Appendix BSymbols, Greek Letters, and Terms
This appendix collects in one place the Greek letters, mathematical symbols, and technical terms used throughout this book—a “quick reference.” If you come across an unfamiliar symbol or word while reading the main text, look it up here first. Every symbol and term comes with a link to the chapter and section where it is first properly explained in the main text, so you can go back there and reread the definition and concrete examples.
B.1 List of Greek Letters
Greek letters are used in great numbers in university mathematics as the names of variables, constants, and operators. The same letter often changes its meaning depending on the context, so simply learning how each letter is read (its name) will considerably lighten the burden of following lectures and textbooks. Note that 13 of the uppercase letters happen to have the same shape as Latin (alphabet) letters; such letters are hardly ever used in mathematics as independent symbols (and this book contains no examples of their use either). For the letters marked “not used in this book,” too, we have listed at least the name, so that you will be able to read other textbooks and papers in the future.
| Uppercase | Lowercase | Name | Main use in this book |
|---|---|---|---|
| $A$ (same shape as the Latin letter A) | $α$ | alpha | The most basic letter, used for the limit value of a sequence or function, for an angle, for a general coefficient, and so on (Chapter 1: the limit value α in the ε-N argument). |
| $B$ (same shape as the Latin letter B) | $β$ | beta | A general symbol for angles, exponents, and the like (Chapter 2, Section 2.5: the addition formulas for trigonometric functions, $\sin(\alpha+\beta)$). |
| $\Gamma$ | $γ$ | gamma | The uppercase $\Gamma(x)$ is the gamma function (Chapter 32). The lowercase $\gamma$ is a general coefficient in linear relations (Chapter 10, Section 10.6). |
| $\Delta$ | $δ$ | delta | The uppercase $\Delta$ denotes a difference or an increment (Chapter 1: $\Delta y/\Delta x$). The lowercase $\delta$ is the tolerance in the ε-δ argument (Chapter 1, Section 1.2) and also the delta function $\delta(x)$ (Chapter 30). |
| $E$ (same shape as the Latin letter E) | ε ($\varepsilon$) | epsilon | Used in the ε-N and ε-δ arguments as “a positive number that can be taken as small as we like” (Chapter 1, Section 1.1). |
| $Z$ (same shape as the Latin letter Z) | $ζ$ | zeta | A general symbol used for an integration variable (Chapter 26: Cauchy's integral formula). |
| $H$ (same shape as the Latin letter H) | $η$ | eta | Not used as a specific symbol in this book (a letter often used for efficiency, the coefficient of viscosity, and so on). |
| $\Theta$ | $θ$ | theta | The most basic letter for denoting an angle (Chapter 2: Euler's formula $e^{i\theta}$). The uppercase $\Theta$ is also used as a symbol for the argument of the image point under a mapping (Chapter 22). |
| $I$ (same shape as the Latin letter I) | $ι$ | iota | Not used in this book (a letter that is rarely used in formulas because it is easily confused with the imaginary unit $i$). |
| $K$ (same shape as the Latin letter K) | $κ$ | kappa | Not used in this book (a letter often used for curvature, thermal diffusivity, and so on). |
| $\Lambda$ | $λ$ | lambda | One of the letters used most heavily in this book: it first appears as the Lagrange multiplier (Chapter 6, Section 6.7) and later denotes an eigenvalue of a matrix (Chapter 12, Section 12.1). |
| $M$ (same shape as the Latin letter M) | $μ$ | mu | A general coefficient that appears in the conditions for linearity and the like (Chapter 11). |
| $N$ (same shape as the Latin letter N) | $ν$ | nu | Used as an arbitrary constant in Lagrange's partial differential equation (Chapter 42, Section 42.3). |
| $\Xi$ | $ξ$ | xi (pronounced ‘ksee’) | The intermediate value that appears in the remainder term of Taylor's theorem (Chapter 4, Section 4.1). |
| $O$ (same shape as the Latin letter O) | $ο$ | omicron | Not used in this book (a letter that is rarely used in formulas because it is easily confused with the Landau symbol $O$). |
