Math cheatsheet
Calculus
| Product Rule | \(\left( uv \right)' = u'v + uv'\) |
| Quotent Rule | \(\left( \frac{u}{v} \right)' = \frac{vu' - uv'}{v^2}\) |
| Chain Rule | \(\frac{d}{dt}\left[ u\left( v(t) \right)\right] = u'(v(t))v'(t)\) |
| Integration by Parts | \(\int u dv = uv - \int vdu\) |
Useful derivatives
| \(\frac{d}{dt}t^n\) | \(=\) | \(nt^{n-1}\) |
| \(\frac{d}{dt}e^t\) | \(=\) | \(e^{t}\) |
| \(\frac{d}{dt}ln(t)\) | \(=\) | \(\frac{1}{t}\), \(t > 0\) |
| \(\frac{d}{dt}ln(u(t))\) | \(=\) | \(\frac{u'(t)}{u(t)}\) |
| \(\frac{d}{dt}sin(t)\) | \(=\) | \(cos(t)\) |
| \(\frac{d}{dt}cos(t)\) | \(=\) | \(-sin(t)\) |
Trig Identities
| \(\tan \theta\) | \(=\) | \(\frac{\cos\theta}{\sin\theta}\) |
| \(\sin^2 \theta + \cos^2 \theta\) | \(=\) | \(1\) |
| \(\sin \frac{\theta}{2}\) | \(=\) | \(\pm \sqrt{\frac{1 - \cos\theta}{2}}\) |
| \(\cos \frac{\theta}{2}\) | \(=\) | \(\pm \sqrt{\frac{1 + \cos\theta}{2}}\) |
| \(\tan \frac{\theta}{2}\) | \(=\) | \(\frac{1 - \cos\theta}{\sin\theta}\) |
3-space vectors
Multiplying unit vectors: \(\begin{align} \hat{i}\hat{j} &= -\hat{j}\hat{i} = \hat{k} \\ \hat{j}\hat{k} &= -\hat{k}\hat{j} = \hat{i} \\ \hat{k}\hat{i} &= -\hat{i}\hat{k} = \hat{j} \end{align}\)
Given two vectors: \(\begin{align} \hat{a} &= a_0\hat{i} + a_1\hat{j} + a_2\hat{k} \\ \hat{b} &= b_0\hat{i} + b_1\hat{j} + b_2\hat{k} \end{align}\)
The cross product is given by: \(\begin{align} \hat{a} \times \hat{b} &= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{vmatrix} \\ &= (a_1b_2 - a_2b_1)\hat{i} + (a_2b_0 - a_0b_2)\hat{j} + (a_0b_1 - a_1b_0)\hat{k} \end{align}\)