Notation

Every symbol the book uses, in the order you meet it, with what it means and the chapter that introduces it. Search works too: press Ctrl K.

Symbol	Means	Introduced in
$P \land Q$	P and Q	0.1 How to read a theorem
$P \lor Q$	P or Q (or both)	0.1 How to read a theorem
$\neg P$	not P	0.1 How to read a theorem
$P \Rightarrow Q$	if P then Q	0.1 How to read a theorem
$P \Leftrightarrow Q$	P if and only if Q	0.1 How to read a theorem
$\forall x$	for every $x$	0.1 How to read a theorem
$\exists x$	there exists an $x$	0.1 How to read a theorem
$\mathbb Z$	the integers $\dots, -2, -1, 0, 1, 2, \dots$	0.2 Proof techniques
$\mathbb N$	the natural numbers $1, 2, 3, \dots$	0.2 Proof techniques
$\blacksquare$ , $\square$	end of proof	0.2 Proof techniques
$x \in A$ , $x \notin A$	$x$ is an element of $A$ ; $x$ is not an element of $A$	0.3 Sets, functions, and counting infinity
$A \subseteq B$	every element of $A$ is in $B$ (subset, equality allowed)	0.3 Sets, functions, and counting infinity
$A \cup B$ , $A \cap B$	union and intersection	0.3 Sets, functions, and counting infinity
$A^c$ , $A \setminus B$	complement of $A$ ; the elements of $A$ not in $B$	0.3 Sets, functions, and counting infinity
$A \times B$	all ordered pairs $(a, b)$ with $a \in A$ , $b \in B$	0.3 Sets, functions, and counting infinity
$\varnothing$	the empty set	0.3 Sets, functions, and counting infinity
$f: A \to B$	a function with domain $A$ and codomain $B$	0.3 Sets, functions, and counting infinity
$g \circ f$ , $f^{-1}$	composition (first $f$ , then $g$ ); inverse function	0.3 Sets, functions, and counting infinity
$\mathbb N, \mathbb Z, \mathbb Q, \mathbb R$	counting numbers $1, 2, 3, \dots$ ; integers; rationals; reals	0.3 Sets, functions, and counting infinity
$a \sim b$	$a$ is related to $b$ (read 'a tilde b')	0.4 Equivalence relations and partitions
$[a]$	the equivalence class of $a$ : everything related to $a$	0.4 Equivalence relations and partitions
$A / {\sim}$	the set of all equivalence classes (the quotient set)	0.4 Equivalence relations and partitions
$n \mid m$	$n$ divides $m$ : $m = nk$ for some integer $k$	0.4 Equivalence relations and partitions
$a \mid b$	$a$ divides $b$ : $b = ak$ for some integer $k$	0.5 Integers mod n, gcd and Bézout
$\gcd(a, b)$	the greatest common divisor of $a$ and $b$	0.5 Integers mod n, gcd and Bézout
$a \equiv b \pmod n$	$n \mid (a - b)$ : $a$ and $b$ leave the same remainder on division by $n$	0.5 Integers mod n, gcd and Bézout
$a \bmod n$	the remainder of $a$ on division by $n$ , a number from $0$ to $n - 1$	0.5 Integers mod n, gcd and Bézout
$\mathbb Z_n$	the integers mod $n$ : the $n$ residue classes $[0], \dots, [n - 1]$	0.5 Integers mod n, gcd and Bézout
$a^{-1}$	the inverse of $a$ mod $n$	0.5 Integers mod n, gcd and Bézout
$\mathbb C$	the complex numbers	0.6 Complex numbers and polynomials, done properly
$z = a + bi$	a complex number with real part $a$ and imaginary part $b$	0.6 Complex numbers and polynomials, done properly
$\operatorname{Re} z,\ \operatorname{Im} z$	the real part $a$ and the imaginary part $b$ of $z = a + bi$	0.6 Complex numbers and polynomials, done properly
$\bar z$	the conjugate of $z$	0.6 Complex numbers and polynomials, done properly
$\|z\|$	the modulus of $z$	0.6 Complex numbers and polynomials, done properly
$\theta$ , $\arg z$	an angle (in radians) giving the direction of $z$	0.6 Complex numbers and polynomials, done properly
$e^{i\theta}$	shorthand for $\cos\theta + i\sin\theta$ (Euler's formula)	0.6 Complex numbers and polynomials, done properly
$\deg f$	the degree of a polynomial $f$	0.6 Complex numbers and polynomials, done properly
$\blacksquare$	end of proof (also written Q.E.D. or $\square$ )	0.7 Writing a proof a grader accepts
$\mathbb Q$	the rational numbers, fractions $p/q$ of integers with $q \ne 0$	1.1 The real numbers and completeness
$\sup S$	the supremum (least upper bound) of $S$	1.1 The real numbers and completeness