| $\Pi$ | $π$ | pi | The lowercase letter denotes the ratio of a circle's circumference to its diameter (Chapter 2, Section 2.1). The uppercase $\Pi$ is the product symbol (Chapter 12, Section 12.2). |
| $P$ (same shape as the Latin letter P) | $ρ$ | rho | A general symbol for density (Chapter 7: computing mass by multiple integrals). |
| $\Sigma$ | $σ$ | sigma | The uppercase $\Sigma$ is the summation symbol (Chapter 1). The lowercase $\sigma$ is used for permutations (Chapter 8, Section 8.2) and for the standard deviation in probability and statistics. |
| $T$ (same shape as the Latin letter T) | $τ$ | tau | A general symbol for time-related quantities such as the mean lifetime (Chapter 14, Section 14.6). |
| $\Upsilon$ | $υ$ | upsilon | Not used in this book (even among the Greek letters, one that is hardly ever used in modern mathematics). |
| $\Phi$ | φ ($\varphi$) | phi | The lowercase $\phi,\varphi$ are general symbols for angles and phases (Chapter 3, Section 3.4; Chapter 20, Section 20.4). The uppercase $\Phi$ denotes magnetic flux (Chapter 20, Section 20.2). |
| $X$ (same shape as the Latin letter X) | $χ$ | chi | Not used in this book (a letter often used in, for example, the $\chi^2$ distribution of statistics). |
| $\Psi$ | $ψ$ | psi | The lowercase $\psi$ appears as a general symbol in the parametric representation of a curve (Chapter 25, Section 25.3) and is also used for the wave function $\psi(x)$ of quantum mechanics (Chapter 45, Section 45.1). The uppercase $\Psi$ is the time-dependent wave function (Chapter 47, Section 47.4). |
| $\Omega$ | $ω$ | omega | The lowercase $\omega$ is a general symbol for angular frequency (Chapter 3, Section 3.4). The uppercase $\Omega$ is not used as a specific symbol in this book (a letter often used for the sample space in probability theory; in this book the whole event is written $U$). |
B.2 List of Symbols (by Field)
Here we have grouped by field the mathematical symbols used in this book, which we picked out by actually searching the body-text files ch*.html. The “First appears in” column gives a link to the chapter or section in which the symbol first appears in the body text (for a symbol that first appears in a passage that comes before the section numbers begin, such as a lead paragraph, the link points to the beginning of the chapter).
Logic and Sets
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\forall$ | for all | Universal quantifier. “For all …” | Chapter 1 |
| $\exists$ | there exists | Existential quantifier. “There exists …” | Chapter 1 |
| $\in$ | is an element of | “Is an element of …” | Chapter 1 |
| $\notin$ | is not an element of | “Is not an element of …” | Chapter 10, Section 10.4 |
| $\subset$ | is a subset of | Being a subset (whether the case of equality is included or excluded depends on the context) | Chapter 2, Section 2.1 |
| $\cap$ | cap / intersection | The set of elements that belong to both of two sets | Chapter 10 |
| $\cup$ | cup / union | The set of elements that belong to at least one of the two sets | Chapter 10, Section 10.5 |
| $\emptyset$ | empty set | A set that has no elements at all | Chapter 13 |
| $\bar A$ | A-bar / complement | The set of elements of the universal set that do not belong to $A$. In probability theory, the complementary event | Chapter 13 |
| $\setminus$ | set difference (“set minus”) | “The set obtained by removing $B$ from $A$,” $A\setminus B$ | Chapter 13, Section 13.1 |
| $\Rightarrow$ | implies | The logical implication “if P, then Q” | Chapter 1 |
| $\Leftrightarrow$ ($\iff$) | if and only if | “P and Q are equivalent.” A necessary and sufficient condition | Chapter 1, Section 1.1 |
Numbers and Functions
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\R$ | R / the set of all real numbers | The set consisting of all real numbers | Chapter 1, Section 1.1 |
| $\C$ | C / the set of all complex numbers | The set consisting of all complex numbers | Chapter 10, Section 10.2 |
| $\N$ | N / the set of all natural numbers | The set consisting of all natural numbers | Chapter 1 |
