🕹️ Practice arcade

📐 Orthogonality & Coefficients Sprint

Evaluate a sine/cosine orthogonality integral, then the mean-value coefficient $a_0$ it feeds into -- both in one run.

🌊 Fourier Series Family Sprint

A sine-series ramp coefficient, a square wave's odd harmonic, or a complex coefficient's magnitude -- three faces of the same series.

📏 Gibbs Overshoot Sprint

Apply the course's own Gibbs constant (8.95% of the jump) to predict the partial-sum overshoot, as many times as you can.

⚡ Parseval & Bessel Energy Sprint

Sum a Fourier series' full energy by Parseval, or find how much energy the higher harmonics must still account for by Bessel.

🔀 Fourier Transform Basics Sprint

A pulse's DC value, a time-shift's phase angle, or a Gaussian's own transform -- the transform's core moves, back to back.

🔁 Convolution Sprint

Find two rectangular pulses' overlap height at $t=0$, or multiply magnitudes straight from the convolution theorem.

🎚️ Sampling, DFT & FFT Sprint

The Nyquist rate, a sampled signal's apparent frequency after aliasing, a DFT's DC bin, or the FFT's own $N\log_2N$ operation count.

🪞 Even/Odd & Inner Product Sprint

Collapse a symmetric-interval integral using even/odd symmetry, then build a function inner product straight from its definition.

📐 Norms & Orthonormal Family Sprint

A rectangular pulse's $L^1$ or $L^2$ norm, or a scaled family member's own norm on $[-\pi,\pi]$ -- three faces of "how big is this function."

📉 Uniform Convergence Sprint

Find the sup-norm bound that decides whether a function sequence converges uniformly -- two of the course's own worked examples, back to back.

🎯 Orthogonal Projection Sprint

A Fourier coefficient is just an orthogonal-projection ratio -- divide the inner product by the basis function's own squared norm, $\pi$.

🌡️ Heat & Wave Modes Sprint

The heat equation's per-mode decay rate, or a vibrating string's own standing-wave frequency -- separation of variables' two payoffs.

📏 Harmonic Spacing Sprint

As the period grows, the Fourier series' harmonic spacing $\omega_0=2\pi/T$ shrinks toward the continuous spectrum -- find it fast.

⚡ Differentiation & Delta Sprint

Apply $i\omega$ twice for a second derivative's transform, or apply the delta's sifting property directly -- three fast closed forms.

🔄 Inverse DFT & Conjugate Symmetry Sprint

Recover a time-domain sample from DFT bins, or use conjugate symmetry to find a real signal's mirrored bin without recomputing it.

🔁 Circulant & Circular Convolution Sprint

A circulant matrix's own DC eigenvalue, or the discrete convolution theorem's magnitude product -- the DFT's matrix-algebra side.

🖼️ Multidimensional Transform Sprint

A separable 2D transform's product form, or a 2D DFT's own DC bin -- the same one-dimensional facts, one axis at a time.

🎲 Characteristic Functions Sprint

An Exponential's or Normal's own characteristic function magnitude, or the independence product rule $\varphi_{X+Y}=\varphi_X\varphi_Y$.

🔔 CLT Approximation Sprint

Evaluate the CLT derivation's own finite-$n$ approximation $(1-t^2/2n)^n$, the exact quantity that converges to $e^{-t^2/2}$.

💀 One-Sided Limits Survival

Read a piecewise function from the correct side. Pick the wrong branch once and the run ends.

💀 Dirichlet Jump Survival

At a jump, the series converges to the MIDPOINT of the two one-sided limits -- one wrong pick ends the run.

⚖️ 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.

🎯 Geometric Series Check

A geometric distribution's own validity leans on this convergence test — get it right four times in a row.