With polar angle measured from positive and azimuth , the Cartesian formulas , , have Jacobian determinant . For and this gives the unsigned volume element shown. The inverse determinant is off the origin and polar axis.
In the usual physics convention for spherical coordinates, the polar angle is the angle between the positive axis and the position vector. For , it satisfies with . The north and south axes have and , respectively; the plane has . The angle is undefined at the origin. In plane polar coordinates, the same phrase usually denotes the oriented in-plane angle instead, so the reference axis and convention should always be stated. The azimuth supplies the other angular coordinate in three dimensions.
For radial mass density within radius , integration against the spherical volume element gives . The dimensionless quantity hasThus is positive, strictly decreasing and convex, with and . Its asymptotic forms are near zero and at infinity. The density's central divergence is integrable against the factor .
At fixed angular coordinates, differentiating the Cartesian coordinates with respect to radial distance gives this unit vector. It is the local direction of increasing radius in the spherical coordinate system.
Its coordinate expression is part of del in cylindrical and spherical coordinates.