$\inf S$	the infimum (greatest lower bound) of $S$	1.1 The real numbers and completeness
$\varepsilon$	epsilon: by convention, a small positive number chosen by the reader	1.1 The real numbers and completeness
$(a_n)$	the sequence $a_1, a_2, a_3, \dots$	1.2 Sequences and limits, from zero
$a_n \to L$ , $\lim_{n \to \infty} a_n = L$	the sequence converges to $L$	1.2 Sequences and limits, from zero
$\|x - y\|$	the distance between $x$ and $y$	1.2 Sequences and limits, from zero
$(a_{n_k})$	a subsequence of $(a_n)$	1.2 Sequences and limits, from zero
$\lim_{x \to a} f(x) = L$	$f(x)$ tends to $L$ as $x$ approaches $a$	1.3 Limits and continuity with epsilon and delta
$\delta$	delta: the input tolerance, chosen by the prover after $\varepsilon$	1.3 Limits and continuity with epsilon and delta
$[a, b]$	the closed interval $\{x : a \le x \le b\}$	1.3 Limits and continuity with epsilon and delta
$f'(a)$ , $\frac{df}{dx}(a)$	the derivative of $f$ at $a$	1.4 The derivative as best linear approximation
$f^{(n)}$	the $n$ -th derivative of $f$	1.4 The derivative as best linear approximation
$T_n(x)$	the Taylor polynomial of degree $n$	1.4 The derivative as best linear approximation
$P = \{x_0, \dots, x_n\}$	a partition of $[a, b]$ : points $a = x_0 < x_1 < \dots < x_n = b$	1.5 The Riemann integral: area as a limit
$\Delta x_i = x_i - x_{i-1}$	the width of the $i$ -th piece	1.5 The Riemann integral: area as a limit
$M_i, m_i$	the supremum and infimum of $f$ on the $i$ -th piece	1.5 The Riemann integral: area as a limit
$U(f, P), L(f, P)$	the upper and lower sum of $f$ for the partition $P$	1.5 The Riemann integral: area as a limit
$\int_a^b f$	the integral of $f$ over $[a, b]$ (the common value of the lower and upper integrals)	1.5 The Riemann integral: area as a limit
$\sum_{k=1}^\infty a_k$	the series: the limit of the partial sums, if it exists	1.6 Series, power series, and uniform convergence
$S_n = a_1 + \dots + a_n$	the $n$ -th partial sum	1.6 Series, power series, and uniform convergence
$f_n \to f$ uniformly	$\sup_x \|f_n(x) - f(x)\| \to 0$	1.6 Series, power series, and uniform convergence
$R$	the radius of convergence of a power series	1.6 Series, power series, and uniform convergence
$\mathbb R$	the real numbers	2.1 Vector spaces and subspaces
$\mathbb R^n$	ordered lists of $n$ real numbers	2.1 Vector spaces and subspaces
$x \in S$	$x$ is an element of the set $S$	2.1 Vector spaces and subspaces
$\{x : \text{condition}\}$	the set of all $x$ satisfying the condition	2.1 Vector spaces and subspaces
$S \subseteq V$	every element of $S$ is in $V$	2.1 Vector spaces and subspaces
$O$	the zero vector	2.1 Vector spaces and subspaces
$-x$	the negative of $x$ , equal to $(-1)x$	2.1 Vector spaces and subspaces
$P_n$	polynomials of degree at most $n$	2.1 Vector spaces and subspaces
$\mathbb R^{m \times n}$	$m$ by $n$ matrices of real numbers	2.1 Vector spaces and subspaces
$U \cap W$	the vectors in both $U$ and $W$	2.1 Vector spaces and subspaces
$L(S)$ , $\operatorname{span}(S)$	the span of $S$ : all linear combinations	2.2 Span, independence, basis, dimension
$\dim V$	the dimension of $V$	2.2 Span, independence, basis, dimension
$\sum_{i=1}^k c_i x_i$	$c_1x_1 + \dots + c_kx_k$	2.2 Span, independence, basis, dimension
$U + W$	all sums $u + w$ with $u \in U$ , $w \in W$	2.2 Span, independence, basis, dimension
$T: V \to W$	a function from $V$ to $W$	2.3 Linear maps, null space, range, rank–nullity
$x \mapsto T(x)$	$x$ is sent to $T(x)$	2.3 Linear maps, null space, range, rank–nullity
$T(V)$ , $R(T)$	the range of $T$	2.3 Linear maps, null space, range, rank–nullity
$N(T)$ , $\ker T$	the null space (kernel) of $T$	2.3 Linear maps, null space, range, rank–nullity
$ST$ , $S \circ T$	do $T$ first, then $S$	2.4 Inverses, one-to-one maps, isomorphism
$I_V$	the identity map on $V$ : $I_V(x) = x$	2.4 Inverses, one-to-one maps, isomorphism
$T^{-1}$	the inverse of $T$	2.4 Inverses, one-to-one maps, isomorphism
$V \cong W$	$V$ and $W$ are isomorphic	2.4 Inverses, one-to-one maps, isomorphism