| $\Z$ | Z / the set of all integers | The set consisting of all integers | Chapter 23, Section 23.1 |
| $\circ$ | small circle / composition | “The composite map of $f$ and $g$,” $g\circ f$ | Chapter 11 |
| $\to$ | to (approaches … / a map to …) | The value approached in a limit, or the destination of a map | Chapter 1 |
| $\mapsto$ | maps to (sends … to …) | “The map that sends $x$ to $f(x)$,” $x\mapsto f(x)$ | Chapter 12, Section 12.2 |
Limits and Calculus
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\lim$ | lim / limit | The value obtained when a variable is made to approach its target arbitrarily closely | Chapter 1 |
| $\limsup$ | lim sup / limit superior (upper limit) | The smallest of the suprema that (a part of) a sequence can approach arbitrarily closely | Chapter 29, Section 29.1 |
| $\sup$ | sup / supremum | The smallest of the numbers that bound a set from above | Chapter 29, Section 29.1 |
| $\Sigma$ ($\sum$) | sigma / summation | The symbol for “numbering the terms and adding them up,” $\sum_{k}a_k$ | Chapter 1 |
| $\Pi$ ($\prod$) | pi / product | The symbol for “numbering the factors and multiplying them together,” $\prod_k a_k$ | Chapter 12, Section 12.2 |
| $\partial$ | round d / partial differentiation | The symbol for differentiating a multivariable function with respect to just one variable, $\partial f/\partial x$ | Chapter 6, Section 6.2 |
| $\nabla$ | nabla | An operator for writing partial derivatives together (common to the gradient, divergence, and curl) | Chapter 6 |
| $\Delta$ | delta / difference, increment | The symbol for an “amount of change,” $\Delta x,\Delta y$. It is also used to mean the Laplacian (Chapter 19) | Chapter 1 |
| $\infty$ | infinity | The symbol for the state of being arbitrarily large | Chapter 1 |
| $\int$ | integral / integration | The operation of dividing an area, a volume, and so on into small pieces and adding them up | Chapter 3, Section 3.5 |
| $\oint$ | closed-path integral | An integral along a closed path (curve or surface) | Chapter 19, Section 19.1 |
| $\iint$ | double integral | A multiple integral over a plane or surface region | Chapter 7 |
| $\iiint$ | triple integral | A multiple integral over a spatial region | Chapter 7 |
| $\|\cdot\|$ | norm | The symbol for the “size” of a vector or a function | Chapter 28 |
Linear Algebra
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\det$ | det / determinant | A single number determined by a square matrix. It represents the factor by which areas and volumes are scaled | Chapter 7 |
| ${}^{t}A$ | A-transpose / transpose | The matrix with its rows and columns interchanged | Chapter 8 |
| $\sgn$ | sign / signature of a permutation | $+1$ if the permutation is even, $-1$ if it is odd | Chapter 8 |
| $\rank$ | rank | The essential “number of independent rows (columns)” of a matrix | Chapter 9 |
| $\dim$ | dim / dimension | The number of vectors in a basis of a vector space | Chapter 10 |
| $\oplus$ | direct sum | The sum of two subspaces whose intersection consists of the zero vector alone | Chapter 10 |
| $\mathrm{Im}\,f$ | im / image | The set of all values that the linear map $f$ actually takes | Chapter 11 |
| $\mathrm{Ker}\,f$ | ker / kernel | The set of all elements that the linear map $f$ sends to the zero vector | Chapter 11 |
| $\langle f,g\rangle$ | inner product | An operation that produces a single number from two vectors (functions) | Chapter 5, Section 5.2 |
Probability
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $P(A)$ | probability | A number, at least $0$ and at most $1$, expressing how likely the event $A$ is to occur | Chapter 13 |
| $E(X)$ | expected value | The average value of the random variable $X$ | Chapter 14, Section 14.1 |
| $V(X)$ | variance | A quantity expressing how much the random variable $X$ scatters around its expected value | Chapter 14, Section 14.6 |
| $\langle v\rangle$ | ensemble average | A symbol used in statistical mechanics to denote an expectation value (in place of $E$) | Chapter 14, Section 14.6 |
Vector Analysis
