any subset of either contravariant or covariant indices and symmetrize the components with respect to the chosen set of indices. This yields an object T0 which is symmetric in the chosen set of indices. Is T0 again a tensor? (In other words, is the concept of a tensor that is symmetric with respect to a given set of indices (either contravariant or covariant) a meaningful concept?) And how about antisymmetry? And what happens if we do not separate contravariant and covariant indices?
9. let V be a vector space with an inner product. Show that the definition of the Hodge star oper- ator is independent of the chosen orthonormal basis, provided that the two bases have the same orientation. What happens if the orientation is different?
Chapter VII. Differential Geometry.
1a. Show that both cilindrical and spherical coordinates are regular coordinates on the subset U ⊂ R3 that one gets by omitting some (closed) half-plane that has the x3-axis as its boundary.
b. Give the components of the metric tensor for cylindrical and for spherical coordinates in R3. c. Let f be a differentiable function on U . Give the components of the (contravariant) gradient ∇f
of f both in cylindrical and in spherical coordinates.
2. Let x1, . . . , xnbe Cartesian coordinates on Rnand let y1, . . . , ynbe regular coordinates on U ⊂ Rn. Let P ∈ U . On the cotangent space (TPRn)∗ we define an inner product by (dxi, dxj) = δij. Let g = gijdyi⊗ dyj be the metric tensor on U . Show that gij = (dyi, dyj).
3. Prove that the covariant derivative of the metric tensor is zero, i.e. ∇igjk = 0.
4. Give the values of the Christoffel symbols Γkij for polar coordinates in R2.
5. Give the expression of the Laplacian ∆f of a function f in cilindrical and spherical coordinates.
6. Let B = {x ∈ R3 : kxk = 1} be the unit sphere in R3. Let N, S be the points (0, 0, 1) and (0, 0, −1) respectively, and U1= B\{N }, U2= B\{S}. The maps φ1: U1→ R2and φ2: U2→ R2 that project a point P ∈ B onto the intersection point of the line through P and N (and the line through P and S, respectively) with the plane x3 = 0 are homeomorphisms between U1 and R2 (U2and R2 resp.). Show that the transition function φ2◦ φ−11 : R2→ R2 maps the point (x1, x2) onto ( x1
x21+ x22, x2
x21+ x22) and argue that it is a diffeomorphism. This shows that the sphere is a differentiable manifold.
7. Let M = Rnand let S = {x ∈ M : kxk = 1} be the unit sphere. S = f−1(0) where f (x) = kxk2−1.
Show that df (x) 6= 0 for all x ∈ S. Conclude that S is a subvariety of M .
8. Let M, N be differentiable manifolds with dimensions m and n respectively. Let x1, . . . , xm and y1, . . . , yn be local coordinates about P and f (P ) on M and N respectively. Show that, for any tangent vector XP = Xi ∂∂xi in TPM ,
f∗(X)f (P ) = Xi∂fj
∂xi
∂
∂yj = X(fj) ∂
∂yj.
Furthermore, if m = n and ω = g(y)dy1∧ . . . ∧ dyn is an n-form in some neighbourhood of f (P ), then show that
f∗ω(x) = (g ◦ f )(x)
¯¯
¯¯∂f
∂x
¯¯
¯¯ dx1∧ . . . ∧ dxn. (7.130)
9. Let d :Vp
M →Vp+1
M be the exterior differentiation operator on M . Prove that d2= 0.
10. Let ω be a 1-form on a differentiable manifold M and X, Y vector fields on M . Prove that dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]).
11. x, y are Cartesian coordinates on R2. Let the vector field X on R2 be given by X = −y∂x∂ + x∂y∂ . Show that the flow of X through the point (p, q) is given by
x(t) = p cos t − q sin t, y(t) = p sin t + q cos t.
(Thus the integral curves of X are circles).
