243 problems.
Linear Algebra
- Conic sections of the cone \(z^2=x^2+y^2\) by the planes \(z-ax=1\) — Problem 1
- Powers, inverse, eigenvalues and diagonalization of \(\begin{pmatrix}1&2\\5&4\end{pmatrix}\) — Problem 2
- Linear independence of \((1,2,3)\), \((1,3,5)\), \((5,13,15)\) — Problem 3
- Determinant under a row interchange — Problem 4
- Determinants of \(A^T\), \(A^{-1}\) and \(cA\) — Problem 5
- \(\log\det A=\operatorname{Tr}\log A\) — Problem 6
- First-order expansion of \(\det(I+\epsilon B)\) — Problem 7
- Determinant of a \(2\times 2\) block matrix — Problem 8
- Determinant of the all-ones matrix with zeros on the diagonal after the first entry — Problem 9
- Eigenvalues and eigenvectors of the cyclic shift matrix — Problem 10
- Vandermonde determinant — Problem 11
- Wronskian: Abel’s formula, change of variables and scaling — Problem 12
- Rotation and reflection matrices in \(\mathbb{R}^2\) — Problem 13
- Orthogonality of real matrices with \((\det A)^2=1\) — Problem 14
- Number of independent entries of an orthogonal matrix — Problem 15
- Reflection matrices in \(\mathbb{R}^3\) across a point, an axis and a plane — Problem 16
- Which reflections in \(\mathbb{R}^n\) are rotations — Problem 17
- Rotation matrix about an axis and its trace — Problem 18
- Pauli matrices as a basis of \(2\times 2\) Hermitian matrices — Problem 19
- Commutators and anticommutators of the Pauli matrices — Problem 20
- No commutator of real matrices is a nonzero multiple of the identity — Problem 21
- Common eigenvectors and commuting transformations — Problem 22
- Matrices of stretching transformations of \(\mathbb{R}^3\) and their eigenvectors — Problem 23
- Translations are not linear — Problem 24
- Diagonalization and invariant subspaces of a symmetric \(3\times 3\) matrix — Problem 25
- Linear transformations preserving a sphere, a light cone and a hyperboloid — Problem 26
- Orthogonal transformation sending \((1,\sqrt2,\sqrt3)\) to \((0,\sqrt2,2)\) — Problem 27
- Orthonormal basis adapted to two vectors in \(\mathbb{R}^2\) — Problem 28
- Continuous path from the identity to the unilateral shift — Problem 29
- Number of independent real parameters of a unitary matrix — Problem 30
- Exponentials of the infinitesimal rotation generators of \(SO(3)\) — Problem 31
- \(e^Ae^B=e^{A+B}\) for commuting matrices — Problem 32
- \(e^A=(e^{A/N})^N\) — Problem 33
- Expansion of \(e^{-\lambda B}Ae^{\lambda B}\) in nested commutators — Problem 34
- Change of basis for a matrix exponential — Problem 35
- Lie–Trotter product formula — Problem 36
- Polar decomposition of \(2\times 2\) matrices — Problem 37
- Rotation–scaling–shear decomposition of \(SL_2(\mathbb{R})\) matrices — Problem 38
- Solving a \(3\times 3\) linear system — Problem 39
- Homogeneous linear systems and their solution spaces — Problem 40
- Consistency of a \(3\times 3\) linear system — Problem 41
- Coplanarity of four points in \(\mathbb{R}^3\) — Problem 42
- Relative position of two lines given as intersections of planes — Problem 43
- Canonical form, rank and signature of a ternary quadratic form — Problem 44
- Positive definiteness of \(9x_1^2+6x_1x_2+6x_2^2+8x_2x_3+4x_3^2\) — Problem 45
- Canonical forms, rank and signature of three quadratic forms — Problem 46
- Simultaneous diagonalization of two binary quadratic forms — Problem 47
- Arranging three stars by spaceship rotations and boosts — Problem 48
Real Analysis
