of the area improves. Taking , the sum becomes the Riemann sum for the above integral. A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral. Generalization Using Cartesian coordinates, an infinitesimal area element can be calculated as dA = dx dy. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered: Hence, an area element in polar coordinates can be written as Now, a function, that is given in polar coordinates, can be integrated as follows: Here, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b. The formula for the area of R is retrieved by taking f identically equal to 1. A more surprising application of this result yields the Gaussian integral: Vector calculus Vector calculus can also be applied to polar coordinates. For a planar motion, let be the position vector , with r and φ depending on time t. We define the unit vectors in the direction of and in the plane of the motion perpendicular to the radial direction, where is a unit vector normal to the plane of the motion. Then This equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors. For a curve in 2D with the parameter is the previous equation simplify to: Centrifugal and Coriolis terms The term is sometimes referred to as the centripetal acceleration, and the term as the Coriolis acceleration. For example, see Shankar. Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's second law of motion in a rotating frame of reference. Here these extra terms are often called fictitious forces; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame. Co-rotating frame For a particle in planar motion, one approach to attaching physical significance