12. Give the flow of the vector field X = x2 ∂∂x+ xy∂y∂ through the point (p, q).
13a. Let M be a differentiable manifold and X a vector field on M . Show that for a 1-form ω (LXω)i= Xj∂jωi+ ωj∂iXj.
b. Show that LXdxi= dXi.
c. Let ω = a1dx1+ a2dx2+ a3dx3=: a · ds (where ds = (dx1, dx2, dx3)T) be a 1-form on R3 and let v be a vector field. Prove that
Lvω = ((∇ × a) × v + ∇(a · v)) · ds
d. Let ω = b1dx2∧dx3+b2dx3∧dx1+b3dx1∧dx2=: b·dσ (where dσ = (dx2∧dx3, dx3∧dx1, dx1∧dx2)T) be a 2-form on R3and let v be a vector field. Prove that
Lvω = (∇ × (b × v) + v∇ · b) · dσ.
e. Suppose that M is Riemannian with metric tensor g. Calculate the components (LXg)ij.
14 Let M be a differentiable manifold, P ∈ M , and X, Y ∈ TPM . The commutator [X, Y ] is defined by [X, Y ](f ) = X(Y (f )) − Y (X(f )) where f : M → R is a differentiable function.
a. Show that [X, Y ] ∈ TPM (use the definition of a tangent vector given in §7.4) b. Let N be another manifold and φ : M → N be a differentiable map. Show that
φ∗[X, Y ] = [φ∗X, φ∗Y ].
c. Let X be a vector field on M with flow ft. Show that, for g : M → R a differentiable function, and P ∈ M ,
t→0lim
(g ◦ ft)(P ) − g(P )
t = XP(g).
15. Let V be a vector space. Show that the inner product iX with respect to a vector X is an antiderivation, i.e.
d(α ∧ β) = dα ∧ β + (−1)pα ∧ dβ.
16. Show that the covariant derivative of the metric tensor g on a Riemannian manifold is zero. (Note that this result holds for the affine connection, not in general).
17. Let C = {x21+ x22= r2} be a cylinder in R3. a. Show that C is a submanifold of R3.
b. Give the geodesic equation for C in terms of the cylindrical coordinates φ, z.
c. What are the geodesics on C?
d. What is the result of parallel displacement of a vector from a point on C along a circle x3= constant?
18. Let K = {x21+ x22= x23, x3> 0} be a cone in R3. a. Is K a submanifold of R3?
b. Give the geodesic equations for K. Choose suitable coordinates.
c. What are the geodesics on K?
d. What is the result of parallel displacement of a vector from a point on K along a circle x3= constant?
19. Consider the curve γ on the cylinder C = {x21+ x22= 1} in R3 with the parametric equations x1= cos φ, x2= sin φ, x3= aφ for 0 ≤ φ ≤ 2φ
where φ, z are cylindrical coordinates.
a. What is the length of C?
b. Show that the angle between the curve and the curves φ = φ0 is constant.
c. Displace the vector ∂z parallel along C from the point φ = 0, z = 0. What is the result?
20. On the unit sphere S2 = {kxk = 1} in R3 the metric tensor in spherical coordinates is given by ds2= dθ2+ sin2θdφ2.
a. Give all Christoffel symbols Γijk.
b. Show that for a point on the equator θ = π/2 the coordinates θ, φ are normal coordinates.
c. Give the geodesic equation for S2.
d. Explain why the equation of a great circle (i.e. a circle which has its center in the center of the sphere) is given by A cos φ + B sin φ + C cot θ = 0 where A, B, C are not all zero.
d. Show that the great circles are exactly the geodesics on on S2.
e. What is the result of parallel displacement of the vector ∂φ along the circle θ = π/4?
f. What are the Killing fields on S2?
21a. Give all Killing fields on Euclidian space E3. b. Give all Killing fields on Minkowski space M4.
22. Show that L[X,Y ]= LX◦ LY − LY ◦ LX if X, Y are vector fields on some manifold M . Conclude that, if M is Riemannian and X and Y are Killing fields on M , then [X, Y ] is a Killing field.
