Comprehensive Guidelines for Mathematical Typesetting in Scientific Writing
Font Usage and Symbol Distinctions
The following points outline how different types of mathematical entities should be formatted to ensure clarity and distinction in scientific documents:
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Variables in Italic Font
- Variables such as \(t\), \(T\), \(m\), etc., are written in italic. This style distinguishes them as mathematical entities that represent quantities which can vary or hold specific values in equations.
- Example: In the acceleration formula \(a = \frac{dv}{dt}\), both \(v\) (velocity) and \(t\) (time) are in italics.
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Bold Symbols for Different Entities
- Symbols in bold, such as \(\mathbf{m}\), often represent matrices or vectors, which are distinct from scalar variables.
- Example: In vector notation \(\mathbf{m} = \begin{bmatrix} m_1 \\ m_2 \\ m_3 \end{bmatrix}\), \(\mathbf{m}\) represents a vector.
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Real Numbers and Constants in Regular Font
- Real numbers and constants are written in regular, non-italic font to differentiate them from variables.
- Example: The number \(\pi\) is often shown in regular font: \(\text{Circumference} = 2\pi r\).
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Units in Regular Font
- Units such as meters (m), seconds (s), etc., are written in regular font to clearly separate them from variable names.
- Example: In the force formula \(F = ma\), "kg" for kilograms is in regular font.
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Functions in Regular Font
- Mathematical functions like sin, cos, log, etc., are written in regular font.
- Example: In the sine function \(y = \sin(x)\).
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Matrices in Bold Font
- Matrices are always in bold and never italicized or slanted.
- Example: For a matrix \(\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\).
Case Sensitivity in Units
It is crucial not to capitalize units improperly as it changes their meaning:
- Incorrect capitalization like "Kg" instead of "kg" would imply Kelvin multiplied by grams, which is incorrect. Proper use of case in units ensures the correct interpretation of scientific data.
| Unit | Correct Usage | Incorrect Usage | Implication of Incorrect Usage |
|---|---|---|---|
| Kilogram | kg | Kg | Kg could be misinterpreted as Kelvin multiplied by gram. |
| Meter | m | M | M is a prefix for mega (\(10^6\)), not meter. |
| Second | s | S | S denotes Siemens, a unit of electrical conductance. |
| Ampere | A | a (if used incorrectly) | "a" isn't used, but small case could confuse as it isn't standard. |
| Watt | W | w | Lowercase "w" is not standard and can lead to confusion. |
| Newton | N | n | Lowercase "n" could be mistaken for nano when used as a prefix. |
| Joule | J | j | Lowercase "j" is not recognized as the standard unit for energy. |
| Volt | V | v | Lowercase "v" could lead to confusion as it isn't the standard symbol. |
| Gram | g | G | G often represents the gravitational constant in physics. |
Examples of Guidelines in Use
These examples illustrate how the guidelines are applied in scientific formulas and equations:
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Variable and Unit Distinction
- Example: \(m = (250.0 \frac{a}{m}) \, \text{kg}\) shows \(m\) as a mass variable in italic, \(a\) as a length variable in italic, and \(m\) in the fraction as meters in regular font.
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Gradient and Divergence Operations
- Gradient of a Vector Field \(\mathbf{u}\): $$ \nabla \mathbf{u} = \begin{bmatrix} \frac{\partial u_1}{\partial x_1} & \cdots & \frac{\partial u_1}{\partial x_d} \ \vdots & \ddots & \vdots \ \frac{\partial u_d}{\partial x_1} & \cdots & \frac{\partial u_d}{\partial x_d} \end{bmatrix} $$ This tensor contains all partial derivatives of a vector field \(\mathbf{u}\), showing the vector's change across different dimensions.
- Divergence of a Second-Rank Tensor \(\boldsymbol{\sigma}\): $$ \nabla \cdot \boldsymbol{\sigma} = (\nabla \cdot \sigma_1, \ldots, \nabla \cdot \sigma_d)^T $$ This operation computes the divergence of each row in the tensor \(\boldsymbol{\sigma}\), providing a vector field of divergences.