Posts
-
Gauss–Markov Theorem: Why OLS Is BLUE [Basic Statistics I Studied #11]
A careful proof of the Gauss–Markov theorem: the assumptions OLS needs, what BLUE really means, and why normal errors are not required.
6 min read -
Macaulay Duration and Modified Duration: Bond Price Sensitivity
Derive Macaulay and modified duration from a bond's cash flows, interpret the weights, and test the price-change estimate with a worked bond example.
6 min read -
Market Risk, Firm-Specific Risk, Beta, and Alpha [Basic Investing I Studied #11]
Separate market exposure from firm-specific surprises with a one-factor return model, then see exactly when beta and residual risk add up to total variance.
6 min read -
TIPS Explained: Real Returns, Inflation Adjustment, and Risks [Basic Investing I Studied #8]
A practical guide to nominal and real returns, the exact Fisher relation, and how TIPS principal, coupons, maturity protection, taxes, and risks work.
7 min read -
Characteristic Polynomials and the Cayley–Hamilton Theorem [Linear Algebra I Studied #18]
We chase down which matrices can actually be diagonalized — starting with characteristic polynomials, hitting complex eigenvalues, and landing on the Cayley–Hamilton theorem.
11 min read -
The Orbit Equation via Energy Methods [Classical Mechanics I Studied #10]
We rederive the orbit equation using energy conservation — kinetic plus potential — then sneak in angular momentum and the u=1/r substitution to nail the same result.
5 min read -
Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 2) [Classical Mechanics I Studied #9]
We derive the polar-coordinate equation of an ellipse from scratch to confirm, in proper 'Ahhhh~~~~' fashion, that the orbit equation from last time really is an ellipse.
3 min read -
Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 1) [Classical Mechanics I Studied #8]
Newton connected Galileo's earthly motion with Kepler's celestial laws through universal gravitation—and the shell theorem tells us when a spherical body can be treated as a point mass.
9 min read -
Ampère's Law: Three Symmetry Examples [Electromagnetism I Studied #18]
Apply Ampère's law to an infinite straight wire, an infinite current sheet, and an ideal infinite solenoid, with explicit symmetry assumptions and sign conventions.
6 min read -
Ampère's Law [Electromagnetism I Studied #17]
In magnetostatics, Ampère's law relates magnetic-field circulation around a closed contour to the signed current through a spanning surface; Maxwell's displacement-current term gives the time-dependent generalization.
6 min read -
The Biot–Savart Law [Electromagnetism I Studied #16]
Derive the Biot–Savart law and use it to find the magnetic field of a finite straight wire, an infinite wire, and a circular current loop.
4 min read -
Lorentz Force and Magnetostatics [Electromagnetism I Studied #15]
An introduction to magnetostatics, the Lorentz force, magnetic-force direction, current densities, forces on steady currents, and charge continuity.
6 min read -
Linear Dielectrics (Part 2) [Electromagnetism I Studied #14]
How a linear dielectric changes a capacitor, followed by worked examples on layered dielectrics, bound charge, and two partial-filling geometries.
5 min read -
Linear Dielectrics [Electromagnetism I Studied #13]
Linear dielectrics, electric susceptibility and permittivity, followed by a complete solution for a charged conducting sphere surrounded by a dielectric shell.
4 min read -
Electric Displacement [Learning Electromagnetism, Part 12]
Deriving the electric displacement field D, its free-charge Gauss law, and its use in cylindrical and spherical dielectric examples.
5 min read -
Bound Charge Density [Learning Electromagnetism, Part 11]
An intuitive derivation of the bound surface charge density σ_b = P·n̂ and bound volume charge density ρ_b = −∇·P in a polarized dielectric.
5 min read -
Polarization [Learning Electromagnetism, Part 10]
An intuitive introduction to induced atomic dipoles, polarization density, and the bound surface and volume charges produced by a polarized dielectric.
5 min read -
Multipole Expansion [Learning Electromagnetism, Part 9]
An intuitive derivation of the far-field multipole expansion for a localized charge distribution, including monopole, dipole, quadrupole, origin dependence, and the ideal-dipole field.
6 min read -
Separation of Variables: Strips and Pipes [Electromagnetism #8]
Solve Laplace's equation in a grounded strip and rectangular pipe using separated modes, Fourier sine coefficients, and carefully stated boundary conditions.
9 min read -
Momentum of a System of Particles [Classical Mechanics I Studied #14]
Derive the momentum of a fixed-mass particle system, show exactly when internal forces cancel, and connect net external impulse to center-of-mass motion.
5 min read