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<!DOCTYPE html>
<html lang="en">
<head>
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<title>Remarks on notation | UWA MATH3022</title>
</head>
<body>
<h1>Remarks on notation</h1>
<noscript><strong class="noscript">Enable JavaScript for equation rendering to work.</strong></noscript>
<p>
<a href="/">Return to the home page.</a>
</p>
<h2 id="derivatives"><a href="#derivatives" class="permalink" aria-label="Permalink"></a>Derivatives</h2>
<span class="js-maths">\gdef \unscaled #1 {\colg{#1}}
\gdef \scaled #1 {\colv{#1'}}
\gdef \scale #1 {\colb{#1}}</span>
<p>
You may have noticed two ways to write derivatives:
<b>subscripts</b> and <b>fractions</b>.
Nev prefers subscripts while I prefer fractions.
</p>
<p>
E.g. for <span class="js-maths">T (x, t)</span>:
</p>
<table>
<thead>
<tr>
<th>Subscripts</th>
<th>Fractions</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="js-maths">T_x</span></td>
<td><span class="js-maths">\dfrac{\pd T}{\pd x}</span></td>
</tr>
<tr>
<td><span class="js-maths">T_x (0, t)</span></td>
<td><span class="js-maths">\eval{\dfrac{\pd T}{\pd x}}_{x = 0}</span></td>
</tr>
<tr>
<td><span class="js-maths">T_x (x, 0)</span></td>
<td><span class="js-maths">\eval{\dfrac{\pd T}{\pd x}}_{t = 0}</span></td>
</tr>
<tr>
<td><span class="js-maths">T_{xx}</span></td>
<td><span class="js-maths">\dfrac{\pd^2 T}{{\pd x}^2}</span></td>
</tr>
</tbody>
</table>
<p>
<strong>Both are fine; just pick one and be consistent</strong>.
</p>
<p>
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.
</p>
<p>
For example, consider moving the term
</p>
<div class="js-maths">\frac{\pd^2 \unscaled{T}}{{\pd \unscaled{x}}^2}
</div>
<p>
from <span class="colour-g">unscaled variables</span> to <span class="colour-v">scaled variables</span>
according to
</p>
<div class="js-maths">\begin{aligned}
\unscaled{T} &= T_0 + \scale{\Theta} \scaled{T}, \\
\unscaled{x} &= \scale{L} \scaled{x}.
\end{aligned}
</div>
<p>
Just by <em>looking</em> at the term,
we see that it is temperature divided by length squared.
<strong>Immediately we can write down</strong>
</p>
<div class="js-maths">\frac{\pd^2 \unscaled{T}}{{\pd \unscaled{x}}^2} =
\frac{\scale{\Theta} \pd^2 \scaled{T}}{\scale{L}^2 \,{\pd \scaled{x}}^2}
</div>
<p>
because the temperature scale is <span class="js-maths">\scale{\Theta}</span>
and the length scale is <span class="js-maths">\scale{L}</span>.
<strong>That's it.</strong>
(We can ignore <span class="js-maths">T_0</span> because it is an <em>offset</em>,
which doesn't affect the derivative
— the derivative of an added constant is zero.)
</p>
<h2 id="functions"><a href="#functions" class="permalink" aria-label="Permalink"></a>Functions</h2>
<p>
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.
</p>
<p>
I will also write things like <span class="js-maths">T = T (x, t)</span>,
which to a pure mathematician is an abuse of notation.
</p>
<p>
The reason is that <strong>in physics & applied maths,
functions are viewed as <em>expressions</em> rather than as <em>maps</em></strong>.
</p>
<p>
This is best demonstrated by <a href="https://sites.science.oregonstate.edu/math/bridge/ideas/functions/">Corinne's Shibboleth</a>
(slightly paraphrased here):
</p>
<blockquote>
Suppose the temperature on a rectangular slab of metal
is given by <span class="js-maths">T (x, y) = k (x^2 + y^2)</span> where <span class="js-maths">k</span> is a constant.
What is <span class="js-maths">T (r, \theta)</span>?
<ul>
<li>
Pure mathematician: <span class="js-maths">T (r, \theta) = k (r^2 + \theta^2)</span>
</li>
<li>
Applied mathematician: <span class="js-maths">T (r, \theta) = k r^2</span>
</li>
</ul>
</blockquote>
<p>
In MATH3022 (and in physics & applied maths more generally),
we choose the applied mathematician's answer.
When we write <span class="js-maths">T = T (x, y)</span>,
we are simply asserting that the temperature <span class="js-maths">T</span>
is to be expressed in terms of the coordinates <span class="js-maths">x</span> and <span class="js-maths">y</span>.
In particular:
</p>
<ul>
<li>
<span class="js-maths">T</span> is the <em>physical</em> temperature, NOT a map.
</li>
<li>
<span class="js-maths">x</span> and <span class="js-maths">y</span> are <em>physical</em> coordinates, NOT replaceable dummy variables.
</li>
</ul>
<p>
When we write <span class="js-maths">T (r, \theta)</span>,
the <span class="js-maths">T</span> is the <em>same physical temperature profile</em>
as when we write <span class="js-maths">T (x, y)</span>.
The only difference is the coordinate system used to express it.
</p>
<p>
For a rough analogy,
1000 metres and 1 kilometre are the <em>same physical length</em>
— they are <em>equal</em>
— even though they have been expressed in terms of different units.
</p>
<div class="end">
END
</div>
<p>
<a href="/">Return to the home page.</a>
</p>
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