Foundations

A linear restoring force, damping and periodic drive determine response.

A local differential equation determines small string disturbances.

Linear disturbances add; their relative phase determines reinforcement.

Boundary conditions select standing-wave frequencies.

Angles at an interface depend on refractive indices.

A paraxial lens relates object and image distances.

An aperture's finite width selects dark far-field directions.

An ideal analyzer transmits the field component along its axis.

Scalar and vector potentials encode electric and magnetic fields.

A linear two-conductor arrangement stores separated charge.

Changing current changes linked magnetic flux and induced emf.

The rate of field-energy change balances outward energy flow and power transferred to charges.

Lumped circuit models balance node currents and loop voltages.

A simple resistor–capacitor circuit has an exponential timescale.

Sinusoidal response combines resistance and net reactance.

A temperature gradient drives conductive heat flux.

Energy changes phase at the specified coexistence conditions.

Vapor pressure changes with temperature through latent enthalpy.

Thermal equilibrium fixes an ideal absorber's total emitted power.

Fixed-temperature equilibrium weights states by energy.

Microstate probabilities quantify statistical entropy.

A state and observable determine the mean of repeated outcomes.

A commutator constrains two outcome spreads in the same state.

A spin one-half state has two outcomes along a chosen axis.

A density operator represents quantum statistical states.

Projectors determine outcome probabilities and conditional states.

A joint pure state can fail to factor into subsystem states.

Boson and fermion states respond differently to exchange.

Free test motion follows the metric connection.

Spacetime curvature responds to stress and energy.