1.7: Notation
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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 \(\mathbf{E}\). Quantities not in boldface are scalars. When writing by hand, it is common to write “\(\overline { E }\)” or “\( \vec { E }\) ” in lieu of “\(\mathbf{E}\).”
- 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 \(\hat { \mathbf { x } }\). In discussion, the quantity “ \(\hat { \mathbf { x } }\)” is typically spoken “\(x\) hat.”
- Time : The symbol \(t\) is used to indicate time.
- Position : The symbols \((x, y, z), (\rho, \phi, z)\), and \((r, \theta, \phi )\) 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 \(\mathbf { r }\). For example, \(\mathbf { r } = \hat { \mathbf { x } } x + \hat { \mathbf { y } } y + \hat { \mathbf { z } } z\) 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 \(\mathbf { E }\) will be indicated as \(\tilde{\mathbf{E}}\).
- Curves, surfaces, and volumes : These geometrical entities will usually be indicated in script; e.g., an open surface might be indicated as \(\mathcal { S }\) and the curve bounding this surface might be indicated as \(\mathcal { C }\). Similarly, the volume enclosed by a closed surface \(\mathcal { S }\) may be indicated as \(\mathcal { V }\).
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Integrations over curves, surfaces, and volumes
will usually be indicated using a single integral sign with the appropriate subscript. For example:
\[\int _ { \mathcal { C } } \cdots d l \nonumber \text{ is an integral over the curve } \mathcal { C } \nonumber \]
\[\int _ { \mathcal { S } } \cdots d s\nonumber \text{ is an integral over the surface } \mathcal { S } \nonumber \]
\[\int _ { \mathcal {V } } \cdots d s\nonumber \text{ is an integral over the volume } \mathcal { V }. \nonumber \] -
Integrations over
closed
curves and surfaces
will be indicated using a circle superimposed on the integral sign. For example:
\[\oint _ { \mathcal { C } } \ldots d l\nonumber \text{ is an integral over the closed curve } \mathcal { C } \nonumber \]
\[\oint _ { \mathcal { S } } \ldots ds \nonumber \text{ is an integral over the closed surface } \mathcal { S } \nonumber \]
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 “\(\cong\)” 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 \(\pi \cong 3.14\).
- 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, \(e ^ { x } = 1 + x + x ^ { 2 } / 2 + \ldots \) as a infinite series, but \(e ^ { x } \approx 1 + x\) for \(x \ll 1\). Using this approximation \(e ^ { 0.1 } \approx 1.1\), which is in good agreement with the actual value \(e ^ { 0.1 } \cong 1.1052\).
- The symbol “\(∼\)” 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, \(\mu \sim 10 ^ { 5 }\) for a class of iron alloys, with exact values being being larger or smaller by a factor of 5 or so.
- The symbol “\(\triangleq\)” means “is defined as” or “is equal as the result of a definition.”
- Complex numbers: \(j \triangleq \sqrt { - 1 }\).
- See Appendix C for notation used to identify commonly-used physical constants.