Double Integrals in Polar Coordinates and the Change-of-Variables Jacobian

The Black-Scholes formula is, at its core, the integral of a lognormal density over a half-line — and the proof that $\int_{-\infty}^\infty e^{-t^2/2}\,dt=\sqrt{2\pi}$ (the Gaussian normalisation) requires converting a double integral over $\mathbb{R}^2$ from Cartesian to polar coordinates. The polar area element $dA=r\,dr\,d\theta$ (rather than $dr\,d\theta$) is not arbitrary: it is the value of the Jacobian determinant of the polar-to-Cartesian transformation. This lesson develops polar integration, proves the Gaussian integral identity, and then derives the general change-of-variables formula that governs every multivariate density transformation in statistics and finance.

Learning objectives. By the end of this note you should be able to: