Foundations

Describe a quantity as a numerical value with a unit, and distinguish its physical kind from its dimension. Build length, area, volume and rate units, check meaningful combinations, and separate exact unit relations from measurement uncertainty.

Convert quantities with exact unit ratios, track cancellation through compound units, and apply the correct powers to area and volume. Distinguish temperature readings from differences, check dimensions and context, and retain measurement uncertainty when reporting.

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

Shannon entropy measures uncertainty in a specified probability distribution.

No total algorithm decides halting for every program and input in a general computation model.

Big O bounds eventual growth up to a fixed constant factor.

Combine input uncertainty through a measurement model.

Use dimensions to detect impossible quantity equations.

Lorentz transformations preserve a combined space-time separation.

Stationary action supplies coordinate equations of motion.

Canonical coordinates and momenta evolve from the Hamiltonian.

Linear disturbances add; their relative phase determines reinforcement.

Free test motion follows the metric connection.

Shared input variation belongs in the joint uncertainty description.

For an autonomous smooth system ẋ=f(x), equilibria satisfy f(x*)=0 and nearby perturbations obey δẋ≈Jδx, J_ij=∂f_i/∂x_j at x*.

A bifurcation is a qualitative change in a parameterized dynamical system.