polar coordinates

Instead of locating a point in the plane by how far right and how far up it is, polar coordinates locate it by how far it is from the origin and in what direction. The pair (r, theta) gives the distance r out from the origin (the pole) and the angle theta measured from the positive x-axis. A circle, hopelessly awkward as x^2 + y^2 = a^2 in Cartesian terms, becomes the beautifully simple r = a in polar. For anything round, polar coordinates are the natural language.

The conversion is x = r cos(theta), y = r sin(theta), with r = sqrt(x^2 + y^2) going back the other way. The thing that matters for integration is the area element. A small polar cell bounded by radii r and r + dr and angles theta and theta + d-theta is not a rectangle of sides dr and d-theta — it is a curved patch whose two arc edges have unequal length. Its area is (r d-theta) times dr, because an arc subtending angle d-theta at radius r has length r d-theta. So the polar area element is dA = r dr d-theta. That extra factor of r — the Jacobian of the polar map — is the single most-forgotten symbol in multivariable calculus, and dropping it silently corrupts every answer.

Polar coordinates turn disks, annuli, circular sectors, and regions with rotational symmetry into integrals with constant or simple limits. They are how you compute the area of a cardioid r = 1 + cos(theta), the volume under a paraboloid over a disk, and — most famously — the Gaussian integral: squaring it produces a double integral over the whole plane that the substitution to polar makes trivial because the r in r dr d-theta is exactly what you need to integrate e^{-r^2}. The honest caveat is that polar coordinates are not unique: the origin has every angle theta at once (r = 0 swallows theta), and adding 2pi to theta gives the same point, so a careless setup can double-count or miss a region.