🕹️ Practice arcade

🔀 PDE Classification & Characteristics Sprint

Classify a 2nd-order PDE by B^2-4AC, or transport a linear initial condition along a characteristic -- two chapter-1-and-2 fundamentals, fast.

🌊 Fourier Coefficients & Orthogonality Sprint

Compute a sine or cosine coefficient for f(x)=x, or evaluate a sine-sine orthogonality integral on [-L,L] -- the same integration-by-parts and orthogonality facts every series formula leans on.

📐 Heat, Wave & Whole-Line Closed-Form Sprint

Read a rod's steady state off its two boundary values, a single mode's conserved energy, or d'Alembert's formula at a point -- three chapters' own boxed results, evaluated fast.

🌡️ Heat Equation Dirichlet & 2D Sprint

Evaluate a Dirichlet rod's single surviving mode, or a rectangular plate's own 2D product mode -- the same separated-solution shape, one and two dimensions at once.

🔷 Laplace Rectangle & Annulus Sprint

Evaluate a single-mode Laplace solution on a rectangle, or on an annulus with both a growing and a decaying radial term -- two geometries, the same separation payoff.

🌀 Heat Kernel & Green's Function Images Sprint

Read the whole-line heat kernel's own value off its bell-curve formula, or place a ball's Kelvin-inversion image point -- two ways a fundamental solution is actually built.

🔀 System Matrix & Hyperbolicity Sprint

Read a coupling matrix's own entry off two equations, or find the eigenvalue that confirms a system is strictly hyperbolic -- ch.11's own opening two facts, fast.

💥 Nonlinear Conservation Laws Sprint

Find the speed a weak solution's jump actually needs, the Rankine-Hugoniot speed a general quadratic flux's shock uses, or a Burgers rarefaction fan's own value -- ch.12's three payoffs, back to back.

🧮 Numerical Schemes & Stability Sprint

Compute the heat scheme's own worst-case amplification factor, or hand-compute the wave scheme's one-time bootstrap step -- the arithmetic behind ch.13's own stability thresholds.

🔁 Fourier Series Type Survival

Match a boundary condition to the series type it actually forces -- one Dirichlet/Neumann mix-up ends the round.

⚠️ Heat, Wave & Laplace Concept-Check Survival

Call an insulated rod's long-time limit, heat-vs-wave propagation speed, a disk's angular periodicity, or a wave's domain of dependence -- one wrong verdict ends the round.

⚠️ Sturm-Liouville & Heat Setup Survival

Keep the weight in a Sturm-Liouville orthogonality integral, shift nonhomogeneous boundary data before separating, or track a heat source's own time-dependent coefficient -- one dropped step ends the round.

⚠️ Wave, Duhamel & Huygens Survival

Match a vibrating string's initial velocity, inject a wave-equation forcing term at the right derivative order, or call Huygens' sharp-edge property by dimension parity -- one wrong verdict ends the round.

🎯 Bessel & Legendre Domain Survival

Pick which pair of Bessel functions is actually orthogonal, or match a domain's own symmetry to Bessel vs. Legendre -- one mixed-up special function ends the round.

⚖️ Weighted Inner Product Mastery

Compute a weighted dot product, then judge whether the weights still give a valid inner product — five in a row, no clock.

🎯 Rayleigh Quotient & Parseval Mastery

Plug a trial function into a boxed Rayleigh-quotient bound, or sum squared Fourier coefficients under Parseval's equality -- careful substitution, not speed, five clean reads in a row.

🧮 Bessel, Green's Function & Explicit Scheme Mastery

Sum three terms of a Bessel series, evaluate the 2D fundamental solution, or hand-compute one explicit finite-difference step -- multi-step arithmetic where getting it right matters more than getting it fast.

🥁 Vibrating Membrane Frequency Mastery

Compute a rectangular membrane's own mode frequency from its two mode numbers -- five clean reads in a row, no clock.

🧭 System Characteristics & Riemann Invariants Mastery

Trace a canonical variable back along its own characteristic family, or rescale a Riemann invariant affinely and still land on the right value -- four clean answers in a row, no clock.