Appendix F — Notation Reference
Mathematics is a written language, and like any language it has spelling and grammar. This appendix is the dictionary. Every symbol used in this book appears here, grouped by the area of calculus where you first meet it, with a plain-English meaning and an example or note in the third column. The notation here matches the conventions locked in for the whole textbook, so a symbol means the same thing in Chapter 2 as it does in Chapter 39.
A few global conventions apply everywhere and are worth stating once. Variables are set in italic ($x$, $y$, $t$), while named functions are set in roman type ($\sin$, $\cos$, $\ln$, $\exp$) so that $\sin x$ is never confused with a product $s \cdot i \cdot n \cdot x$. Vectors are bold lowercase ($\mathbf{v}$, $\mathbf{r}$, $\mathbf{F}$). Inside an integral, the differential element always carries a thin space — written \,dx in LaTeX — so we print $\int f(x)\,dx$, never $\int f(x)dx$. And we always name the variable of integration: $\int f(x)\,dx$, never the ambiguous $\int f$.
F.1 Sets and Logic
| Symbol |
Meaning |
Example / Notes |
| $\mathbb{R}$ |
The real numbers |
The default domain for most functions in this book |
| $\mathbb{R}^n$ |
$n$-dimensional real space |
$\mathbb{R}^2$ is the plane; $\mathbb{R}^3$ is space; a point is an $n$-tuple |
| $\mathbb{N}$ |
The natural numbers |
$\{1, 2, 3, \dots\}$ (this book starts at $1$); used for series indices |
| $\mathbb{Z}$ |
The integers |
$\{\dots, -2, -1, 0, 1, 2, \dots\}$ |
| $\mathbb{Q}$ |
The rational numbers |
Ratios $p/q$ with $p, q \in \mathbb{Z}$, $q \neq 0$ |
| $\mathbb{C}$ |
The complex numbers |
Numbers $a + bi$; appears with Euler's formula in Ch. 24 |
| $\in$ |
Is an element of |
$3 \in \mathbb{N}$; "$x \in [a,b]$" reads "$x$ is in the interval" |
| $\notin$ |
Is not an element of |
$\tfrac{1}{2} \notin \mathbb{Z}$ |
| $\subset$ |
Is a subset of |
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ |
| $\cup$ |
Union |
$A \cup B$ — everything in $A$ or $B$ (or both) |
| $\cap$ |
Intersection |
$A \cap B$ — everything in both $A$ and $B$ |
| $\varnothing$ |
The empty set |
The set with no elements; $A \cap B = \varnothing$ means disjoint |
| $\forall$ |
For all |
"$\forall \varepsilon > 0$" reads "for every positive $\varepsilon$" |
| $\exists$ |
There exists |
"$\exists \delta > 0$" reads "there is a positive $\delta$" |
| $\implies$ |
Implies |
"$x > 2 \implies x > 1$" — if the left holds, so does the right |
| $\iff$ |
If and only if |
Logical equivalence; both directions of implication hold |
| $[a, b]$ |
Closed interval |
All $x$ with $a \le x \le b$; endpoints included |
| $(a, b)$ |
Open interval |
All $x$ with $a < x < b$; endpoints excluded |
Half-open intervals $[a, b)$ and $(a, b]$ combine the two; $(-\infty, b]$ and $[a, \infty)$ use $\infty$ as a placeholder, never as an endpoint to be reached.
