Quizzes · Mathematical Methods I

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

Conceptual

Q1. Why does the curl of any gradient field vanish identically?

Answer

\((\nabla\times\nabla\phi)_i = \epsilon_{ijk}\partial_j\partial_k\phi\): mixed partials are symmetric in \(j,k\) while \(\epsilon_{ijk}\) is antisymmetric — the contraction is zero.

Q2. What property of an operator guarantees real eigenvalues, and why does physics insist on it?

Answer

Hermiticity (\(A = A^\dagger\)). Measured quantities are real numbers, and the eigenvalues of an observable are its possible measurement outcomes — see Hermitian matrices.

Q3. When does separation of variables fail for a PDE?

Answer

When the equation (or boundary) couples the variables inseparably — e.g. coefficients depending on both variables jointly, or boundaries not aligned with a coordinate system in which the PDE separates.

Q4. Why do Fourier methods turn ODEs into algebra?

Answer

\(e^{ikx}\) are eigenfunctions of \(d/dx\): differentiation acts as multiplication by \(ik\). In the transform domain a linear ODE with constant coefficients becomes a polynomial equation — see Fourier transform.

Computational

Q5. Eigenvalues of \(\begin{pmatrix}4&2\\1&3\end{pmatrix}\)?

Answer

\(\lambda^2 - 7\lambda + 10 = 0 \Rightarrow \lambda = 5, 2\) (trace \(= 7\) ✓, det \(= 10\) ✓).

Q6. Fourier sine coefficient \(b_3\) for \(f(x) = x\) on \([-\pi, \pi]\)?

Answer

\(b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} x\sin nx\,dx = \frac{2(-1)^{n+1}}{n}\), so \(b_3 = \frac{2}{3}\).

Q7. Solve \(y'' + y = \delta(x - \xi)\) on \([0, \pi]\) with \(y(0) = y(\pi) = 0\) — what jump condition does the Green's function satisfy at \(x = \xi\)?

Answer

\(G\) continuous at \(\xi\); \(G'\) jumps by \(+1\): \(G'(\xi^+) - G'(\xi^-) = 1\) (integrate the equation across the delta). See Green's functions.

Multiple choice

Q8. The theorem converting \(\oint_S \vec A\cdot d\vec S\) into \(\int_V \nabla\cdot\vec A\,dV\): (a) Stokes (b) Divergence (c) Fundamental theorem of calculus

Answer

(b) — divergence theorem. Stokes converts a line integral to a surface integral of the curl.

Q9. If \(A\) is \(3\times3\) with \(\det A = -4\), then \(\det A^{-1}\) is: (a) \(-4\) (b) \(-1/4\) (c) \(4\)

Answer

(b) — \(\det(A^{-1}) = 1/\det A\).

Q10. The Laplace transform of \(\sin(\omega t)\): (a) \(\dfrac{\omega}{s^2+\omega^2}\) (b) \(\dfrac{s}{s^2+\omega^2}\) (c) \(\dfrac{1}{s^2+\omega^2}\)

Answer

(a) — and (b) is \(\cos(\omega t)\); remember by checking \(t \to 0\) behavior.