Kaoyan Mathematics II - independent mathematical foundations cheat sheet

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|.

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.

∫(1/x) dx = ln|x| + C, and ∫f′/f dx = ln|f| + C.

Integration by parts: ∫u dv = uv − ∫v du.

Split the interval at axis crossings before computing area.

Volume of revolution about the x-axis is π∫y² dx.

The trapezium rule overestimates when the curve is concave up.

The derivative is a linear map, not merely a list of partial derivatives.

C¹ regularity makes the Jacobian a reliable total derivative.

Inverse function theorem needs an invertible Jacobian at the point.

Implicit function theorem needs invertibility of the partial Jacobian in the dependent variables.

Both theorems are local and yield C¹ objects under C¹ hypotheses.

Linear first order: μ = exp(∫P dx) turns the left side into (μy)'.

Repeated characteristic root: multiply the second term by x.

Complementary + particular, then apply initial conditions.

Resonance: multiply the trial by x raised to the relevant characteristic-root multiplicity.

Sign of f(y) between equilibria determines stability on the phase line.

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.