42 · Chalk Notes II

Polar coordinates: don't mix up polar coordinates and Argand diagrams. Polar is just a different method of representing points — a distance from origin (the modulus, a scalar radius) and an angle. The whole mechanism is a compass and ruler: elementary school mathematics with new clothes. The curve families to memorise: the limaçon, r = a + b·cosθ, which may or may not have a dimple depending on the boundary conditions — cardioids, inner loops, dimples; the rose petal curves; the circle forms; the lemniscate, r² = a²cos2θ — Bernoulli's infinity shape; and the Archimedean spiral, r = aθ, where radius grows with angle. When you see a polar curve, identify which family it is; the recurring patterns are few.

Hyperbolic functions: cosh, sinh, tanh — derivatives map to each other like their trig counterparts, but where sin and cos swap with a sign, here they swap without; tanh differentiates to sech². For the reciprocals — cosech, sech, coth — all derivative signs are negated. These are reciprocals of the trinity, not the inverses (arcsinh and friends). Don't drag Osborn's rule into the reciprocals. When an integrand contains a square number and a square of x — added or subtracted, under a root or not — the formula booklet almost certainly already has it. Go straight to the booklet. Read and memorise the entire formulae booklet; it takes reading it every day, and it's that important.

Partial fractions in the integrand: split so each denominator factor's powers are covered up to the highest. First-order differential equations: separate the variables — Moses' Arc the x's to one side and y's to the other — and integrate both. When the reverse product rule doesn't apply, arrange into the standard non-homogeneous first-order linear form and multiply by the integrating factor: e raised to the integral of the function multiplying y. Integrate both sides, adding the constant on the right so the left stays the clean product of y and the integrating factor. Second-order: damped harmonic motion — heavy damping, critical damping, light damping. Three cases. Simple harmonic, damped, forced — modelling is on the syllabus and I was slacking on it; that's the bottleneck habit of keeping a list of weaknesses instead of dealing with them now.

Physics. Threshold frequency: the minimum frequency of incident light for electrons to be released from the metal surface. Photons: discrete packets of electromagnetic waves with energy E = hf. The de Broglie wavelength is not just any wavelength — it is associated with a particle: Planck's constant over the particle's momentum. The photoelectric effect literally RELEASES electrons — they are ejected, liberated, not trapped between energy levels. Not just any intensity does it — only above the threshold. Atomic absorption: a photon's energy exactly matches an energy gap and an electron moves up within the atom — no photon out, just an excited atom. Atomic emission: an electron falls down within the atom and a photon is emitted. Emission's input-output relation is the opposite of the photoelectric effect. Spectrum questions: the dark line among the bright is the absorption spectrum — absorption of photons. Einstein's photoelectric equation: know the what, why and how of each variable; the big caveat is that the work function is the product of Planck's constant and the threshold frequency.

Waves. Phase difference formulae apply only to PROGRESSIVE waves — including longitudinal ones, where the oscillation is parallel to the direction of energy transfer. Stationary waves do not transfer energy; the phase-path relationship does not apply, and distance from a node is irrelevant to being in phase or antiphase — they are discrete. Why does diffraction occur better for smaller wavelengths? Because when the wavelength is similar in size to the gap, diffraction is stronger. Why does intensity decrease with distance? The power distributes over a larger area, so power per unit area falls. Higher frequency means smaller wavelength (v = fλ), and the smaller wavelength is the direct reason for greater resolution. Ultrasound comes in pulses; coupling gel is used to reduce reflection at the skin boundary.

Mechanics and materials. The conditions for the equations of motion: constant acceleration. The principle of linear momentum applies only in a closed system with no external forces. A Newton's third law pair: equal magnitude, opposite direction — and if A acts on B, the pair is B acting on A; the agent, or source, is the object enacting the force. Springs in series versus parallel: the combined spring constant follows the same inverse pattern as resistors — series springs stretch more, so stiffness halves; parallel springs are stiffer. Same force applied, different extensions. Know what is constant and what changes as a byproduct. Work done in stretching: looks deceive — don't trust the graph alone; depend on the verbal conditions given. Upthrust: by Archimedes' principle, upthrust equals the weight of the fluid displaced. For a fully submerged object it stays constant — so when something floats upward it's usually the weight decreasing (the Galilean thermometer's bulbs), and for a rising balloon the surrounding air density falls, upthrust decreases at constant weight, and the balloon decelerates to a stop. Electricity: a filament bulb's resistance rises with voltage so the IV graph plateaus; the NTC thermistor's gradient goes to the moon; the diode perks up after a threshold. Vivid, stupid mnemonics work — when something is connected visually with a strong emotion, memorisation spikes.

The dot product and planes: r·n = a·n, where r is the general position vector, a is a known point on the plane, and n is the normal vector. Three dimensions, n dimensions — the plane lives in them.