F.2 Functions
| Symbol |
Meaning |
Example / Notes |
| $f, g, h$ |
Functions (italic lowercase) |
The standard names for functions in this book |
| $f(x)$ |
The function $f$ evaluated at $x$ |
A single output number, not the function itself |
| $f : A \to B$ |
A function from set $A$ to set $B$ |
$A$ is the domain, $B$ the codomain |
| domain |
The set of allowed inputs |
For $\ln x$, the domain is $(0, \infty)$ |
| range |
The set of achieved outputs |
For $\sin x$, the range is $[-1, 1]$ |
| $f \circ g$ |
Composition: $(f \circ g)(x) = f(g(x))$ |
Apply $g$ first, then $f$; key to the chain rule (Ch. 7) |
| $f^{-1}$ |
The inverse function |
$f^{-1}(f(x)) = x$; not the reciprocal $1/f$ |
| $\lvert x \rvert$ |
Absolute value |
$\lvert{-3}\rvert = 3$; distance from $0$ on the number line |
| $\lfloor x \rfloor$ |
Floor (greatest integer $\le x$) |
$\lfloor 2.7 \rfloor = 2$, $\lfloor -1.2 \rfloor = -2$ |
| $\lceil x \rceil$ |
Ceiling (least integer $\ge x$) |
$\lceil 2.1 \rceil = 3$ |
| $\sin, \cos, \tan$ |
Trigonometric functions (roman) |
Written $\sin x$, not $sin\,x$ |
| $\ln, \exp$ |
Natural log and exponential |
$\ln x = \log_e x$; $\exp x = e^x$ |
F.3 Limits and Continuity
| Symbol |
Meaning |
Example / Notes |
| $\displaystyle\lim_{x \to a} f(x)$ |
Two-sided limit as $x \to a$ |
Both one-sided limits must agree for this to exist |
| $\displaystyle\lim_{x \to a^+} f(x)$ |
Right-hand limit |
Approach $a$ from values greater than $a$ |
| $\displaystyle\lim_{x \to a^-} f(x)$ |
Left-hand limit |
Approach $a$ from values less than $a$ |
| $\displaystyle\lim_{x \to \infty} f(x)$ |
Limit at infinity |
End behavior; describes horizontal asymptotes |
| $\to$ |
Tends to / approaches |
"$x \to a$"; also "$a_n \to L$" for sequences |
| $\infty$ |
Infinity |
A direction/size, not a real number; never write $\infty \in \mathbb{R}$ |
| $\varepsilon$ |
A small positive target tolerance |
The "$\varepsilon$" in the $\varepsilon$–$\delta$ definition of a limit |
| $\delta$ |
A small positive input tolerance |
Chosen in response to $\varepsilon$: $0 < \lvert x - a\rvert < \delta \implies \lvert f(x) - L\rvert < \varepsilon$ |
A function is continuous at $a$ when $\lim_{x \to a} f(x) = f(a)$ — the limit exists, the value exists, and they match.
F.4 Derivatives
| Symbol |
Meaning |
Example / Notes |
| $f'(x)$ |
First derivative (Lagrange) |
Read "$f$ prime of $x$"; the instantaneous rate of change |
| $f''(x)$ |
Second derivative (Lagrange) |
Rate of change of the rate of change; concavity |
| $f^{(n)}(x)$ |
$n$-th derivative |
The parentheses prevent confusion with a power $f^n$ |
| $\dfrac{df}{dx}$ |
Derivative (Leibniz) |
Emphasizes "change in $f$ per change in $x$" |
| $\dfrac{d^2 f}{dx^2}$ |
Second derivative (Leibniz) |
The $2$ on top is a count of operations, not a square |
| $Df$ |
Derivative (operator) |
Thinks of $D$ as a machine that turns $f$ into $f'$ |
| $\dot{x}$ |
Time derivative (Newton dot) |
$\dot{x} = dx/dt$; standard in physics and ODEs |
| $\dfrac{\partial f}{\partial x}$ |
Partial derivative |
Differentiate in $x$, holding other variables fixed (Ch. 27+) |
| $f_x, f_y$ |
Partial derivatives (subscript form) |
$f_x = \partial f/\partial x$; compact for second partials $f_{xy}$ |
| $\nabla f$ |
Gradient |
The vector $\langle f_x, f_y, f_z\rangle$ of all first partials |
| $\nabla \cdot \mathbf{F}$ |
Divergence |
A scalar measuring outflow of a vector field |
| $\nabla \times \mathbf{F}$ |
Curl |
A vector measuring rotation of a vector field |
| $D_{\mathbf{u}} f$ |
Directional derivative |