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\bm{a}$ | boldface vector symbol | Boldface notation indicating that the quantity is a vector | Chapter 7, Section 7.7 |
| $\times$ | cross product | An operation that produces, from two vectors, a new vector perpendicular to both | Chapter 15 |
| $\operatorname{grad}f$ | grad / gradient | The vector field giving the direction in which the scalar field $f$ increases most steeply, together with the magnitude of that increase. Also written $\nabla f$ | Chapter 6, Section 6.6 |
| $\operatorname{div}\vec f$ | div / divergence | The scalar field expressing the strength of the “source” of a vector field. Also written $\nabla\cdot\vec f$ | Chapter 19 |
| $\rot\vec f$ | rot / curl | The vector field expressing the strength and the axis of the “vortex” of a vector field. Also written $\nabla\times\vec f$ | Chapter 15 |
Complex Analysis
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\operatorname{Re}z$ | real part | The real part $a$ of the complex number $z=a+bi$ | Chapter 21, Section 21.1 |
| $\operatorname{Im}z$ | imaginary part | The real part $b$ of the complex number $z=a+bi$ | Chapter 21, Section 21.1 |
| $\bar z$ | z-bar / complex conjugate | The complex number obtained by reversing only the sign of the imaginary part | Chapter 21 |
| $|z|$ | absolute value (modulus) | The distance of the complex number $z$ from the origin | Chapter 21 |
| $\arg z$ | argument | The angle that $z$ makes with the real axis in the complex plane | Chapter 21 |
| $\operatorname{Log}z$ | Log / principal value of the (complex) logarithm | The function obtained by taking only the principal value of the multivalued complex logarithm | Chapter 23, Section 23.3 |
Fourier and Laplace
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $\mathcal{L}[f(t)]$ | Laplace transform | An integral transform that converts a function of $t$ into a function of $s$. $\mathcal{L}$ is the script $L$ | Chapter 31, Section 31.1 |
| $\mathcal{F}[f(x)]$ | Fourier transform | An integral transform that converts a function of $x$ into a function of the frequency $k$. $\mathcal{F}$ is the script $F$ | Chapter 30 |
| $\delta(x)$ | delta function (Dirac delta) | A “distribution” (generalized function) concentrated infinitely at the origin alone. It has the sifting property | Chapter 30 |
Differential Equations and Special Functions
| Symbol | How it is read | Meaning | First appears in |
|---|---|---|---|
| $D$ | differential operator | The symbol that stands for the operation of “differentiating with respect to $x$” itself. $Dy=dy/dx$ | Chapter 38, Section 38.1 |
| $W(x)$ | Wronskian | The determinant that tests whether two solutions are linearly independent | Chapter 35, Section 35.2 |
| $\Gamma(x)$ | gamma function | A function that extends the factorial $n!$ to real and complex numbers | Chapter 32 |
| $B(m,n)$ | beta function | A special function of two variables that is related to the gamma function | Chapter 32 |
| $\mathrm{erf}(x)$ | error function | A special function, built by integrating the Gaussian function, that is closely related to the normal distribution | Chapter 7, Section 7.5 |
| $J_n(x)$ | Bessel function of the first kind | A special function that appears in Laplace's equation in cylindrical coordinates | Chapter 41 |
| $P_n(x)$ | Legendre polynomial | An orthogonal polynomial that appears in Laplace's equation in spherical coordinates | Chapter 40 |
| $L_m(x)$ | Laguerre polynomial | An orthogonal polynomial that appears in the radial wave function of the hydrogen atom | Chapter 41 |
| $H_n(u)$ | Hermite polynomial | An orthogonal polynomial that appears in the wave functions of the harmonic oscillator | Chapter 46 |
B.3 Index of Terms (Alphabetical Order)
This is an index of the principal technical terms that are introduced as “definitions” in the body text of this book. In this English edition the terms are arranged in alphabetical order of their English names (the Japanese original lists them in the order of their Japanese readings, together with the English names given in the body text). Each link leads to the chapter and section in which the term is first defined.