- Derivatives of \(\log\frac{a}{x}\), \(\cos\arcsin x\) and \(\frac{x^2+1}{x^3+1}\) — Problem 1
- Derivative of \(x^x\) — Problem 2
- Derivatives of \(e^{-1/x^2}\) at \(0\) — Problem 3
- \(f^{(100)}(0)\) for \(f(x)=(x^{100}+x)e^{100x}\) — Problem 4
- Estimating \(\sin^{100} 0.1\) to within \(10\%\) — Problem 5
- Derivative of \(\cosh\sqrt{1-x^2}\) at \(0\) — Problem 6
- Minimum of the tilted double well \(\frac\lambda4(x^2-v^2)^2-\epsilon x\) for small \(\epsilon\) — Problem 7
- Local extrema of \(xy\log(x^2+y^2)\) — Problem 8
- Extrema of \(x^2+y^2+z^2\) on an ellipsoid — Problem 9
- Triple product rule for partial derivatives of an implicit function — Problem 10
- Huygens problem: amplifying velocity through a chain of elastic collisions — Problem 11
- \(\lim_{x\to0}\frac{\sin x}{x}\) — Problem 12
- Limit as \(x\to0\) involving \(\sin\tan x-\tan\sin x\) and \(\arcsin\arctan x-\arctan\arcsin x\) — Problem 13
- Double and iterated limits of \(x+y\sin\frac1x\) — Problem 14
- \(\int\cot x\,dx\) — Problem 15
- \(\int\log x\,dx\) — Problem 16
- \(\int\frac{x^5+2}{x^2-1}\,dx\) — Problem 17
- \(\int e^{2x}\sin x\,dx\) — Problem 18
- \(\int\frac{dx}{\sin x}\) — Problem 19
- \(\int_0^\infty x^ne^{-x}\,dx=n!\) — Problem 20
- Gaussian integral \(\int_0^\infty e^{-x^2}\,dx\) — Problem 21
- \(\int_0^{\pi^2}\cos\sqrt x\,dx\) — Problem 22
- \(\int_0^{x_0}\sqrt{1-x^2/x_0^2}\,dx\) — Problem 23
- \(\int_{-\infty}^\infty e^{-x^{-4}-x^4}\sin^5x\,dx\) — Problem 24
- \(\int_{\sqrt2}^\infty\frac{dx}{x+x^{\sqrt2}}\) — Problem 25
- \(\int_0^1\frac{dx}{(ax+b(1-x))^2}=\frac1{ab}\) — Problem 26
- Convergence of \(\int_0^1 x^{-\alpha}\,dx\) and \(\int_1^\infty x^{-\alpha}\,dx\) — Problem 27
- Convergence of \(\int\frac{dx}{x^\alpha\log^\beta x}\) on \((0,1)\) and \((1,\infty)\) — Problem 28
- Convergence of the pendulum period integral at the turning point — Problem 29
- Asymptotic approximation of the cosine integral \(\operatorname{Ci}x\) — Problem 30
- \(\int_0^\infty e^{-ax}\frac{\sin xy}{x}\,dx\) — Problem 31
- Arc length of \(y=\log\cos x\) on \([0,\pi/6]\) — Problem 32
- Arc length of a helix — Problem 33
- Polar equation and asymptotics of the hyperbolic spiral — Problem 34
- Jacobians of spherical and cylindrical coordinates — Problem 35
- Surface areas of the sphere and the torus — Problem 36
- Area of the surface of revolution of \(y=\sin x\) — Problem 37
- Volume of the ball in \(\mathbb{R}^3\) — Problem 38
- Volume of the solid of revolution of \(y=\arcsin x\) — Problem 39
- Volumes of high-dimensional balls and concentration near the boundary — Problem 40
- Components of \(\nabla\log\frac1r\) in Cartesian, cylindrical and spherical coordinates — Problem 41
- Divergence, Laplacian and curl of \(\nabla\frac1r\) — Problem 42
- Flux of the Coulomb field through a sphere — Problem 43
- Electric field on the axis of a charged ring and a charged disk — Problem 44
- Fourier transforms of a box function and a one-sided exponential — Problem 45
- Inverse Fourier transform of \(-\frac1\pi\frac{k}{a^2+k^2}\) — Problem 46
- Eigenfunctions of the Fourier transform — Problem 47
- Fourier transforms under scaling and translation — Problem 48
- Fourier transform of the Yukawa potential \(\alpha e^{-\mu r}/r\) — Problem 49
- Fourier transform of the indicator of a ball in \(\mathbb{R}^3\) — Problem 50
- Fourier transform of \(f_0(\frac1r+\frac1a)e^{-2r/a}\) in \(\mathbb{R}^3\) — Problem 51
- Divergence of the \(p\)-series \(\sum n^{-\alpha}\) — Problem 52
