How to Read Technical Notation

Notation is compressed language

Every formula is a sentence written in shorthand. Reading it well means expanding the shorthand: saying the formula aloud in words, knowing which symbols are fixed and which vary, and knowing which conventions the author is relying on without stating them. Two questions do most of the work.

• Which symbols are bound? A bound (dummy) variable, such as $x$ in $\int_0^1 x^2\,dx$ or $k$ in $\sum_{k=1}^{n} k$, can be renamed without changing the meaning. The expression does not depend on it.
• Which symbols are free? The result depends on these. In $\sum_{k=1}^{n} k = n(n+1)/2$, the free symbol is $n$.

When a formula confuses you, list its free symbols. Often the confusion is that you were treating a dummy variable as if the answer depended on it, or the reverse.

Derivative operators

The notations $\frac{dy}{dx}$, $y'$, $Dy$ and, in physics, $\dot y$ (for time derivatives) all name the same derivative. Operator notation $D = \frac{d}{dx}$ is useful because operators can be added, multiplied by constants and composed. Powers mean repeated application: $D^2 y = y''$. For constant coefficients, operator polynomials multiply like ordinary polynomials:

An inverse operator such as $(D+b)^{-1}$ means "the operation that undoes $D + b$." It is only meaningful once a space of functions is fixed on which $D + b$ is one-to-one, so a careful text always says which space. The book, for instance, works with a finite-dimensional representation of $(D+b)^{-1}$ in Section 5.2.8, on a space closed under differentiation.

Watch the exponents. The $n$-th derivative is written $f^{(n)}$, with parentheses, and $f^{(0)} = f$. Without parentheses, $f^n$ usually means a power, and $f^{-1}$ means the inverse function, not $1/f$. Trigonometry adds an inconsistency you simply have to memorize: $\sin^2 x = (\sin x)^2$, but $\sin^{-1} x = \arcsin x$.

Partial derivatives, $\partial f/\partial x$ or $f_x$ or $\partial_x f$, differentiate in one variable with all others held fixed. The phrase "held fixed" carries information: when variables are related, you need to know which ones are being held fixed, and a careful text says so.

Integrals: bounds, dummies and parameters

In $\int_a^b f(x,t)\,dx$, the variable $x$ is bound by the integral and the $dx$ tells you so. The result is a function of $a$, $b$ and the parameter $t$, but not of $x$. A standard example makes the dependence explicit:

Differentiating with respect to the parameter, under conditions that justify moving $\frac{d}{dt}$ inside the integral, gives $F'(t) = \int_0^1 x^{t}\ln x\,dx = -1/(t+1)^2$, and in general $F^{(n)}(t) = \int_0^1 x^{t}(\ln x)^n\,dx = (-1)^n n!/(t+1)^{n+1}$. Using definite integrals to organize higher-order derivatives in a parameter is the starting theme of Chapter 1 of the book; the article The structure behind higher-order derivatives discusses it.

When the variable of integration would clash with a limit, rename the dummy. Write $G(x) = \int_0^x f(u)\,du$, not $\int_0^x f(x)\,dx$; the second form uses one letter for two different roles.

Transform notation

The Laplace transform takes a function $f$ and produces a new function of $s$:

Here $t$ is a dummy variable and $s$ is free. Texts often write $\mathcal{L}\{f(t)\}$ or $\mathcal{L}\{t^2\} = 2/s^3$; this is a harmless abuse of notation as long as you remember that the transform acts on the whole function, not on a value. Capital letters are a common convention, $F = \mathcal{L}\{f\}$, and $\mathcal{L}^{-1}$ denotes the inverse transform. A transform formula is incomplete without its region of convergence: $\mathcal{L}\{t^2\}(s) = 2/s^3$ holds for $\operatorname{Re} s > 0$, and the integral diverges for real $s \le 0$.

Convolution is written $(f \ast g)(t)$. In the Laplace setting it means $\int_0^t f(\tau)\,g(t-\tau)\,d\tau$; in the Fourier setting the integral runs over the whole real line. Same symbol, different definition: check which one a text uses.

Matrices, vectors and indices

A matrix $A = (a_{ij}) \in \mathbb{R}^{m\times n}$ has $m$ rows and $n$ columns, and $a_{ij}$ is the entry in row $i$, column $j$: row first, always. Vectors are columns unless stated otherwise, $A^{T}$ is the transpose, and the product is defined entrywise by $(Ax)_i = \sum_{j=1}^{n} a_{ij}x_j$. The Kronecker delta $\delta_{ij}$ equals $1$ when $i = j$ and $0$ otherwise, so the identity matrix is $I = (\delta_{ij})$.

Matrices of this kind, differentiation written as linear algebra on a space closed under differentiation, are the subject of Chapter 5 of the book and of the article From function spaces to matrix representation.

Summation and silent conventions

The general Leibniz rule for the $n$-th derivative of a product shows several conventions at once:

The index $k$ is bound and $n$ is free. The terms $k = 0$ and $k = n$ use $f^{(0)} = f$ and $0! = 1$. For $n = 2$ the sum expands to $f''g + 2f'g' + fg''$, which you can confirm by differentiating $fg$ twice. Other silent conventions worth knowing: an empty sum equals $0$, an empty product equals $1$, and in some physics and geometry texts a repeated index is summed automatically (the Einstein convention), so $a_{ij}x_j$ already means $\sum_j a_{ij}x_j$.

Glossary