A
| Term | Defined in |
|---|---|
| absolute value (magnitude) of a complex number | Chapter 21, Section 21.1 |
| adjugate matrix | Chapter 9, Section 9.1 |
| algebraic multiplicity and geometric multiplicity | Chapter 12, Section 12.2 |
| argument and polar form of a complex number | Chapter 21, Section 21.2 |
| associated Laguerre polynomial | Chapter 41, Section 41.5 |
| augmented matrix (nonhomogeneous case) | Chapter 9, Section 9.4 |
B
| Term | Defined in |
|---|---|
| basis of a vector space | Chapter 10, Section 10.3 |
| basis vectors and the position vector in rectangular coordinates | Chapter 16, Section 16.1 |
| Bernoulli's differential equation | Chapter 33, Section 33.5 |
| Bessel function of the first kind | Chapter 41, Section 41.3 |
| Bessel function of the second kind | Chapter 41, Section 41.3 |
| beta function | Chapter 32, Section 32.4 |
| bijective | Chapter 11, Section 11.1 |
| Bohr radius | Chapter 48, Section 48.3 |
| Boltzmann constant | Chapter 49, Section 49.1 |
| Boltzmann distribution | Chapter 49, Section 49.1 |
C
| Term | Defined in |
|---|---|
| cofactor | Chapter 8, Section 8.4 |
| complement | Chapter 13, Section 13.1 |
| complementary error function | Chapter 32, Section 32.1 |
| complementary event | Chapter 13, Section 13.5 |
| complex exponential function | Chapter 23, Section 23.1 |
| complex function | Chapter 22, Section 22.1 |
| complex integral (definition as a line integral) | Chapter 25, Section 25.1 |
| complex logarithm | Chapter 23, Section 23.3 |
| complex numbers and complex conjugates (review) | Chapter 21, Section 21.1 |
| complex polynomial function of degree $n$ | Chapter 22, Section 22.5 |
| complex power | Chapter 23, Section 23.4 |
| complex trigonometric functions | Chapter 23, Section 23.6 |
| continuity of a function (by the $\varepsilon$-$\delta$ argument) | Chapter 2, Section 2.2 |
| continuity of complex functions | Chapter 24, Section 24.2 |
| convergence of a sequence (the $\varepsilon$-$N$ argument) | Chapter 1, Section 1.1 |
| Coulomb potential of the hydrogen atom | Chapter 48, Section 48.2 |
| cross product (vector product) | Chapter 15, Section 15.3 |
| curl (also called rotation) | Chapter 19, Section 19.4 |
| cylindrical coordinates | Chapter 44, Section 44.3 |
| cylindrical coordinates | Chapter 16, Section 16.2 |
D
| Term | Defined in |
|---|---|
| defining formula of the derivative | Chapter 1, Section 1.6 |
| definite integral (defined by Riemann sums) | Chapter 5, Section 5.3 |
| definite integral of a vector-valued function | Chapter 17, Section 17.3 |
| degeneracy | Chapter 45, Section 45.5 |
| degree of degeneracy | Chapter 46, Section 46.4 |
| degrees of freedom | Chapter 9, Section 9.3 |
| delta function (distribution) | Chapter 30, Section 30.3 |
| determinant | Chapter 8, Section 8.3 |
| diagonalizability | Chapter 12, Section 12.3 |
| differential equation | Chapter 33, Section 33.1 |
| differential operator $D$ | Chapter 38, Section 38.1 |
| differentiation | Chapter 1, Section 1.5 |
| differentiation of vector-valued functions | Chapter 17, Section 17.1 |
| dimension of a vector space | Chapter 10, Section 10.3 |