23. Let M be an n-dimensional Riemannian manifold with metric tensor g. Let g = det(gij) and let x1, . . . , xn be a set of local coordinates. The n-form ω =√
gdx1∧ . . . ∧ dxn is a volume form on M . For a vector field X on M the divergence is (as in §7.7) defined by div(X)ω = d(iXω).
a. Give an expression for the (n − 1)-form iXω in terms of the local coordinates.
b. Let ∇ the metric connection. Show that div(X) = ∇iXi. 24. Let Tµν be a (contravariant) Lorentz tensor van rank 2.
a. Fix ν = α and let vµ= Tµα. Is vµ a Lorentz vector?
b. Show that `i= T0i (i = 1, 2, 3) are the components of a Cartesian vector.
25. The energy momentum tensor for a perfect fluid Tµν has with respect to a certain coordinate system (called its rest system) components
ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p
. How does Tµν transform under a
Lorentz boost
x00 = γ(x0+ v · x), x0 = x + γ2
1 + γ(v · x)v + γx0v?
Here x = (x1, x2, x3)T is the spatial part of the 4-vector xµ, v ∈ R3 is the velocity vector and γ = (1 − v2)−1/2. Express the components of Tµν in terms of the 4-velocity uµ= (γ, γv) and show that Tµν= (ρ + p)uµuν− pηµν. (Hint: write the transformation matrices Λµν in terms of uµ.) 26. Vector fields and orthogonal surfaces.
Let v(x) = (v1, v2, v3) be some vector field in Ω ⊂ E3. If v is continuous on Ω and nowhere zero, then the flow of v determines a set of integral curves, i.e. curves that are tangent to v in every point of Ω. These integral curves are solutions of the system of DE x01(t) = v1, x02(t) = v2, x03(t) = v3, or dx1
v1 = dx2
v2 = dx3
v3 . We ask ourselves if there also exists (locally) a family of surfaces F (x1, x2, x3) = c such that the vector field is everywhere orthogonal to the surfaces F = c.
Such surfaces are called orthogonal surfaces of the vector field.
a. Express the condition that F = c are orthogonal surfaces of v in terms of F and v. Why is F a differentiable function of x1, x2, x3?
b. Show that a necessary condition for the existence of a family of orthogonal surfaces is that v· curl(v)= 0. (In fact it can be shown that this condition is also sufficient.)
We now consider the case that the vector field v(x) is nowhere zero and that the integral curves of v are geodesics with respect to some metric (not necessarily the standard Euclidian metric) on Ω. Assume that there exists some surface F (x1, x2, x3) = 0 that is orthogonal to the vector field, so that the geodesics intersect the surface orthogonally. The surface F = 0 is a 2-dimensional submanifold of R3 and so there exists a local parametrisation x(t, u) of the surface. De geodesics can then also be parametrized by t en u: the geodesic γt,u intersects F = 0 in x(t, u); if s is the arc length of the geodesic and we choose s = 0 on the surface F = 0, then s, t, u are regular coordinates.
c. Why is the surface F = 0 a submanifold of R3? d. Show that gst= gsu= 0 en show that curl(∂s) = 0.
e. Show that the surfaces s = s0are orthogonal to the bundle of geodesics and show that the distance between the planes s = s0 en s = s1is everywhere the same.
Remark: Light rays in some medium M ⊂ E3 with isotropic index of refraction n(x) (i.e. the index of refraction is a scalar field - there is no dependence on the direction) are geodesics with respect to the metric ds2 = n(x)2(dx2+ dy2+ dz2). This is a result of Fermat’s principle (light rays follow the path of shortest time; if c is the velocity of light in a vacuum, then s/c is a measure of the time) and the fact that geodesics are (locally) the paths of shortest length, a fact that can be shown with the aid of the theory of calculus of variations (for which see chapter 10). A bundle of light rays originating in a point P has an orthogonal surface (an infinitesimally small sphere
