Modelling and Units

What are the seven SI base quantities and their units?

• Mass: kilogram $\mathrm{kg}$

• Length: metre $\mathrm{m}$

• Time: second $\mathrm{s}$

• Temperature: kelvin $\mathrm{K}$

• Electric current: ampere $\mathrm{A}$

• Amount of substance: mole $\mathrm{mol}$

• Luminous intensity: candela $\mathrm{cd}$

What is the principle of dimensional homogeneity?

If $a=b+c$ then $[a]=[b]=[c]$

What does the notation $[q]$ mean?

$[q]$ denotes the units (dimensions) of the quantity

What form must the units of any physical quantity take?

Every physical quantity has units of the form $[q]=\mathrm{kg}^{\alpha_1}\,\mathrm{m}^{\alpha_2}\,\mathrm{s}^{\alpha_3}\,\mathrm{K}^{\alpha_4}\,\mathrm{A}^{\alpha_5}\,\mathrm{mol}^{\alpha_6}\,\mathrm{cd}^{\alpha_7}$ with $\alpha_1,\dots,\alpha_7\in\mathbb{R}$ constants.

Why must the argument of $e^{x}$, $\sin x$ or $\ln x$ be dimensionless?

Expanding e.g. $e^{t}=1+t+\tfrac{t^{2}}{2!}+\cdots$ would add terms with units $1,\ \mathrm{s},\ \mathrm{s^{2}},\dots$, violating dimensional homogeneity. So we need e.g. $e^{ct}$ with $[c]=\mathrm{s^{-1}}$, making $[ct]=1$.

What are the dimensions of $\dfrac{dy}{dt}$, $\dfrac{d^{n}y}{dt^{n}}$, $\displaystyle\int y\,dt$, $\nabla$ and $\nabla^{2}$?

• $$\left[\frac{dy}{dt}\right]=\frac{[y]}{[t]}$$

• $$\left[\frac{d^{n}y}{dt^{n}}\right]=\frac{[y]}{[t]^{n}}$$

• $\left[\int y\,dt\right]=[y][t]$ , $[\nabla]=\mathrm{m^{-1}}$

• $$[\nabla^{2}]=\mathrm{m^{-2}}$$

Why is an angle dimensionless?

$\theta=\dfrac{s}{r}$ (arc length over radius), so $[\theta]=\dfrac{[s]}{[r]}=\dfrac{\mathrm{m}}{\mathrm{m}}=1$

What does it mean for quantities to be dimensionally independent?

Quantities $X_1,\dots,X_r$ such that none of their dimension functions $[X_i]$ can be built as a product of powers of the others' dimension functions.