Kaoyan Mathematics III cheat sheet

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

Consistency requires rank A = rank [A|b]; uniqueness also needs full column rank.

det A = 0 means no unique solution, not necessarily no solution.

Trace = sum of eigenvalues; determinant = product of eigenvalues.

a×b is perpendicular to both vectors; its magnitude is the parallelogram area.

Distance from point to plane: |n·P − d| / |n|.

Bayes: posterior ∝ likelihood × prior; always compute P(+) first.

Standard error of the mean is σ/√n, not σ.

p-value = P(data at least as extreme | H₀), never P(H₀ true).

Binomial variance np(1 − p); Poisson mean and variance both λ.

Sample variance uses n − 1 to be unbiased.

Logistic equilibria: 0 unstable, K stable; r is rate, K is capacity.

LP optimum lies at a vertex; check every intersection of two constraints.

Sensitivity analysis shows how fragile a conclusion is to parameter change.

Residual patterns, not residual size, reveal model misspecification.

A Markov steady state must satisfy πP = π and sum to 1.