$m(T)$	the matrix of $T$ in the chosen bases (Apostol's notation)	2.5 Matrices as maps, multiplication, change of basis
$a_{ij}$	the entry of $A$ in row $i$ , column $j$	2.5 Matrices as maps, multiplication, change of basis
$AB$	the matrix product: do $B$ first, then $A$	2.5 Matrices as maps, multiplication, change of basis
$I_n$	the $n$ by $n$ identity matrix	2.5 Matrices as maps, multiplication, change of basis
$A^{\mathsf T}$	the transpose of $A$ : rows become columns	2.5 Matrices as maps, multiplication, change of basis
$C^{-1}AC$	the same map as $A$ , written in a new basis	2.5 Matrices as maps, multiplication, change of basis
$[A \mid b]$	the augmented matrix of $Ax = b$	2.6 Systems of equations and elimination
$C(A)$	the column space of $A$	2.6 Systems of equations and elimination
$A^{-1}$	the inverse matrix	2.6 Systems of equations and elimination
$\det A$ , $\|A\|$	the determinant of $A$	2.7 Determinants
$\operatorname{sgn}(\sigma)$	the sign of a permutation, $\pm 1$	2.7 Determinants
$C_{ij}$	the $(i, j)$ cofactor	2.7 Determinants
$\lambda$	an eigenvalue	2.8 Eigenvalues and eigenvectors
$E(\lambda)$	the eigenspace of $\lambda$	2.8 Eigenvalues and eigenvectors
$\det(\lambda I - A)$	the characteristic polynomial	2.8 Eigenvalues and eigenvectors
$\operatorname{tr} A$	the trace: the sum of the diagonal entries	2.8 Eigenvalues and eigenvectors
$\langle x, y \rangle$ , $(x, y)$	the inner product of $x$ and $y$	2.9 Inner products, orthogonality, Gram–Schmidt, projection
$\lVert x \rVert$	the norm (length) of $x$	2.9 Inner products, orthogonality, Gram–Schmidt, projection
$x \perp y$	$x$ and $y$ are orthogonal	2.9 Inner products, orthogonality, Gram–Schmidt, projection
$S^\perp$	the orthogonal complement of $S$	2.9 Inner products, orthogonality, Gram–Schmidt, projection
$\hat x$	a least squares solution	2.10 Least squares and regression as projection
$A^{\mathsf T}A\hat x = A^{\mathsf T}b$	the normal equations	2.10 Least squares and regression as projection
$P = A(A^{\mathsf T}A)^{-1}A^{\mathsf T}$	the projection matrix onto the column space	2.10 Least squares and regression as projection
$\pi$	a stationary distribution: $P\pi = \pi$	2.11 Markov chains and PageRank
$\mathbf 1$	the vector of all ones	2.11 Markov chains and PageRank
$G = dP + \frac{1-d}{n}\mathbf 1\mathbf 1^{\mathsf T}$	the Google matrix with damping $d$	2.11 Markov chains and PageRank
$A^*$	the adjoint of $A$ ; for real matrices the transpose $A^{\mathsf T}$ , for complex the conjugate transpose	3.1 Symmetric matrices and the spectral theorem
$Q^{\mathsf T}Q = I$	the columns of $Q$ are orthonormal	3.1 Symmetric matrices and the spectral theorem
$A = Q\Lambda Q^{\mathsf T}$	orthogonal diagonalization: $\Lambda$ diagonal, $Q$ orthogonal	3.1 Symmetric matrices and the spectral theorem
$q(x) = x^{\mathsf T}Ax$	the quadratic form of the symmetric matrix $A$	3.2 Quadratic forms and principal axes
$A \succ 0$	$A$ is positive definite	3.2 Quadratic forms and principal axes
$Q^{\mathsf T}Q = I$	the columns of $Q$ are orthonormal; $Q$ is orthogonal	3.3 Orthogonal matrices, rotations and reflections
$R_\theta$	rotation of the plane by angle $\theta$	3.3 Orthogonal matrices, rotations and reflections
$U^*U = I$	$U$ is unitary (complex version)	3.3 Orthogonal matrices, rotations and reflections
$A = U\Sigma V^{\mathsf T}$	the SVD: $U, V$ orthogonal, $\Sigma$ diagonal with $\sigma_1 \ge \sigma_2 \ge \dots \ge 0$	3.4 The singular value decomposition and PCA
$\sigma_i$	the $i$ -th singular value	3.4 The singular value decomposition and PCA
$A_k$	the best rank- $k$ approximation of $A$	3.4 The singular value decomposition and PCA
$\lVert A \rVert$	the operator norm: the largest stretch of $A$ , equal to $\sigma_1$	3.5 Matrix norms, conditioning, and why numerics fail
$\kappa(A)$	the condition number $\sigma_{\max}/\sigma_{\min}$	3.5 Matrix norms, conditioning, and why numerics fail
