Pre-course requirements

Physics knowledge

Dimensions and Units

It should come as no surprise to you that we’ll be (mostly) using SI units in this course and throughout your degree. So as a reminder I’ve provided a list of the 7 base units, as defined in the International System of Units

Base units

Basic unit	SI unit	Unit symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

I’ve included all seven for completeness but in practice the candela is rarely used in most areas of physics.

Mathematical knowledge

There are some core mathematical functions and tools that I’ll expect you to be familiar with at the very outset of this course. You don’t have to memorise them before you start, but having them all in a single place is hopefully a helpful resource for you.

Mathematical symbols

Symbols

Symbol	Definition
\(\approx\)	Approximately
\(\therefore\)	Therefore
\(|x|\)	Modulus (size) of \(x\)

Trigonometric formulae

Relation between trigonometric functions

\[\begin{align*} \frac{\sin\theta}{\cos\theta} &= \tan\theta & \frac{1}{\cos\theta} &= \sec\theta & \frac{1}{\sin\theta} &=\csc\theta \end{align*}\]

\[\begin{align*} \sin^2\theta + \cos^2\theta &= 1 \end{align*}\]

Addition formulae

\[\begin{align*} \sin(\theta \pm \phi) &= \sin\theta\cos\phi \pm \cos\theta\sin\phi\\ \cos(\theta \pm \phi) &= \cos\theta\cos\phi \mp \sin\theta\sin\phi\\ \tan(\theta \pm \phi) &= \frac{\tan\theta \pm \tan\phi}{1 \mp \tan\theta\tan\phi} \end{align*}\]

Addition formula: special cases

\[\begin{align*} \sin 2\theta &= 2\sin\theta\cos\theta\\ \cos 2\theta &= \cos^2\theta-\sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\\ \tan 2\theta &= \frac{2\tan\theta}{1-\tan^2\theta}\\ \sin 3\theta &= 3\sin\theta - 4\sin^3\theta\\ \cos 3\theta &= 4\cos^3\theta - 3\cos\theta \end{align*}\]

The following two sets of identities can all be derived from these addition formulae. I’d encourage you to try and derive the following as it’s a good exercise in algebra (especially after the long summer!).

Product formulae

\[\begin{align*} \sin\theta\sin\phi &= \frac{1}{2}[\cos(\theta-\phi)-\cos(\theta+\phi)]\\ \cos\theta\cos\phi &= \frac{1}{2}[\cos(\theta-\phi)+\cos(\theta+\phi)]\\ \sin\theta\cos\phi &= \frac{1}{2}[\sin(\theta-\phi)+\sin(\theta+\phi)]\\ \end{align*}\]

Product formulae: special cases

\[\begin{align*} \sin^2\theta &= \frac{1}{2}(1-\cos 2\theta)\\ \cos^2\theta &= \frac{1}{2}(1+\cos 2\theta)\\ \sin^3\theta &= \frac{1}{4}(3\sin\theta - \sin 3\theta)\\ \cos^3\theta &= \frac{1}{4}(3\cos\theta - \cos 3\theta) \end{align*}\]

Differentiation

Definition

You’ll cover lots of different methods and tricks for differentiating functions during your studies but just as a reminder

(1)\[\begin{align} f(x) &= x^n \\ \therefore \frac{\mathrm{d}f(x)}{\mathrm{d}x} &= nx^{n-1} \end{align}\]

Some common derivatives that may be used in this course are included in the dropdown box below for quick reference.

Common derivatives

\(f(x)\)	\(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\)
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2 x\)
\(e^x\)	\(e^x\)
\(e^{ax}\) where \(a\) is constant	\(ae^{ax}\)
\(\ln x\), \(x>0\)	\(\frac{1}{x}\)

Basic differentiation rules

For two functions \(u(x)\) and \(v(x)\):

\[f(x)\]	\[\frac{\mathrm{d}f(x)}{\mathrm{d}x}\]
\[u(x)v(x)\]	\[u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}\]	\[\text{Product rule}\]
\[\frac{\mathrm{d}u(x)}{\mathrm{d}v(x)}\]	\[\frac{1}{v^2}\left(v\frac{\mathrm{d}v}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}\right)\]	\[\text{Quotient rule}\]
\[\frac{1}{v(x)}\]	\[-\frac{1}{v^2}\frac{\mathrm{d}v}{\mathrm{d}x}\]
\[y(u(x))\]	\[\frac{\mathrm{d}y}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}\]	\[\text{Chain rule}\]

Notation

‘Prime’ notation

Note

Here we are assuming that \(y\) is a function of \(x\) only, and no other variables (no dependence on time \(t\) or anything else). When dealing with functions that have multiple input variables the prime notation isn’t great.

