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1.7: Notation

Last updated

Jul 7, 2024
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- 1.6: Units
- 2: Electric and Magnetic Fields

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The list below describes notation used in this book.

Vectors: Boldface is used to indicate a vector; e.g., the electric field intensity vector will typically appear as $E$ . Quantities not in boldface are scalars. When writing by hand, it is common to write “ $\overset{―}{E}$ ” or “ $\vec{E}$ ” in lieu of “ $E$ .”
Unit vectors: A circumflex is used to indicate a unit vector; i.e., a vector having magnitude equal to one. For example, the unit vector pointing in the $+ x$ direction will be indicated as $\hat{x}$ . In discussion, the quantity “ $\hat{x}$ ” is typically spoken “ $x$ hat.”
Time: The symbol $t$ is used to indicate time.
Position: The symbols $(x, y, z), (ρ, ϕ, z)$ , and $(r, θ, ϕ)$ indicate positions using the Cartesian, cylindrical, and polar coordinate systems, respectively. It is sometimes convenient to express position in a manner which is independent of a coordinate system; in this case, we typically use the symbol $r$ . For example, $r = \hat{x} x + \hat{y} y + \hat{z} z$ in the Cartesian coordinate system.
Phasors: A tilde is used to indicate a phasor quantity; e.g., a voltage phasor might be indicated as $\tilde{V}$ , and the phasor representation of $E$ will be indicated as $\tilde{E}$ .
Curves, surfaces, and volumes: These geometrical entities will usually be indicated in script; e.g., an open surface might be indicated as $S$ and the curve bounding this surface might be indicated as $C$ . Similarly, the volume enclosed by a closed surface $S$ may be indicated as $V$ .
Integrations over curves, surfaces, and volumes will usually be indicated using a single integral sign with the appropriate subscript. For example:
$\begin{matrix} \int_{C} \dots d l is an integral over the curve C \end{matrix}$
$\begin{matrix} \int_{S} \dots d s is an integral over the surface S \end{matrix}$
$\begin{matrix} \int_{V} \dots d s is an integral over the volume V . \end{matrix}$
Integrations over closed curves and surfaces will be indicated using a circle superimposed on the integral sign. For example:
$\begin{matrix} \oint_{C} \dots d l is an integral over the closed curve C \end{matrix}$
$\begin{matrix} \oint_{S} \dots d s is an integral over the closed surface S \end{matrix}$
A “closed curve” is one which forms an unbroken loop; e.g., a circle. A “closed surface” is one which encloses a volume with no openings; e.g., a sphere.
The symbol “ $≅$ ” means “approximately equal to.” This symbol is used when equality exists, but is not being expressed with exact numerical precision. For example, the ratio of the circumference of a circle to its diameter is $π$ , where $π ≅ 3.14$ .
The symbol “ $\approx$ ” also indicates “approximately equal to,” but in this case the two quantities are unequal even if expressed with exact numerical precision. For example, $e^{x} = 1 + x + x^{2} / 2 + \dots$ as a infinite series, but $e^{x} \approx 1 + x$ for $x ≪ 1$ . Using this approximation $e^{0.1} \approx 1.1$ , which is in good agreement with the actual value $e^{0.1} ≅ 1.1052$ .
The symbol “ $\sim$ ” indicates “on the order of,” which is a relatively weak statement of equality indicating that the indicated quantity is within a factor of 10 or so the indicated value. For example, $μ \sim 10^{5}$ for a class of iron alloys, with exact values being being larger or smaller by a factor of 5 or so.
The symbol “ $≜$ ” means “is defined as” or “is equal as the result of a definition.”
Complex numbers: $j ≜ \sqrt{- 1}$ .
See Appendix C for notation used to identify commonly-used physical constants.

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