Rate of change of $f$ in the direction of unit vector $\mathbf{u}$; $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$ |
| $Jf$ or $D\mathbf{f}$ |
Jacobian matrix |
Matrix of all first partials of a vector-valued $\mathbf{f}$ |
| $Hf$ |
Hessian matrix |
Matrix of all second partials $f_{ij}$; tests for max/min in $\mathbb{R}^n$ |
| $\Delta x, \Delta y$ |
Finite differences |
An actual nonzero change; $f'(x) \approx \Delta y / \Delta x$ |
| $dx, dy$ |
Differentials |
Infinitesimal changes; the building blocks of integrals |
F.5 Integrals
| Symbol |
Meaning |
Example / Notes |
| $\int f(x)\,dx$ |
Indefinite integral (antiderivative) |
Result includes "$+ C$"; note the thin space before $dx$ |
| $\displaystyle\int_a^b f(x)\,dx$ |
Definite integral from $a$ to $b$ |
A number: signed area under $f$ on $[a,b]$ |
| $\displaystyle\oint$ |
Integral around a closed curve |
Used in circulation and Green's/Stokes' theorems |
| $\displaystyle\iint_R f\,dA$ |
Double integral over region $R$ |
$dA$ is the area element ($dx\,dy$ or $r\,dr\,d\theta$) |
| $\displaystyle\iiint_V f\,dV$ |
Triple integral over solid $V$ |
$dV$ is the volume element ($dx\,dy\,dz$, etc.) |
| $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r}$ |
Line integral of a vector field |
Work done by $\mathbf{F}$ along curve $C$ |
| $\displaystyle\iint_S \mathbf{F}\cdot d\mathbf{S}$ |
Surface (flux) integral |
Flow of $\mathbf{F}$ through surface $S$ |
| $\displaystyle\oiint$ |
Integral over a closed surface |
Total flux out of a closed surface (Divergence Theorem) |
| $dA$ |
Area element |
Infinitesimal patch of area in a double integral |
| $dV$ |
Volume element |
Infinitesimal chunk of volume in a triple integral |
| $dS$ |
Surface-area element |
Scalar patch of area on a surface |
| $ds$ |
Arc-length element |
$ds = \lvert d\mathbf{r}\rvert$; used in scalar line integrals |
| $d\mathbf{r}$ |
Vector line element |
Tangent displacement along a curve, $d\mathbf{r} = \mathbf{r}'(t)\,dt$ |
F.6 Sequences and Series
| Symbol |
Meaning |
Example / Notes |
| $\{a_n\}$ |
A sequence |
The ordered list $a_1, a_2, a_3, \dots$ |
| $\displaystyle\sum_{i=1}^{n} a_i$ |
Finite sum |
Add $a_1$ through $a_n$ |
| $\displaystyle\sum_{n=1}^{\infty} a_n$ |
Infinite series |
The limit of the partial sums, if it exists |
| $s_n$ |
$n$-th partial sum |
$s_n = \sum_{i=1}^{n} a_i$; the series converges iff $\{s_n\}$ converges |
| $\displaystyle\prod_{i=1}^{n} a_i$ |
Finite product |
Multiply $a_1$ through $a_n$ |
| $n!$ |
Factorial |
$n! = n(n-1)\cdots 2 \cdot 1$; $0! = 1$; appears in Taylor series |
| $\dbinom{n}{k}$ |
Binomial coefficient |
$\dfrac{n!}{k!\,(n-k)!}$; "$n$ choose $k$" |
F.7 Vectors
| Symbol |
Meaning |
Example / Notes |
| $\mathbf{v}, \mathbf{r}, \mathbf{F}$ |
Vectors (bold lowercase) |
$\mathbf{r}$ is position, $\mathbf{F}$ is force or a vector field |
| $\langle a, b, c\rangle$ |
Vector in component form |
Angle brackets distinguish a vector from a point $(a,b,c)$ |
| $\lVert \mathbf{v}\rVert$ |
Norm (magnitude/length) |
$\lVert\langle 3,4\rangle\rVert = 5$; uses double bars |
| $\mathbf{u} \cdot \mathbf{v}$ |
Dot product |
A scalar; $\mathbf{u}\cdot\mathbf{v} = \lVert\mathbf{u}\rVert\lVert\mathbf{v}\rVert\cos\theta$ |
| $\mathbf{u} \times \mathbf{v}$ |
Cross product |
A vector perpendicular to both (only in $\mathbb{R}^3$) |
| $\mathbf{i}, \mathbf{j}, \mathbf{k}$ |
Standard basis vectors |
Unit vectors along the $x$-, $y$-, $z$-axes |
F.8 Greek Letters in Calculus
Greek letters are not decoration — each tends to carry a job. Here are the ones this book uses, with their typical calculus meanings.
