to these terms is based on the concept of an instantaneous co-rotating frame of reference. To define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ2, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only. Thus, using these forces in Newton's second law we find: where over dots represent time differentiations, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes: which can be compared to the equations for the inertial frame: This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame. For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force. Differential geometry In the modern terminology of differential geometry, polar coordinates provide coordinate charts for the differentiable manifold , the plane minus the origin. In these coordinates, the Euclidean metric tensor is given byThis can be seen via the change of variables formula for the metric tensor, or by