\[ \newcommand{\vect}[1]{\mathbf{\mathrm{#1}}} \newcommand{\rmd}[0]{\mathrm{d}} \]

Key terms and concepts for sections 13.6-13.8

Mass in two and three dimensions by integrating density.

First moments (dreiemoment) about coordinate planes, center of mass (massesenter).

Cylindrical coordinates (sylinderkoordinater) \((r,\theta,z)\). Related to cartesian coordinates \((x,y,z)\) by \[ x=r\cos\theta \quad\quad y=r\sin\theta \quad\quad z=z. \] Integration measure: \(r\,\rmd r\,\rmd z\,\rmd\theta\).

Spherical coordiantes (kulekoordinater) \((\rho,\varphi,\theta)\). Related to cartesian coordinates \((x,y,z)\) by \[ x=\rho\sin\varphi\cos\theta \quad\quad y=\rho\sin\varphi\sin\theta \quad\quad z=\rho\cos\varphi. \] Integration measure: \(\rho^2 \sin\varphi\, \rmd\rho\, \rmd\varphi\, \rmd\theta\).

Substitutions in multiple integrals, new measure given by Jacobian determinant (Jacobideterminanten). The measures given for cylindrical and spherical coordinates above are just special cases of this.

2013-04-03, spreeman