Mathematical communication
Part of your grade for all homework and exams will be based on the quality of your mathematical communication. Effective mathematical communication enables others to read and evaluate (and learn from) your work. Effective communication aids your own thinking, and supports your collaborations with other scientists.
Mathematical communication aids thinking by facilitating external cognition. Our brains are powerful but have limitations. To solve challenging physics problems we need to offload our ideas and store them for later use. Learning to organize your reasoning clearly on paper will enable you to more easily and accurately solve more challenging problems. You can read and evaluate your own work, and streamline the process of finding errors that you may have made.
Diagrams
Diagrams are a powerful tool for visualizing a physical system and connecting its details to your mathematical analysis. A well-drawn diagram can clarify relationships, streamline problem-solving, and often replace lengthy explanations. Make it a habit to sketch diagrams; they can save you a thousand words!
Defining algebraic variables
Any variables you use must be defined. This can happen early in the problem, or in a figure, or right after an equation with a phrase like “where \(m\) is the mass of the particle”.
Declaring numerical quantities
Option 1: You could say “Assume that the specific energy density of gasoline is 40 MJ/kg”.
Option 2: You could declare your assumptions in within an equality. In the example below, I've written enough to communicate my assumptions about the battery size, and the type of charger:
\begin{align}
\text{time to charge the battery}
= \frac{\text{storage capacity of battery}}{\text{charging rate}}
= \frac{5\times 10^7\text{ J}}{5\times 10^{3}\text{ J/s}}&= 10^4\text{ s}
\end{align}
Units
When a physicist writes down the numerical value of a quantity, they write the units next to the number (unless it's a dimensionless quantity). See the examples above.
Significant figures
Use an appropriate number of significant figures. If you give an answer 6.70931309283 kg, you are probably claiming a level of precision that is impossible/unrealistic. If the problem is a zeroth-order estimate, use 1 or 2 sig. fig. to convey a precision of 10 to 20%. If you have a more refined answer, use 2 or 3 sig. fig. to indicate the appropriate level of precision.
Starting equation
A starting equation is an equation that is not deduced from equations you already have written in your answer. You may have several starting equations in a given problem. Often, starting equations will be physical laws. In some cases your starting equations may simply be conversions. When the source of a starting equation is not obvious, you must describe where it came from (or why it is true) in words. For example, “From the Stefan-Boltzmann law...”.
If you find yourself writing a lot more words than math, please check in with the instructor.
Following equation
A following equation is an equation that can be mathematically shown to be true based on equations already present. In many cases, following equations do not require words to explain them. However, if it is not obvious what you did, then some concise words of explanation are important (like a comment in a computer code). For example, “Change the integration variable to \(u \equiv 1/r\)” or “Assume that all heat entering the water came from the rock”.
For algebraic manipulations, it is sufficient to simple write the lines of algebra. It is unnecessary (and a nuisance) to write words such as "We now divide both sides of the equation by the \(m\)."
If you find yourself writing a lot more words than math, please check in with the instructor.
Equation sequences
The equation sequence is a common idiom in mathematical communication. One side of the equation does not change:
\begin{align}
\vec{F} &= m\vec{a} \\
&= m \frac{d^2\vec r}{dt^2}
\\
&= -m x_0\omega^2 \sin\omega t
\end{align}
In this idiom, we do not rewrite the left-hand side of the equation, but it is taken to be identical. Each expression on the right-hand side in this example is equal to the force.
Equate the leftside to the rightside
Variables and numerical quantities must always be part of either a sentence or a mathematical eqution. For example, do not write:
\begin{align}
\frac{5\times 10^3\text{ J}}{5\times 10^{7}\text{ J/s}}
\end{align}
In the example above, there is no equals sign (it is not an equation). It is not a statement that conveys meaning. It is analogous to a sentence with no verb.
Always check that the leftside and rightside of an equation have the same dimensions. For example, if the leftside has dimensions of length, the rightside must also have dimensions of length.