
# 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.