- Sum of the alternating harmonic series — Problem 53
- \(\sum_{n\ge1}\frac{x^{n-1}}{n(n+1)}\) — Problem 54
- Closed form and small-\(a\) expansion of \(\sum_n ne^{-an}\) — Problem 55
- \(\sum_{n\in\mathbb{Z}}\lvert 2n+1\rvert e^{-a\lvert 2n+1\rvert}\) — Problem 56
- Finite geometric sums \(\sum r^ke^{ik\phi}\), \(\sum r^k\cos k\phi\) and \(\sum r^k\sin k\phi\) — Problem 57
- Ant on a stretching rubber rope — Problem 58
- Solutions of the exponential functional equation \(f(x)f(y)=f(x+y)\) — Problem 59
- Integrable smooth function with a single nonzero derivative at \(0\) — Problem 60
- \(\int_0^\pi\log(1-2a\cos x+a^2)\,dx=0\) for \(\lvert a\rvert<1\) — Problem 61
- Borwein integrals — Problem 62
Complex Analysis
- Polar forms of \(3i\), \(1-i\), \(2+i\), \(-4\), \(2-i\sqrt3\) and \(\frac{2-i}{1+4i}\) — Problem 1
- Cartesian forms of \(e^{i\theta}\), \(3e^{i\pi/4}\), \(\frac12 e^{i\pi}\), \(2e^{2i\pi/3}\) and \(e^{-3i\pi/4}\) — Problem 2
- Complex roots of \(z^2+3z+12\), \(z^4+5z^2+4\), \(z^6=1\) and \(z^3=-1\) — Problem 3
- Laurent series of \(e^z/(z-1)^2\) about \(z=1\) — Problem 4
- Laurent series of \(1/(z-3)^3\) about \(z=i\) — Problem 5
- Laurent series of \(z/((z+1)(z-1))\) about \(z=1\) — Problem 6
- Laurent series of \(\sin\frac{z}{1-z}\) about \(z=1\) — Problem 7
- Laurent series of \(1/((z+1)(z+2))\) on its annuli of convergence — Problem 8
- Conformal map from the unit disk onto the left half-plane — Problem 9
- Image of a disk under the principal branch of \(\log z\) — Problem 10
- Image of the sector \(\operatorname{Re} z>\operatorname{Im} z>0\) under \(e^{z^2}\) — Problem 11
- \(\int_0^\infty \frac{dx}{1+x^2}\) by residues — Problem 12
- \(\int_{-\infty}^\infty \frac{dx}{1+x^6}\) by residues — Problem 13
- Fourier transform of \(1/(x^2+m^2)\) by residues — Problem 14
- \(\int_{-\infty}^\infty \frac{e^{i\alpha x}}{(3-ix)(1+ix)}\,dx\) by residues — Problem 15
- The Dirichlet integral \(\int_0^\infty\sin x/x\,dx\) — Problem 16; exact existing canonical owner
- \(\int_{-\infty}^\infty \frac{x^2}{x^4-2x^2\cos 2\theta+1}\,dx\) by residues — Problem 17
- \(\int_0^{2\pi}\frac{d\theta}{1+a\cos\theta}\) for \(|a|<1\) — Problem 18
- \(\int_0^\pi \cos^{2n}\theta\,d\theta\) — Problem 19
- \(\int_0^\infty \frac{dx}{(x+i)\sqrt x}\) by a keyhole contour — Problem 20
- \(\int_0^\infty \frac{x^p}{x^2+1}\,dx\) for \(0<p<1\) by a keyhole contour — Problem 21
- \(\int_0^1 (x^2-x^3)^{-1/3}\,dx\) by a dogbone contour — Problem 22
- \(\int_0^\infty \frac{\log x}{1+x^\alpha}\,dx\) by a wedge contour — Problem 23
- \(\int_0^\infty \frac{\log^2 x}{1+x^2}\,dx\) by contour integration — Problem 24
- Matsubara sums giving the Bose and Fermi distributions — Problem 25
- Saddle-point asymptotics of \(\int_{-\infty}^\infty \frac{e^{-t(z^2-1/4)}\cos tz}{1+z^2}\,dz\) — Problem 26
- Large-\(x\) asymptotics of the modified Bessel function \(K_\nu\) — Problem 27
- Large-\(x\) asymptotics of the Hankel function \(H^{(1)}_\nu\) — Problem 28
- Stirling’s approximation from the Gamma integral — Problem 29
- Stirling asymptotics of \(\binom{2N}{N}\) and \(\log\binom{N}{m}\) — Problem 30
- Analytic continuation of \(\int_0^\infty t e^{-zt}\,dt\) — Problem 31
- Integral representation and functional equation of the Riemann zeta function — Problem 32
- Poles and residues of the Gamma function at nonpositive integers — Problem 33
- \(\int_0^\infty \frac{dx}{1+x^\alpha}\) by contour integration of \(\frac{\log z}{1+z^\alpha}\) — Problem 34