$\varepsilon_{\text{mach}}$	machine epsilon, about $1.1 \times 10^{-16}$ in double precision	3.5 Matrix norms, conditioning, and why numerics fail
$A = UTU^*$	Schur decomposition: $U$ unitary, $T$ upper triangular	3.6 Cayley–Hamilton, Schur, and the Jordan form
$m_A(\lambda)$	the minimal polynomial of $A$	3.6 Cayley–Hamilton, Schur, and the Jordan form
$J_k(\lambda)$	the $k \times k$ Jordan block with eigenvalue $\lambda$	3.6 Cayley–Hamilton, Schur, and the Jordan form
$V^*$	the dual space of $V$ : all linear functionals $V \to \mathbb R$ (not the adjoint of 3.1, which is a map $V\to V$ )	3.7 Dual spaces and quotient spaces
$\varphi_1, \dots, \varphi_n$	the dual basis of a basis $v_1, \dots, v_n$ : $\varphi_i(v_j) = 1$ if $i = j$ and $0$ otherwise	3.7 Dual spaces and quotient spaces
$T^{\mathsf T}$	the dual map of $T:V\to W$ , a map $W^\to V^$ , defined by $T^{\mathsf T}\psi = \psi\circ T$	3.7 Dual spaces and quotient spaces
$U^0$	the annihilator of $U$ : the functionals that are $0$ on all of $U$	3.7 Dual spaces and quotient spaces
$v + U$	the coset of $v$ : all vectors $v + u$ with $u \in U$ , a copy of $U$ moved by $v$	3.7 Dual spaces and quotient spaces
$V/U$	the quotient space, read 'V mod U': the set of all cosets of $U$ in $V$	3.7 Dual spaces and quotient spaces
$\nabla f$	the gradient of a scalar function $f: \mathbb R^n \to \mathbb R$ , the column of partial derivatives	3.8 Matrix calculus and backpropagation
$J_g$	the Jacobian matrix of $g: \mathbb R^n \to \mathbb R^m$ , the $m \times n$ matrix of partial derivatives	3.8 Matrix calculus and backpropagation
$\sigma$	an activation function applied entry by entry (in this chapter only; elsewhere $\sigma$ is a singular value)	3.8 Matrix calculus and backpropagation
$(G, \ast)$ or $G$	a group: a set $G$ with an operation $\ast$ , often written as a product $ab$	3.9 Groups, rings and fields: a preview built from what you know
$e$ , $a^{-1}$	the identity element of a group, and the inverse of $a$ (in additive groups: $0$ and $-a$ )	3.9 Groups, rings and fields: a preview built from what you know
$H \le G$	$H$ is a subgroup of $G$	3.9 Groups, rings and fields: a preview built from what you know
$\lvert G\rvert$ , $\operatorname{ord}(g)$	the number of elements of $G$ , and the order of the element $g$	3.9 Groups, rings and fields: a preview built from what you know
$gH$	the left coset $\{gh : h \in H\}$ of the subgroup $H$	3.9 Groups, rings and fields: a preview built from what you know
$\ker\varphi$	the kernel of a homomorphism $\varphi$ : the elements sent to the identity (the null space of 2.3, in group language)	3.9 Groups, rings and fields: a preview built from what you know
$\mathrm{GL}_n$ , $\mathrm{SL}_n$ , $S_n$ , $A_n$	invertible $n\times n$ matrices; those of determinant $1$ ; all permutations of $n$ objects; the even permutations	3.9 Groups, rings and fields: a preview built from what you know
$d(x, y)$	the distance between the points $x$ and $y$ of a metric space	4.1 Metric and normed spaces; open and closed sets
$B(x, r)$	the open ball $\{y : d(x, y) < r\}$	4.1 Metric and normed spaces; open and closed sets
$\lVert x \rVert_1,\ \lVert x \rVert_2,\ \lVert x \rVert_\infty$	the 1-norm (sum of sizes), 2-norm (usual length) and max-norm (largest size) of a vector	4.1 Metric and normed spaces; open and closed sets
$\operatorname{int} A,\ \overline A,\ \partial A$	the interior, closure and boundary of the set $A$	4.1 Metric and normed spaces; open and closed sets
$x_n \to x$	the sequence $(x_n)$ converges to $x$ in a metric space: $d(x_n, x) \to 0$	4.2 Convergence, completeness and the contraction mapping theorem
$T x$ , $T^n x$	the map $T$ applied to $x$ , and applied $n$ times	4.2 Convergence, completeness and the contraction mapping theorem
$q$	a contraction constant: $d(Tx, Ty) \le q\,d(x, y)$ with $0 \le q < 1$	4.2 Convergence, completeness and the contraction mapping theorem
$x^*$	a fixed point: $T x^* = x^*$	4.2 Convergence, completeness and the contraction mapping theorem