In addition to the standard notation when differentiating the function \(y(x)\) with respect to \(x\) there is another notation style that will be used in this course and indeed throughout your degree, often known as the “prime” notation[1] because we use the symbol \('\) otherwise known as prime.

So the expression \(y'(x)\) is read as “\(y\) prime of \(x\)” and is the first derivative of \(y(x)\) with respect to \(x\). Similarly \(y''(x)\) is “\(y\) double prime of \(x\)” and means the second derivative of \(f(x)\) with respect to \(x\).

‘Dot’ notation

It’s worth noting a typical shorthand convention in physics when dealing with terms that have a time dependence such as velocity (rate of change of position). As differentiating things with respect to time is so common in physics and astronomy we often use a different type of notation whereby we put a dot above the symbol. In other words

\[\begin{align*} \dot{x} &= \frac{\mathrm{d}x}{\mathrm{d}t}=v \\ \text{and} \qquad \dot{v}' = \frac{\mathrm{d}v}{\mathrm{d}t} = a = \ddot{x} \end{align*}\]

Unsuprisingly this is often called the dot notation of differentiation.[2]

Integration

Indefinite Integrals

Some common indefinite integrals that may be used in this course are included in the dropdown box below for quick reference.

Common integrals

\(f(x)\)	\(\int f(x)\,\mathrm{d}x\)
\(x^n\)	\(\frac{x^{n+1}}{n+1} + c\)
\(\frac{1}{x}\)	\(\ln |x| +c\)
\(e^x\)	\(e^x + c\)
\(\sin x\)	\(-\cos x + c\)
\(\cos x\)	\(\sin x + c\)
\(\tan x\)	\(\ln\left|\cos x\right| + c\)

Definite Integrals

In contrast to the indefinite integral case above, a definite integral is one for which we know the range of \(x\) values we are integrating over.

As a quick reminder the integral of the function \(x^n\) between the limits \(x=a\) and \(x=b\) is

\[\begin{align*} \int_a^b x^n \mathrm{d}x = \left[\frac{x^{n+1}}{n+1}\right]_a^b = \frac{b^{n+1}}{n+1} - \frac{a^{n+1}}{n+1} \end{align*}\]

Vectors

You will be taught vectors in great detail within PHY129 because they’re incredibly important for physics. Here though I’ll just give a quick reminder about the notation convention.

When a parameter in an expression is a vector quantity then I will indicated this using bold font. So if we are considering the velocity of something, \(\textbf{v}\) would indicate the velocity vector whereas \(v\) (no bold) would be the scalar quantity (speed).
More formally \(v=|\textbf{v}|\) and \(\textbf{v} = v\hat{\textbf{v}}\) where \(\hat{\textbf{v}}\) is a unit vector.

Unit vectors

A unit vector is a helpful quantity that allows us to define a direction without changing the magnitude, because a unit vector has a ‘size’ of 1. In the example above the velocity vector \(\textbf{v}\) is defined by the magnitude of the velocity \(v\) and this unit vector \(\hat{\textbf{v}}\).

We typically define unit vectors with respect to the coordinate system we are using. So \(\hat{\textbf{i}}\), \(\hat{\textbf{j}}\) and \(\hat{\textbf{k}}\) are the unit vectors the \(x\), \(y\) and \(z\) directions respectively.

\[\begin{align*} |\hat{\textbf{i}}|=|\hat{\textbf{j}}|=|\hat{\textbf{k}}|=1 \end{align*}\]

You’ll see in the PHY129 course that these are not the only unit vectors that exist but for this course you only need one other set for when we discuss non-Cartesian coordinate systems.

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[1]

Historical note: prime is also known as Lagrange’s notation, though most people just say prime. For completeness, the more ‘traditional’ fraction form \(\frac{\mathrm{d}y}{\mathrm{d}x}\) is known as Leibniz’s notation.

[2]

Historical note: this type of notation is also known as Newton’s notation.