This chapter is the master notation table for the whole book. The conventions here are field-standard and override the courseware wherever they differ; the “Course writes” column records each deviation so you can cross-reference Zhang Bo’s slides. The choice of notation policy is recorded in docs/adr/0002-pragmatic-notation-standardization.md, and the canonical word for each concept in CONTEXT.md.
Sets, spaces, and general symbols
Symbol
Meaning
Course writes
\(\mathbb{R}, \mathbb{R}^n, \mathbb{R}_+\)
reals, \(n\)-space, non-negative reals
same
\(X \subseteq \mathbb{R}^n\)
feasible set / admissible decisions
\(\aleph\)
\(f:\mathbb{R}^n\to\mathbb{R}\)
static objective
\(F\), with codomain \(\Im\)
\(\nabla f,\ \nabla^2 f\)
gradient, Hessian
same
\(\langle a,b\rangle,\ a^\top b\)
inner product
same
\(a \odot b\)
componentwise (Hadamard) product
\(a\circ b\)
\(:=\)
“is defined as”
\(\triangleq\) / \(=\)
Static optimization (Ch. 1)
Symbol
Meaning
Course writes
\(g(x)\ge 0\)
inequality constraints (sign convention kept)
same (\(g\ge 0\))
\(h(x)=0\)
equality constraints
same
\(\lambda \ge 0\)
KKT multipliers on \(g\)
\(\lambda\)
\(\mu\)
multipliers on \(h\)
\(\mu\)
\(\lambda_0\)
Fritz–John multiplier on the objective
\(\lambda_0\)
\(\mathcal{L}(x,\lambda,\mu)\)
Lagrangian
\(L\)
\(V(\theta)\)
value function of a parameter \(\theta\) (envelope)
same
Dynamics and ODEs (Ch. 2)
Symbol
Meaning
Course writes
\(\dot x = \mathrm{d}x/\mathrm{d}t\)
time derivative
same (also \(x'\))
\(A\), \(\lambda_i\), \(\xi_i\)
system matrix, eigenvalues, eigenvectors
same
\(\tau=\operatorname{tr}A,\ \Delta=\det A\)
trace and determinant (phase classification)
\(\sigma,\ \tau\) (swapped!)
\(e^{At}\)
matrix exponential
same
steady state \(\bar x\) (or \(x^\ast\))
equilibrium of \(\dot x=F(x)\)
\(x^\ast\)
The course labels the trace \(\sigma\) and the determinant \(\tau\) in the planar-system classification. We use the more common \(\tau=\operatorname{tr}A\) and \(\Delta=\det A\) with discriminant \(\tau^2-4\Delta\). Watch this when comparing the \(\sigma\)–\(\tau\) diagrams.
Calculus of variations (Ch. 3)
Symbol
Meaning
Course writes
\(J[x]=\int_0^T F(t,x,\dot x)\,\mathrm{d}t\)
variational objective (functional)
\(\int_0^T F\,\mathrm{d}t\)
\(F\), \(F_x\), \(F_{\dot x}\)
integrand and its partials
same
\(\mathcal{A}\)
class of admissible paths (a subset of \(C^1[0,T]\), not of \(\mathbb{R}^n\))
\(\theta\) (clashes with the parameter \(\theta\))
The course reuses \(\theta\) both for the multiplier on an algebraic constraint (§4.5) and for an exogenous parameter in comparative statics (§4.7). We write the multiplier as \(\eta\) and keep \(\theta\) for the parameter.
The discount trap. The course writes the discrete-time Bellman discount and the continuous-time discount with the same symbol \(\rho\), even though one is a factor in \((0,1)\) and the other a rate in \((0,\infty)\). Throughout these notes: \(\rho\) is the continuous-time rate (\(e^{-\rho t}\), Ch. 3–4 and continuous DP) and \(\beta=1/(1+\rho)\) is the discrete-time factor (Ch. 5). Here \(\beta=1/(1+\rho)\) is the per-period (simple) factor, whereas \(e^{-\rho}\) is the continuous-compounding factor; the two agree only to first order, \(\beta=1-\rho+O(\rho^2)=e^{-\rho}+O(\rho^2)\), and the \(\rho\) in each is a distinct object (a discrete per-period rate vs. a continuous-time rate).