$K$	a subset of a metric space, usually the one we call compact	4.3 Compactness
$\{U_\alpha\}$	a family (cover) of open sets, indexed by $\alpha$	4.3 Compactness
$\operatorname{diam} Q$	the diameter of a box $Q$ : the largest distance between two of its points	4.3 Compactness
$A = U \cup V$	a separation of $A$ when $U$ and $V$ are disjoint, non-empty and open in $A$	4.4 Connectedness and path-connectedness
$\gamma : [0, 1] \to X$	a path in $X$ : a continuous map from the interval $[0, 1]$ into $X$	4.4 Connectedness and path-connectedness
$f^{-1}(W)$	the preimage of $W$ : all points $x$ with $f(x) \in W$	4.5 Continuity between metric spaces
$d_X,\ d_Y$	the distances of the metric spaces $X$ and $Y$	4.5 Continuity between metric spaces
$\pi_i(x)$	the $i$ -th coordinate of $x$ : the coordinate projection	4.5 Continuity between metric spaces
$f: S \to \mathbb R$ , $F: S \to \mathbb R^m$	a scalar field and a vector field defined on a set $S \subseteq \mathbb R^n$	5.1 Scalar and vector fields, limits and continuity
$L(c)$	the level set $\{x : f(x) = c\}$ of a scalar field $f$	5.1 Scalar and vector fields, limits and continuity
$\lVert x - a \rVert \to 0$	the point $x$ approaches $a$ , measured by the usual length	5.1 Scalar and vector fields, limits and continuity
$f'(a; y)$	the derivative of the scalar field $f$ at $a$ with respect to the vector $y$ (Apostol); the directional derivative when $\lVert y \rVert = 1$	5.2 Directional derivatives, partial derivatives and the gradient
$D_k f(a)$ , $\dfrac{\partial f}{\partial x_k}(a)$ , $f_{x_k}(a)$	the partial derivative of $f$ in the direction of the $k$ -th coordinate axis	5.2 Directional derivatives, partial derivatives and the gradient
$\nabla f(a)$ , $\operatorname{grad} f(a)$	the gradient: the vector $(D_1 f(a), \dots, D_n f(a))$ ('del f')	5.2 Directional derivatives, partial derivatives and the gradient
$F'(a)$ , $DF(a)$ , $T_a$	the total derivative of $F$ at $a$ : a linear map $\mathbb R^n \to \mathbb R^m$	5.3 The total derivative, the Jacobian matrix and tangent planes
$J_F(a)$	its matrix, the $m \times n$ Jacobian matrix of partial derivatives	5.3 The total derivative, the Jacobian matrix and tangent planes
$\lVert v \rVert E(a, v)$	the error of the linear approximation, with $E(a, v) \to 0$ as $v \to 0$	5.3 The total derivative, the Jacobian matrix and tangent planes
$h = f \circ g$	the composite field $h(x) = f(g(x))$ : apply $g$ first, then $f$	5.4 The chain rule for fields
$Dh(a) = Df(g(a))\,Dg(a)$	the chain rule: the Jacobian of the composite is the Jacobian of $f$ at the image point times the Jacobian of $g$ at $a$	5.4 The chain rule for fields
$f_x,\ f_{xy}$	partial derivatives written as subscripts, read left to right: $f_{xy} = (f_x)_y$ means differentiate in $x$ first, then in $y$ . Apostol writes this $D_{2,1}f = D_2(D_1f)$ . Spagnolie's boards are not consistent about the order (his Final Review writes $h_{yx}$ for $(h_x)_y$ ), so this book fixes one reading; for the fields met here it never matters (5.5)	5.4 The chain rule for fields
$f_{xy}$ , $D_{2,1}f$ , $\dfrac{\partial^2f}{\partial y\,\partial x}$	differentiate in $x$ first, then in $y$ : $(f_x)_y$ . This book reads subscripts left to right, so it writes $f_{xy}$ for it; Apostol writes $D_{2,1}f = D_2(D_1f)$ , and Spagnolie's boards use both readings	5.5 Higher partial derivatives and the equality of mixed partials
$f_{yx}$ , $D_{1,2}f$ , $\dfrac{\partial^2f}{\partial x\,\partial y}$	differentiate in $y$ first, then in $x$ : $(f_y)_x$	5.5 Higher partial derivatives and the equality of mixed partials
$C^k$	all partial derivatives of order at most $k$ exist and are continuous	5.5 Higher partial derivatives and the equality of mixed partials
$H_f(a)$ , $\nabla^2f(a)$ , $[D_{ij}f(a)]$	the Hessian matrix of $f$ at $a$ : the symmetric $n \times n$ matrix of second partial derivatives. Its $(i, j)$ entry is written $D_{ij}f$ ; here it is $D_jD_if$ , the Jacobian of the gradient (Apostol and Lecture 24 write $D_i(D_jf)$ , the transpose, which is the same matrix when $f$ is $C^2$ )	5.6 Taylor's formula, the Hessian and extrema