Variational Principle
- Fubini instanton for the potential \(-\frac\lambda4 f^4\) — Problem 1
- Extremal of \(\int_0^{\pi/2}(f'^2-f^2)\,dx\) with fixed endpoints — Problem 2
- Brachistochrone — Problem 3
- Minimal surface of revolution — Problem 4
- Extremal of \(\int_0^1(360x^2f-f''^2)\,dx\) with clamped endpoints — Problem 5
- Euler–Lagrange equations for a particle in a potential — Problem 6
- Dirichlet energy and Laplace’s equation — Problem 7
- Variational principle for Poisson’s equation — Problem 8
- Variational principle for the biharmonic equation — Problem 9
- Natural boundary conditions from a boundary term in the functional — Problem 10
- Natural boundary conditions for free-boundary extremals — Problem 11
- Action functionals for the time-dependent and stationary Schrödinger equations — Problem 12
- Gaussian variational ansatz for the harmonic oscillator ground state — Problem 13
- Q-ball ansatz in the Euler–Lagrange equations of a complex scalar field — Problem 14
- Derrick’s theorem — Problem 15
Differential Equations
- Separable equation \(y'=xe^{x^2-2\log y}\) — Problem 1
- Linear system \(x'=-x+6y\), \(y'=2x+3y\) — Problem 2
- First-order linear equation \((x+1)y'=2y+(x+1)^{5/2}\) — Problem 3
- Bernoulli equation \(y'=f(x)y+g(x)y^n\) — Problem 4
- Constant-coefficient equation \(y''-y'-2y=0\) — Problem 5
- Repeated-root equation \(y''-6y'+9y=0\) with boundary conditions — Problem 6
- Third-order equation \(y'''-3y'+2y=0\) — Problem 7
- Forced equation \(y''-3y'+2y=\sin x\) — Problem 8
- Forced equation \(y''+3y'+2y=\tanh x\) — Problem 9
- Separable rate equation \(C'=\alpha(a-C)(b-C)\) with \(C(0)=0\) — Problem 10
- Integrating factor for \((xy^2-y)\,dx+x\,dy=0\) — Problem 11
- Initial value problem \(y'-y=e^{3t}\), \(y(0)=2\) — Problem 12
- Initial value problem \(y''-3y'+2y=e^{3t}\) — Problem 13
- Initial value problem \(y''-6y'+15y=2\sin 3t\) — Problem 14
- First-order equation \(y'+2y=e^{-t}\theta(t)\) with Heaviside forcing — Problem 15
- Oscillator \(y''+16y=\theta(\pi-t)\) with Heaviside forcing — Problem 16
- Green function of the damped harmonic oscillator — Problem 17
- Charge distribution of the Yukawa potential \(e^{-ar}/r\) — Problem 18
- Power series solution of \(x(2-x)y''+3(1-x)y'-y=0\) about \(x=1\) — Problem 19
- Frobenius series of \((\xi u')'+(\frac12E\xi+\alpha-\frac{m^2}{4\xi}-\frac14F\xi^2)u=0\) at \(\xi=0\) — Problem 20
- Perturbative ground-state energy of the quartic anharmonic oscillator — Problem 21
- Reduction of order for \(y''+P y'+Qy=0\) — Problem 22
- Integral solution of the modified Bessel equation of order zero — Problem 23
- Frobenius series for the Bessel functions \(J_{\pm p}\) — Problem 24
- Recurrence relations for Bessel functions of the first kind — Problem 25
- Small-\(x\) asymptotics of the modified Bessel function \(K_\nu\) — Problem 26
- Self-adjoint forms of the Laguerre, Hermite and Chebyshev equations — Problem 27
- Series solution of Laguerre’s equation — Problem 28
- Neumann eigenvalue problem for \(y''+\lambda y=0\) on \([0,\pi]\) — Problem 29
- Eigenvalue problem for the Euler equation \(x^2y''+3xy'+\lambda y=0\) on \([1,e]\) — Problem 30
- Eigenvalues of \((xu')'=-\lambda u/x\) with mixed boundary conditions — Problem 31
- \(\delta(x)\) as the limit of \(\frac1\pi\operatorname{Im}\frac{1}{x-i\epsilon}\) — Problem 32
- Composition of the Dirac delta with a function — Problem 33
- \(\int_{-\infty}^\infty \delta''(x-2)\frac{dx}{1+x^2}\) — Problem 34