| dimensionless variable $u$ and energy variable $\varepsilon$ | Chapter 46, Section 46.1 |
| divergence of a plane vector field | Chapter 19, Section 19.1 |
| domain (open domain) | Chapter 24, Section 24.1 |
E
| Term | Defined in |
|---|---|
| eigenspace | Chapter 12, Section 12.2 |
| eigenvalue equation of an operator | Chapter 45, Section 45.1 |
| eigenvectors and eigenvalues | Chapter 12, Section 12.1 |
| elementary column operation | Chapter 9, Section 9.6 |
| elementary event | Chapter 13, Section 13.3 |
| empty event | Chapter 13, Section 13.3 |
| empty set | Chapter 13, Section 13.1 |
| equation of a plane (via the normal vector) | Chapter 18, Section 18.7 |
| error function | Chapter 32, Section 32.1 |
| Euler's differential equation (Cauchy–Euler equation) | Chapter 36, Section 36.3 |
| even and odd permutations | Chapter 8, Section 8.2 |
| event | Chapter 13, Section 13.3 |
| exact differential equation | Chapter 34, Section 34.2 |
F
| Term | Defined in |
|---|---|
| Fourier transform and inverse Fourier transform | Chapter 30, Section 30.1 |
| function of exponential order | Chapter 31, Section 31.4 |
| fundamental system of solutions | Chapter 37, Section 37.2 |
G
| Term | Defined in |
|---|---|
| gamma function | Chapter 32, Section 32.2 |
| generalized Lagrange partial differential equation | Chapter 42, Section 42.3 |
| gradient vector | Chapter 18, Section 18.4 |
H
| Term | Defined in |
|---|---|
| Hamiltonian operator | Chapter 45, Section 45.1 |
| Hermite's differential equation | Chapter 46, Section 46.1 |
| higher-order exact differential equation (linear exact differential equation) | Chapter 37, Section 37.1 |
| holomorphy and the Cauchy–Riemann equations | Chapter 24, Section 24.2 |
| homogeneous and nonhomogeneous linear differential equations | Chapter 33, Section 33.1 |
| homogeneous-type differential equation | Chapter 33, Section 33.2 |
| hyperbolic functions | Chapter 2, Section 2.5 |
I
| Term | Defined in |
|---|---|
| image ($\mathrm{Im}\,f$) | Chapter 11, Section 11.3 |
| improper integral | Chapter 5, Section 5.4 |
| improper integral over an infinite interval | Chapter 5, Section 5.4 |
| indefinite integral of a vector-valued function | Chapter 17, Section 17.3 |
| independent | Chapter 13, Section 13.5 |
| infinite cylinder potential (three-dimensional stationary state) | Chapter 47, Section 47.4 |
| infinite cylindrical well potential (two-dimensional stationary state) | Chapter 47, Section 47.1 |
| infinite spherical well potential | Chapter 48, Section 48.1 |
| infinite square well potential | Chapter 45, Section 45.3 |
| injective | Chapter 11, Section 11.1 |
| inner product (also called the scalar product) | Chapter 15, Section 15.2 |
| inner product of functions | Chapter 27, Section 27.2 |
| intersection | Chapter 13, Section 13.1 |
| intersection, sum, and direct sum | Chapter 10, Section 10.5 |
| inverse operator $\dfrac{1}{\Phi(D)}$ | Chapter 38, Section 38.3 |
| inverse permutation | Chapter 8, Section 8.2 |
| inverse trigonometric functions $\sin^{-1}x,\cos^{-1}x,\tan^{-1}x$ (principal values) | Chapter 3, Section 3.6 |
J
| Term | Defined in |
|---|---|