$v^{\mathsf T}Hv$	the quadratic form of the Hessian at the step $v$ ; Lecture 24 writes it $vHv^{\mathsf T}$ for a row vector $v$	5.6 Taylor's formula, the Hessian and extrema
$E_2(a, v)$	the error of the second-order Taylor formula: $f(a + v) = f(a) + \nabla f(a)\cdot v + \frac12v^{\mathsf T}Hv + \lVert v \rVert^2E_2(a, v)$ with $E_2 \to 0$	5.6 Taylor's formula, the Hessian and extrema
$D_xF$ , $D_yF$	for $F(x, y)$ with $x \in \mathbb R^n$ , $y \in \mathbb R^m$ : the $m \times n$ and $m \times m$ blocks of the Jacobian matrix $J_F = [D_xF\ \ D_yF]$	5.7 Inverse and implicit functions
$G = F^{-1}$	a local inverse of $F$ , defined on a neighbourhood of $F(a)$	5.7 Inverse and implicit functions
$\dfrac{\partial(F, G)}{\partial(u, v)}$	the Jacobian determinant of $(F, G)$ with respect to $(u, v)$	5.7 Inverse and implicit functions
$L(x, \lambda) = f(x) - \sum_i\lambda_i\big(g_i(x) - c_i\big)$	the Lagrangian; its critical points in $(x, \lambda)$ are the candidates	5.8 Constrained extrema and Lagrange multipliers
$T_a = \ker Dg(a)$	the tangent space of the constraint set at $a$	5.8 Constrained extrema and Lagrange multipliers
$\lambda$ , $\lambda_i$	Lagrange multipliers: $\nabla f(a) = \sum_i\lambda_i\nabla g_i(a)$	5.8 Constrained extrema and Lagrange multipliers
$u_t$ , $u_{tt}$ , $u_{xx}$	partial derivatives of $u(x, t)$ : first in $t$ , second in $t$ , second in $x$	5.9 Partial differential equations
$\nabla^2u = u_{xx} + u_{yy}$	the Laplacian of $u$ (written $\Delta u$ in many books)	5.9 Partial differential equations
$\alpha$ , $c$	the diffusivity in the heat equation and the wave speed in the wave equation	5.9 Partial differential equations
$\eta$	the learning rate (step size) of gradient descent	5.10 Gradient descent, momentum and Newton's method
$L$ , $\mu$	the largest and smallest eigenvalues of the Hessian of a quadratic: $\mu\,I \preceq H \preceq L\,I$ ; $\kappa = L/\mu$ is the condition number	5.10 Gradient descent, momentum and Newton's method
$\beta$	the momentum coefficient: how much of the previous step is carried over	5.10 Gradient descent, momentum and Newton's method
$\int_C F \cdot dr$	the line integral of the vector field $F$ along the oriented curve $C$ : $\int_a^b F(r(t)) \cdot r'(t)\,dt$	6.1 Paths and line integrals
$\oint_C$	the same integral around a closed curve; the circle on the integral sign says $C$ starts and ends at the same point	6.1 Paths and line integrals
$\int_C P\,dx + Q\,dy$	the line integral of $F = (P, Q)$ , with $dx = x'(t)\,dt$ and $dy = y'(t)\,dt$	6.1 Paths and line integrals
$\int_C \varphi\,ds$	the line integral of a scalar field with respect to arc length: $\int_a^b \varphi(r(t))\,\lVert r'(t) \rVert\,dt$	6.1 Paths and line integrals
$F = \nabla\varphi$	$F$ is a gradient field and $\varphi$ is a potential for it	6.2 Gradient fields and potentials
$P_y = Q_x$	the necessary condition $\partial P / \partial y = \partial Q / \partial x$ for the field $F = (P, Q)$ to be a gradient	6.2 Gradient fields and potentials
$\varphi\big\|_A^B$	$\varphi(B) - \varphi(A)$ : the change of the potential, end minus start	6.2 Gradient fields and potentials
$\iint_Q f\,dA$	the double integral of $f$ over the region $Q$ ; $dA$ is the area element, $dx\,dy$ in rectangular coordinates	6.3 Multiple integrals
$\int_c^d\!\int_a^b f\,dx\,dy$	an iterated integral: integrate in $x$ first (inner), then in $y$	6.3 Multiple integrals
$\underline{I}(f),\ \overline{I}(f)$	the lower and upper integral: the sup of step integrals below $f$ and the inf of step integrals above $f$	6.3 Multiple integrals
$\oint_C P\,dx + Q\,dy = \iint_R (Q_x - P_y)\,dA$	Green's theorem; $C$ is the boundary of $R$ run counterclockwise (positively oriented)	6.4 Green's theorem and change of variables
