You may have noticed two ways to write derivatives: subscripts and fractions. Nev prefers subscripts while I prefer fractions.
E.g. for T (x, t):
| Subscripts | Fractions |
|---|---|
| T_x | \dfrac{\pd T}{\pd x} |
| T_x (0, t) | \eval{\dfrac{\pd T}{\pd x}}_{x = 0} |
| T_x (x, 0) | \eval{\dfrac{\pd T}{\pd x}}_{t = 0} |
| T_{xx} | \dfrac{\pd^2 T}{{\pd x}^2} |
Both are fine; just pick one and be consistent.
The reason I prefer fractions is because they explicitly show the dimensions of a derivative. It is very easy to do scaling with derivatives in fractional form.
For example, consider moving the term
from unscaled variables to scaled variables according to
Just by looking at the term, we see that it is temperature divided by length squared. Immediately we can write down
because the temperature scale is \scale{\Theta} and the length scale is \scale{L}. That's it. (We can ignore T_0 because it is an offset, which doesn't affect the derivative — the derivative of an added constant is zero.)
You may also have noticed that I regularly omit the variables which a function depends on, and that I prefer to use vertical-bar notation to denote evaluation of a function at a particular point.
I will also write things like T = T (x, t), which to a pure mathematician is an abuse of notation.
The reason is that in physics & applied maths, functions are viewed as expressions rather than as maps.
This is best demonstrated by Corinne's Shibboleth (slightly paraphrased here):
Suppose the temperature on a rectangular slab of metal is given by T (x, y) = k (x^2 + y^2) where k is a constant. What is T (r, \theta)?
- Pure mathematician: T (r, \theta) = k (r^2 + \theta^2)
- Applied mathematician: T (r, \theta) = k r^2
In MATH3022 (and in physics & applied maths more generally), we choose the applied mathematician's answer. When we write T = T (x, y), we are simply asserting that the temperature T is to be expressed in terms of the coordinates x and y. In particular:
When we write T (r, \theta), the T is the same physical temperature profile as when we write T (x, y). The only difference is the coordinate system used to express it.
For a rough analogy, 1000 metres and 1 kilometre are the same physical length — they are equal — even though they have been expressed in terms of different units.