Polar coordinates represent a planar point by a distance r from a fixed origin and an angle theta from a fixed axis, related to Cartesian coordinates x, y by x = r cos theta, y = r sin theta, r squared = x squared + y squared, and tan theta = y/x. Unlike Cartesian coordinates, this representation is not unique — the same point admits infinitely many (r, theta) names (via theta + 2k*pi or via (−r, theta + pi)) — which has consequences for determining curve membership, differentiation, and area computation in polar form. This belongs to the multivariable/vector calculus treatment of curves and motion in the plane, motivated by central force fields where distance from and angle to a fixed force center are the natural descriptive variables.
Polar coordinates represent a planar point by a distance r from a fixed origin and an angle theta from a fixed axis, related to Cartesian coordinates x, y by x = r cos theta, y = r sin theta, r squar…