Change of Variables

A change of variables is simply $u$-substitution for double integrals. In one variable, $x = g(u)$ came with $dx = g'(u)\,du$: the $g'(u)$ told you how much the substitution stretches lengths. Here the substitution $x = g(u, v)$, $y = h(u, v)$ stretches areas, and the Jacobian $\frac{\partial(x, y)}{\partial(u, v)}$ (a $2\times2$ determinant of partial derivatives) plays the role of $g'(u)$. The big formula is just three moves: rewrite $f$ in $u$ and $v$, describe the region in $u$ and $v$, and replace $dA$ with $\left|\text{Jacobian}\right|du\,dv$. It looks heavy, but it's an old friend in a new outfit.

Why a determinant. Think about where a tiny $du\times dv$ rectangle in the $uv$-plane lands. Up close, any smooth map looks linear (that's what derivatives are for), so the rectangle lands on a tiny parallelogram. Its sides are where you go when you nudge $u$ alone, $\langle x_u, y_u\rangle du$, and $v$ alone, $\langle x_v, y_v\rangle dv$. From linear algebra, the area of a parallelogram spanned by two vectors is the absolute value of the determinant with those vectors as columns. So the patch has area $\left|x_uy_v - x_vy_u\right|du\,dv$. That's linear algebra and calculus meeting.

Choosing the substitution: pick $u$ and $v$ to make the region simple (ideally a rectangle) or the integrand simple. For an ellipse, $x = au$, $y = bv$ turns it into a unit disk, with Jacobian $ab$. When the region's edges are lines like $x + y = c$, let those combinations be your new variables.

A common mistake. Often it's easier to write $u$ and $v$ in terms of $x$ and $y$, which gives $\frac{\partial(u, v)}{\partial(x, y)}$, the stretch going the other way. It's tempting to use that number directly. It isn't the factor you want: the factor in $dA$ is its reciprocal, $dA = \frac{du\,dv}{\left|\partial(u, v)/\partial(x, y)\right|}$, just as $\frac{dx}{du} = 1\big/\frac{du}{dx}$ in one variable.

Polar coordinates fit right in: $x = r\cos\theta$, $y = r\sin\theta$ give Jacobian $r$, the extra factor you've been carrying since polar double integrals. In 3D the Jacobian is a $3\times3$ determinant (a volume instead of an area), and spherical coordinates give $\rho^2\sin\varphi$.

Probability uses Jacobians to find the distribution of transformed random variables, and simulations map messy physical domains onto simple grids with a change of variables.