| Jacobian | Chapter 7, Section 7.3 |
K
| Term | Defined in |
|---|---|
| kernel ($\mathrm{Ker}\,f$) | Chapter 11, Section 11.3 |
L
| Term | Defined in |
|---|---|
| Lagrange's partial differential equation | Chapter 42, Section 42.2 |
| Laguerre polynomial | Chapter 41, Section 41.4 |
| Laplace transform | Chapter 31, Section 31.1 |
| Laplacian | Chapter 19, Section 19.3 |
| Legendre polynomial $P_n(x)$ | Chapter 40, Section 40.4 |
| Legendre's differential equation | Chapter 40, Section 40.1 |
| length of a curve | Chapter 17, Section 17.2 |
| level curve | Chapter 18, Section 18.2 |
| level surface | Chapter 18, Section 18.2 |
| limit of a function ($\varepsilon$-$\delta$ argument) | Chapter 1, Section 1.2 |
| limit of a function of two variables | Chapter 6, Section 6.1 |
| line integral of a planar scalar field | Chapter 20, Section 20.1 |
| linear and nonlinear differential equations | Chapter 33, Section 33.1 |
| linear combination | Chapter 10, Section 10.1 |
| linear fractional function (Möbius transformation) | Chapter 22, Section 22.5 |
| linear independence and linear dependence | Chapter 10, Section 10.2 |
| linear map | Chapter 11, Section 11.2 |
| linear relation (relation of linear dependence) | Chapter 10, Section 10.1 |
| linear transformation | Chapter 11, Section 11.2 |
| local maxima and minima of a function of two variables | Chapter 6, Section 6.6 |
M
| Term | Defined in |
|---|---|
| Maclaurin expansion | Chapter 4, Section 4.2 |
| many-to-one complex function | Chapter 22, Section 22.2 |
| matrix | Chapter 8, Section 8.1 |
| matrix product | Chapter 8, Section 8.1 |
| method of separation of variables | Chapter 43, Section 43.4 |
| momentum operator and energy operator | Chapter 45, Section 45.1 |
| Morse potential | Chapter 46, Section 46.5 |
| multiple integral (double integral) | Chapter 7, Section 7.1 |
| mutually exclusive | Chapter 13, Section 13.5 |
N
| Term | Defined in |
|---|---|
| Napier's constant $e$ | Chapter 1, Section 1.5 |
| neighborhood ($r$-neighborhood) | Chapter 24, Section 24.1 |
| norm of a function | Chapter 27, Section 27.2 |
| normal form (for systems of linear equations) | Chapter 9, Section 9.6 |
| normal form (of a differential equation) | Chapter 36, Section 36.6 |
| $n$th roots of $1$ | Chapter 21, Section 21.5 |
O
| Term | Defined in |
|---|---|
| one-sided limits (left-hand limit and right-hand limit) | Chapter 27, Section 27.4 |
| one-to-one complex function (injective) | Chapter 22, Section 22.2 |
| ordinary and partial differential equations | Chapter 33, Section 33.1 |
| ordinary point | Chapter 39, Section 39.3 |
P
| Term | Defined in |
|---|---|
| partial derivative | Chapter 6, Section 6.2 |
| partial differentiation of vector-valued functions | Chapter 17, Section 17.5 |
| permutation | Chapter 8, Section 8.2 |
| piecewise continuous function | Chapter 27, Section 27.4 |
| piecewise smooth function | Chapter 27, Section 27.4 |
| point at infinity | Chapter 22, Section 22.4 |
| pointwise convergence | Chapter 29, Section 29.3 |
| polar coordinates | Chapter 44, Section 44.2 |
| principal value integral (Cauchy principal value) | Chapter 30, Section 30.6 |