| Symbol |
Name |
Typical Meaning / Notes |
| $\varepsilon$ |
epsilon |
Small positive output tolerance in limit proofs |
| $\delta$ |
delta (lower) |
Small positive input tolerance; partner of $\varepsilon$ |
| $\Delta$ |
Delta (upper) |
A finite change: $\Delta x$, $\Delta y$ |
| $\sum$ |
Sigma (upper) |
Summation operator |
| $\prod$ |
Pi (upper) |
Product operator |
| $\pi$ |
pi (lower) |
The constant $\approx 3.14159$ |
| $\theta$ |
theta |
An angle; the polar angle in 2D |
| $\varphi$ |
phi |
A second angle; the azimuthal/declination angle in spherical coordinates |
| $\rho$ |
rho |
A radial distance (spherical/polar); also density |
| $\lambda$ |
lambda |
A Lagrange multiplier; an eigenvalue; a decay/growth rate |
| $\Gamma$ |
Gamma (upper) |
The Gamma function $\Gamma(n) = (n-1)!$; generalizes factorial |
F.9 Common Constants and Relations
| Symbol |
Meaning |
Example / Notes |
| $e$ |
Euler's number |
$\approx 2.71828$; base of the natural exponential |
| $\pi$ |
Pi |
$\approx 3.14159$; ratio of circumference to diameter |
| $i$ |
Imaginary unit |
$i^2 = -1$; star of Euler's formula $e^{i\pi} + 1 = 0$ |
| $\equiv$ |
Identically equal |
Holds for all values, e.g. $\sin^2 x + \cos^2 x \equiv 1$ |
| $\approx$ |
Approximately equal |
Used for numerical estimates and linearizations |
| $\sim$ |
Asymptotic / behaves like |
$a_n \sim b_n$ when their ratio tends to $1$ |
F.10 Common Abbreviations
| Abbreviation |
Stands For |
Notes |
| FTC |
Fundamental Theorem of Calculus |
The bridge between derivatives and integrals (Ch. 14) |
| MVT |
Mean Value Theorem |
Some point's slope equals the average slope |
| IVT |
Intermediate Value Theorem |
A continuous function hits every value between two outputs |
| EVT |
Extreme Value Theorem |
A continuous function on $[a,b]$ attains a max and a min |
| ODE |
Ordinary Differential Equation |
One independent variable (often time) |
| PDE |
Partial Differential Equation |
Several independent variables; uses partial derivatives |
| LHS / RHS |
Left-/Right-Hand Side |
The two sides of an equation |
| iff |
If and only if |
Spoken form of $\iff$ |
| WLOG |
Without Loss of Generality |
"We may assume...without weakening the argument" |
| QED |
Quod erat demonstrandum |
"Which was to be shown"; marks the end of a proof |
F.11 Notation Traps
A handful of symbols look like things they are not. Watch for these.
- $f^{-1} \neq 1/f$. The superscript $-1$ on a function means the inverse function, not the reciprocal. So $\sin^{-1} x = \arcsin x$, but $(\sin x)^{-1} = 1/\sin x = \csc x$. When in doubt, write $\arcsin$ or $1/\sin$ explicitly.
- $\dfrac{dy}{dx}$ is not a fraction — it is the limit of the difference quotient $\Delta y / \Delta x$, a single symbol for the derivative. Yet it behaves like a fraction in many manipulations (the chain rule, separation of variables, $u$-substitution), which is why Leibniz designed it that way. Treat the fraction-like behavior as a useful convenience backed by theorems, not as literal division.
- $f^{(n)}$ vs. $f^n$. The parentheses matter: $f^{(2)}$ is the second derivative, while $f^2$ usually means $(f(x))^2$. The Leibniz $d^2 f/dx^2$ likewise counts derivatives — nothing is squared.
- $\Delta x$ vs. $dx$. $\Delta x$ is a real, finite, measurable change; $dx$ is an infinitesimal differential. They are cousins, not twins.
- $(a, b)$ is ambiguous out of context — it can be an open interval on the number line or a point/ordered pair in the plane. The surrounding sentence tells you which. Vectors avoid this with angle brackets: $\langle a, b\rangle$.
- $\int f\,dx$ always needs its $dx$. Omitting the differential (writing $\int f$) is incomplete notation in this book and hides which variable you are integrating against — fatal once multiple variables are in play.
Notation is a tool for thinking clearly, not a hurdle to clear. When a symbol confuses you, return here, find its row, and read the example. The fluency comes with use.