- Expansion of \(\delta''(x-x_0)/(1+x-x_0)\) in derivatives of \(\delta\) — Problem 35
- Boundary value problem \(y''-3y'+2y=\delta(x-1)\) — Problem 36
- Green function of \(-\frac{d^2}{dx^2}+m^2\) on the line — Problem 37
- Distributional derivative of \(\log x\) — Problem 38
- First-order PDE \(f_x+af_y+(x-2y)f=0\) by characteristics — Problem 39
- First-order PDE \(xf_x-yf_y=0\) by characteristics — Problem 40
- First-order PDE \((\partial_x+\partial_y+\partial_z)f=x-y\) by characteristics — Problem 41
- Boundary value problem for \(u_{xx}+2u_{xy}+u_{yy}=0\) — Problem 42
- Helmholtz equation in a cube with Dirichlet boundary conditions — Problem 43
- Vibrating rectangular membrane with a single-mode initial profile — Problem 44
- Steady temperature in a sphere with hemispheres at different temperatures — Problem 45
- Heat equation on a half-line with delta initial data — Problem 46
- Heat equation on a rod with fixed end temperatures — Problem 47
- Struck string with fixed ends — Problem 48
- Steady temperature in a semi-infinite cylinder — Problem 49
- Radial harmonic functions in \(\mathbb{R}^d\) — Problem 50
- Grounded conducting sphere in a uniform electric field — Problem 51
Probability
- Probability that five randomly shelved volumes are in order — Problem 1
- Birthday problem — Problem 2
- De Méré’s dice problem — Problem 3
- Probability that the \(j\)th student draws a lucky ticket — Problem 4
- Probability of a derangement of five presents — Problem 5
- Pairwise but not mutual independence: Bernstein’s tetrahedron — Problem 6
- Probability that \(ax^2+bx+c\) has real roots for uniform coefficients — Problem 7
- Reliability of a five-relay bridge circuit — Problem 8
- Meeting problem with a ten-minute wait — Problem 9
- Buffon’s needle — Problem 10
- Exponential waiting time between collisions — Problem 11
- Poisson arrivals at a counter with dead time — Problem 12
- Gambler’s ruin: absorption probabilities and mean duration of a random walk — Problem 13
- No nonnegative solution of \(\int \frac{x(t)\,dt}{(y-t)^2+1}=e^{-y^2}\) — Problem 14
- Random variables with \(\mathbf{E}\xi^2=\mathbf{E}\xi^3=\mathbf{E}\xi^4\) are \(\{0,1\}\)-valued — Problem 15
- Sums of independent normal and of independent Poisson random variables — Problem 16
- Chernoff and Markov tail bounds — Problem 17
- Typical sequences and entropy for Bernoulli trials — Problem 18
- Conditional expectations of sums of independent Bernoulli variables — Problem 19
- Wald’s identities for a random sum — Problem 20
- Expected area of the projection of a randomly oriented unit cube — Problem 21
- Maximum likelihood capture–recapture estimate — Problem 22
- Central limit estimate of accumulated round-off error — Problem 23
- Normal approximation to the number of sixes in 12000 dice throws — Problem 24
- Monte Carlo integration and its sample size — Problem 25
- Confidence interval for a proportion of defective goods — Problem 26
- Outlier test for wheat yield data — Problem 27
- Chi-squared test of a Poisson fit to a cosmic-ray spectrum — Problem 28
- Markov chain of a die rolled onto adjacent faces — Problem 29
- Transition probabilities of a Markov chain on energy levels — Problem 30
- Sample size for testing coin-matching probability \(2/3\) against \(1/2\) — Problem 31
- Almost sure convergence of the random harmonic series \(\sum \xi_n/n\) — Problem 32
- Borel’s normal number theorem — Problem 33