| probability | Chapter 13, Section 13.4 |
| probability density function | Chapter 14, Section 14.2 |
| product event | Chapter 13, Section 13.4 |
R
| Term | Defined in |
|---|---|
| radius of convergence and d'Alembert's ratio test | Chapter 39, Section 39.2 |
| random variables and expected value (in the discrete case) | Chapter 14, Section 14.1 |
| rectangular (Cartesian) coordinates | Chapter 16, Section 16.1 |
| representation matrix | Chapter 11, Section 11.4 |
| Riccati's differential equation | Chapter 33, Section 33.5 |
| Riemann sphere | Chapter 22, Section 22.4 |
| row echelon form and rank | Chapter 9, Section 9.2 |
S
| Term | Defined in |
|---|---|
| sample space | Chapter 13, Section 13.3 |
| scalar field | Chapter 18, Section 18.1 |
| second-order linear differential equations, and homogeneous and nonhomogeneous equations | Chapter 35, Section 35.1 |
| set | Chapter 13, Section 13.1 |
| sifting property of the delta function | Chapter 31, Section 31.5 |
| sign (signature) of a permutation $\sgn$ | Chapter 8, Section 8.3 |
| simply connected and multiply connected domains | Chapter 25, Section 25.4 |
| singular points and regular singular points | Chapter 39, Section 39.3 |
| solution space of a homogeneous system of linear equations | Chapter 10, Section 10.5 |
| source-free field (solenoidal field) | Chapter 19, Section 19.2 |
| spanned subspace and spanning set | Chapter 10, Section 10.5 |
| spherical coordinates | Chapter 7, Section 7.7 |
| spherical coordinates | Chapter 16, Section 16.3 |
| standard form of a first-order linear differential equation | Chapter 33, Section 33.3 |
| streamline | Chapter 18, Section 18.3 |
| subspace | Chapter 10, Section 10.4 |
| sum and scalar multiple of matrices | Chapter 8, Section 8.1 |
| sum event | Chapter 13, Section 13.4 |
| surface integral of a scalar field over a parametrized surface | Chapter 20, Section 20.4 |
| surjective | Chapter 11, Section 11.1 |
T
| Term | Defined in |
|---|---|
| tangential line integral of a planar vector field | Chapter 20, Section 20.2 |
| Taylor expansion | Chapter 4, Section 4.2 |
| total differential equation | Chapter 42, Section 42.4 |
| total differential of a vector-valued function | Chapter 17, Section 17.5 |
| totally differentiable | Chapter 6, Section 6.3 |
| transpose (matrix) | Chapter 8, Section 8.1 |
| trial | Chapter 13, Section 13.3 |
U
| Term | Defined in |
|---|---|
| uniform convergence | Chapter 29, Section 29.3 |
| union | Chapter 13, Section 13.1 |
| unit step function | Chapter 31, Section 31.5 |
| unit vector | Chapter 15, Section 15.1 |
V
| Term | Defined in |
|---|---|
| vector | Chapter 15, Section 15.1 |
| vector field | Chapter 18, Section 18.1 |
| vector space (linear space) | Chapter 10, Section 10.1 |
| vector surface element | Chapter 20, Section 20.5 |
W
| Term | Defined in |
|---|---|
| wave function of the hydrogen atom (in the form containing spherical harmonics) | Chapter 48, Section 48.4 |
| Wronskian | Chapter 35, Section 35.2 |
Z
| Term | Defined in |
|---|---|
| zero $\lambda_m(l)$ of the Bessel function | Chapter 47, Section 47.2 |
| zero vector | Chapter 15, Section 15.1 |