Appendix A

Appendix A: Review of useful mathematical operations for Kinematics

# Review of useful mathematical operations for Kinematics

Appendix A provides an overview of some basic mathematical relations for students of kinematics

## Complex number representation

Complex number notation provides a useful form for representing vectors in 2D. Two forms of complex numbers will be considered: 1) complex cartesian form, 2) complex polar form. As numbers, the traditional operations of addition, multiplication, and differentiation apply.

Further, these operations have physical meaning as shown in the table below.

Vector | complex cartesian | complex polar |

$r$ | $r_x + r_y*i$ | $re^{i\theta}$ |

$r$ | 0 | $r=\sqrt{r_x^2+r_y^2}$ |

$\theta$ | 0 | $\theta=atan2(r_y,r_x}$ |

$r_x$ | $r_x=rcos(\theta)$ | 0 |

$r_y$ | $r_y=rsin(\theta)$ | 0 |

Physical Operation | Mathematical Operation | Preferred representation |

Vector loop | addition | complex cartesian |

pure rotation | Multiplication (unit mag.) | Complex polar |

pure stretching / translation | Multiplication (0 angle) | complex polar |

Complex Polar Notation - a few more tricks

### Compex Cartersian

(1)\begin{equation} r=r_x + r_y*i \end{equation}

page revision: 5, last edited: 08 Oct 2010 17:43