In a purely mathematical treatment, regardless of the frame that the coordinate system is associated with (inertial or non-inertial), extra terms appear in the acceleration of an observed particle when using curvilinear coordinates. For example, in polar coordinates the acceleration is given by (see below for details): a = d v d t = d 2 r d t 2 = ( r ¨ ¨ − − r θ θ ˙ ˙ 2 ) r ^ ^ + ( r θ θ ¨ ¨ + 2 r ˙ ˙ θ θ ˙ ˙ ) θ θ ^ ^ , which contains not just double time derivatives of the coordinates but added terms. This example employs polar coordinates, but more generally the added terms depend upon which coordinate system is chosen (that is, polar, elliptic, or any other system). Sometimes these coordinate-system dependent terms also are referred to as "fictitious forces", introducing a second meaning for "fictitious forces", despite the fact that these terms do not have the vector transformation properties expected of forces. For example, see Shankar and Hildebrand. According to this terminology, fictitious forces are determined in part by the coordinate system itself, regardless of the frame it is attached to, that is, regardless of whether the coordinate system is attached to an inertial or a non-inertial frame of reference. In contrast, the fictitious forces defined in terms of the state of motion of the observer vanish in inertial frames of reference. To distinguish these two terminologies, the fictitious forces that vanish in an inertial frame of reference, the inertial forces of Newtonian mechanics, are called in this article the "state-of-motion" fictitious forces and those that originate in the interpretation of time derivatives in particular coordinate systems are called "coordinate" fictitious forces.