$$\require{cancel}$$

# 1.7: Notation

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 $$\mathcal { V }$$.
• Integrations over curves, surfaces, and volumes will usually be indicated using a single integral sign with the appropriate subscript. For example:
$\int _ { \mathcal { C } } \cdots d l \nonumber \text{ is an integral over the curve } \mathcal { C }$
$\int _ { \mathcal { S } } \cdots d s\nonumber \text{ is an integral over the surface } \mathcal { S }$
$\int _ { \mathcal {V } } \cdots d s\nonumber \text{ is an integral over the volume } \mathcal { V }.$
• Integrations over closed curves and surfaces will be indicated using a circle superimposed on the integral sign. For example:
$\oint _ { \mathcal { C } } \ldots d l\nonumber \text{ is an integral over the closed curve } \mathcal { C }$
$\oint _ { \mathcal { S } } \ldots ds \nonumber \text{ is an integral over the closed surface } \mathcal { S }$
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.