$J = \dfrac{\partial(x, y)}{\partial(u, v)}$	the Jacobian determinant of the map $(u, v) \mapsto (x, y)$ : the old variables in terms of the new, then differentiate	6.4 Green's theorem and change of variables
$W(r; p)$	the winding number of the closed path $r$ about the point $p$ : how many times $r$ goes counterclockwise around $p$	6.4 Green's theorem and change of variables
$a \times b$	the cross product of two vectors in space: a vector perpendicular to both, whose length is the area of the parallelogram they span	6.5 Surfaces, area and flux
$r_u \times r_v$	the fundamental vector product of a parametrized surface $r(u, v)$ : a normal vector to the surface, of length equal to the area stretching factor	6.5 Surfaces, area and flux
$\iint_S f\,dS$	the surface integral of a scalar field: $\iint_T f(r(u, v))\,\lVert r_u \times r_v \rVert\,du\,dv$	6.5 Surfaces, area and flux
$\iint_S F \cdot n\,dS$	the flux of $F$ through the oriented surface $S$ with unit normal $n$ : $\iint_T F(r) \cdot (r_u \times r_v)\,du\,dv$ when $r_u \times r_v$ points the way $n$ does	6.5 Surfaces, area and flux
$\nabla = (\partial_x, \partial_y, \partial_z)$	the vector of derivative operators; $\nabla f$ is the gradient, $\nabla\times F$ the curl, $\nabla\cdot F$ the divergence	6.6 Curl, divergence, Stokes' theorem and Gauss's theorem
$\operatorname{curl}F = \nabla\times F$	$(R_y - Q_z,\ P_z - R_x,\ Q_x - P_y)$ for $F = (P, Q, R)$	6.6 Curl, divergence, Stokes' theorem and Gauss's theorem
$\operatorname{div}F = \nabla\cdot F$	$P_x + Q_y + R_z$ for $F = (P, Q, R)$	6.6 Curl, divergence, Stokes' theorem and Gauss's theorem
$\partial S$	the boundary curve of the surface $S$ (or, for a solid $E$ , its boundary surface $\partial E$ )	6.6 Curl, divergence, Stokes' theorem and Gauss's theorem
$dx$ , $dy$ , $dz$	the basic 1-forms: $dx$ takes a displacement vector $v$ and returns its $x$ component $v_1$	6.7 Differential forms and the one big theorem
$\omega\wedge\eta$	the wedge product of two forms: $dx\wedge dy = -dy\wedge dx$ and $dx\wedge dx = 0$	6.7 Differential forms and the one big theorem
$d\omega$	the exterior derivative of $\omega$ : it raises the degree by one. On functions $df$ is the gradient, on 1-forms it is the curl, on 2-forms it is the divergence	6.7 Differential forms and the one big theorem
$\varphi^*\omega$	the pullback of $\omega$ along a map $\varphi$ : substitute $x = x(u,v)$ and $dx = x_u\,du + x_v\,dv$	6.7 Differential forms and the one big theorem
$\int_{\partial M}\omega = \int_Md\omega$	the generalized Stokes theorem	6.7 Differential forms and the one big theorem
$E$ , $B$	the electric and magnetic field vectors; $E$ is force per unit charge, and $B$ is defined through the force $qv\times B$ on a moving charge	6.8 Maxwell's equations, conservation laws and heat
$\rho$ , $J$	the charge density (charge per unit volume) and current density (charge crossing unit area per unit time) in this chapter; do not confuse $\rho$ with the spherical radius of 6.4	6.8 Maxwell's equations, conservation laws and heat
$\varepsilon_0$ , $\mu_0$	the permittivity and permeability of free space: $\varepsilon_0\approx8.854\times10^{-12}$ F/m and $\mu_0 = 4\pi\times10^{-7}$ N/A $^2$ in the units used here	6.8 Maxwell's equations, conservation laws and heat
$\partial_tu + \nabla\cdot j = 0$	the continuity equation: the density $u$ of a conserved quantity changes only because of the flux $j$ through the boundary	6.8 Maxwell's equations, conservation laws and heat
$y'$ , $y^{(n)}$	the first and the $n$ -th derivative of $y$ with respect to $t$	7.1 Existence and uniqueness for differential equations
$L$	a Lipschitz constant of the right-hand side	7.1 Existence and uniqueness for differential equations
$y_0, y_1, y_2, \dots$	the Picard iterates, each one built from the one before	7.1 Existence and uniqueness for differential equations
$(Ty)(t)$	the Picard operator: $x_0 + \int_{t_0}^{t} f(s, y(s))\,ds$	7.1 Existence and uniqueness for differential equations
$D$	the differentiation operator, $Df = f'$	7.2 Linear differential equations
