Quizzes · Mathematical Methods II

Integrative questions — each concept page has its own quiz too. Study in lecture order.

Conceptual

Q1. Hamilton's principle says the action is stationary, not minimal. What's the difference, and does it matter?

Answer

Stationary means \(\delta S = 0\) — minimum, maximum, or saddle. Real trajectories are often saddle points (e.g. long harmonic-oscillator paths past a conjugate point). The equations of motion only need \(\delta S = 0\).

Q2. What is the geometric meaning of the Cauchy–Riemann equations?

Answer

They force the local map \(z \mapsto f(z)\) to be a pure rotation + scaling (multiplication by \(f'(z)\)) — angle-preserving wherever \(f' \neq 0\). Analyticity is local conformality.

Q3. Why does SU(2) cover SO(3) twice?

Answer

Both \(U\) and \(-U\) produce the same rotation (\(R(U) = R(-U)\)): the map SU(2) → SO(3) is 2-to-1. Physical consequence: spinors pick up a sign under a \(360°\) rotation and need \(720°\) to return — see SU(2) ↔ SO(3).

Q4. A soap film and a light ray both "solve variational problems." What functional does each extremize?

Answer

The film minimizes area (minimal surface, \(H = 0\)); the ray extremizes optical path length \(\int n\,ds\) (Fermat) — both are Euler–Lagrange in different costumes.

Computational

Q5. Find the Euler–Lagrange equation for \(F = y'^2 + V(y)\).

Answer

\(\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = V'(y) - 2y'' = 0 \Rightarrow 2y'' = V'(y)\).

Q6. Residue of \(f(z) = \dfrac{1}{z^2+1}\) at \(z = i\)?

Answer

Simple pole: \(\lim_{z\to i}(z - i)f(z) = \dfrac{1}{2i} = -\dfrac{i}{2}\). (Bonus: closing \(\int_{-\infty}^{\infty}\frac{dx}{1+x^2}\) upstairs gives \(2\pi i \times(-i/2) = \pi\) ✓.)

Q7. Evaluate \(\displaystyle\int_0^{2\pi} \frac{d\theta}{2 + \cos\theta}\) by residues.

Answer

Set \(z = e^{i\theta}\): integral becomes \(\oint \frac{2\,dz}{i(z^2 + 4z + 1)}\) on \(|z|=1\). Poles at \(z = -2 \pm \sqrt3\); only \(-2 + \sqrt3\) lies inside. Residue \(\frac{2}{i}\cdot\frac{1}{2\sqrt3}\) ⇒ integral \(= 2\pi i \cdot \frac{1}{i\sqrt3} = \frac{2\pi}{\sqrt3}\).

Multiple choice

Q8. The Euler–Lagrange equation for \(F = y'^2\): (a) \(y'' = 0\) (b) \(y = 0\) (c) \(y' = 0\)

Answer

(a) — extremals are straight lines.

Q9. The residue of \(\dfrac{1}{(z-2)^2}\) at \(z = 2\): (a) 1 (b) 0 (c) undefined

Answer

(b) — a double pole with no \(1/(z-2)\) term in its Laurent series; the residue (coefficient of the simple-pole term) is zero, so \(\oint = 0\) around it.

Q10. Which group is non-Abelian? (a) \(C_4\) (b) \(S_3\) (c) \(\mathbb{Z}_5\)

Answer

(b) — permutations of 3 objects don't commute (swap-then-cycle ≠ cycle-then-swap); it's the smallest non-Abelian group (order 6).