$L y = R$	a linear equation, with $L = D^n + a_{n-1}D^{n-1} + \dots + a_0$	7.2 Linear differential equations
$P_A(t)$	the characteristic polynomial of the constant-coefficient operator $A = P_A(D)$	7.2 Linear differential equations
$\ker L$	the null space of $L$ : all solutions of $Ly = 0$	7.2 Linear differential equations
$W(f_1, \dots, f_n)(t)$	the Wronskian: the determinant of the matrix whose rows are the functions and their derivatives up to order $n - 1$	7.3 The Wronskian, variation of parameters and undetermined coefficients
$u_k'$	the derivative of the unknown coefficient function in variation of parameters	7.3 The Wronskian, variation of parameters and undetermined coefficients
$y_p$	a particular solution of the inhomogeneous equation	7.3 The Wronskian, variation of parameters and undetermined coefficients
$e^{tA}$	the matrix exponential of $tA$ : $\sum_{k \ge 0} t^kA^k / k!$	7.4 Systems of equations and the matrix exponential
$\lVert A \rVert$	in this chapter, the entrywise sum norm $\sum_{i,j}\lvert a_{ij}\rvert$ used in MATH 376	7.4 Systems of equations and the matrix exponential
$N$	a nilpotent matrix: some power $N^k$ is the zero matrix	7.4 Systems of equations and the matrix exponential
$x' = Ax$	a linear system with constant coefficients; $x(t)$ is a vector function	7.4 Systems of equations and the matrix exponential
$\sum a_n (x - x_0)^n$	a power series solution about the point $x_0$ ; $a_0, a_1, \dots$ are the unknown coefficients	7.5 Power series solutions, Legendre and Bessel
$P_n(x)$	the Legendre polynomial of degree $n$ , normalised so that $P_n(1) = 1$	7.5 Power series solutions, Legendre and Bessel
$F(s)$	the indicial polynomial $s(s-1) + p_0 s + q_0$ of a regular singular point	7.5 Power series solutions, Legendre and Bessel
$J_0(x)$	the Bessel function of order zero, $\sum (-1)^k (x/2)^{2k}/(k!)^2$	7.5 Power series solutions, Legendre and Bessel
$x^*$	an equilibrium of $x' = F(x)$ : a point where $F(x^*) = 0$	7.6 Phase portraits, stability and linearization
$\tau,\ \Delta$	the trace and the determinant of a $2 \times 2$ matrix	7.6 Phase portraits, stability and linearization
$J = DF(x^*)$	the Jacobian matrix of $F$ at the equilibrium $x^*$	7.6 Phase portraits, stability and linearization
$\dot V$	the derivative of $V$ along solutions: $\dot V(x) = \nabla V(x) \cdot F(x)$	7.6 Phase portraits, stability and linearization
$\omega_j,\ v_j$	the natural frequency and the mode shape of the $j$ -th normal mode: $Mv_j = \omega_j^2v_j$	7.7 Applications to springs, circuits, epidemics and qubits
$\alpha,\ \omega_0,\ Q$	for an RLC circuit: the decay rate $R/2L$ , the natural frequency $1/\sqrt{LC}$ and the quality factor $\omega_0L/R$	7.7 Applications to springs, circuits, epidemics and qubits
$\beta,\ \gamma,\ R_0$	SIR model: infection rate, recovery rate and their ratio $\beta/\gamma$	7.7 Applications to springs, circuits, epidemics and qubits
$\psi(t),\ H,\ \hbar$	the state vector, the Hamiltonian matrix and the reduced Planck constant of a quantum system	7.7 Applications to springs, circuits, epidemics and qubits
$(\Omega, \mathcal F, P)$	a probability space: outcomes, events, probabilities	10.1 Probability as measure
$P(A \mid B)$	the probability of $A$ given $B$	10.1 Probability as measure
$F_X(x) = P(X \le x)$	the distribution function of $X$	10.1 Probability as measure
$E[X], \operatorname{Var}(X)$	the expectation and the variance of $X$	10.1 Probability as measure
$\bar X_n$	the average $(X_1 + \dots + X_n)/n$	10.1 Probability as measure
$p_n$	the interpolating polynomial of degree at most $n$	15.1 Numerical analysis
$\ell_j(x)$	the Lagrange basis polynomial that is $1$ at $x_j$ and $0$ at the other nodes	15.1 Numerical analysis
$f[x_0, \dots, x_k]$	the divided difference of $f$ at those nodes	15.1 Numerical analysis
$T_n$	the Chebyshev polynomial of degree $n$ , with $T_n(\cos\theta) = \cos n\theta$	15.1 Numerical analysis
$h$	the width of one subinterval, $h = (b - a)/n$	15.1 Numerical analysis