Change of variables formula

The change-of-variables formula multiplies an integral by the absolute determinant of the Jacobian.

For a smooth local diffeomorphism $a:{\mathbb{R}}^d\to {\mathbb{R}}^d$, each inverse branch contributes an inverse Jacobian determinant in the area formula. Their sum is the weighted inverse multiplicity. A lower determinant bound controls each branch separately, not the number of branches. Bounded weighted inverse multiplicity is sufficient for dispersion with a nonlinear velocity map.

The displayed map is a diffeomorphism of the open unit square onto itself. For fixed $u$, the second coordinate decreases strictly from one to zero as $v$ increases from zero to one. Its inverse is $u=1-x$, $v=(1-y)/[1-(1-x)y]$, and its positive Jacobian determinant is $[1-(1-x)y{]}^2/x$. Composing it with $(t,w)\mapsto ((1-t)/(1-wt),1-w)$ gives determinant $[1-(1-x)y][1-(1-x^2)y{]}^2/[x(1-y)]$. The reciprocal determinant therefore integrates to one over the target square by the change of variables formula, despite boundary singularities.

A hyperbolic change of integration variable uses identities such as $1+{\sinh }^{2}u={\cosh }^{2}u$ to simplify square roots. For an open matter-dominated Friedmann solution, $a=b{\sinh }^2(\eta /2)$ makes the cosmic-time integral elementary.

For a nondecreasing real function $z$ on an interval $I$ and nonnegative measurable function $h$,The pushforward of $z^{\prime }(u)du$ is dominated by Lebesgue measure: on each interval, the integral of the derivative is at most the increase of $z$. This handles jumps and singular parts without a smoothness assumption.

For a locally Lipschitz function $T$ between Euclidean spaces of the same dimension, the area formula integrates its absolute Jacobian determinant against the multiplicity of its fibers. In particular, for an injective $C^1$ map $T:X\to {\mathbb{R}}^d$ and nonnegative measurable $a$,This injective version extends the change of variables formula to maps with a possibly singular derivative.

Polar coordinates use $x=r\cos \theta$, $y=r\sin \theta$ and area element $rdrd\theta$.

The cylindrical coordinate system extends planar polar coordinates by an axial coordinate: $(x,y,z)=(R\cos \varphi ,R\sin \varphi ,z)$.

If the cylindrical components of a vector field have dependence $e^{im\varphi }$, differentiation of the rotating basis adds the displayed cross product term. Scalar-component differentiation alone misses this contribution. For $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, one also has $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. These identities are useful in cylindrical magnetohydrodynamic waves.

The cylindrical radius $R=\sqrt{x^2+y^2}$ is the perpendicular distance from the symmetry axis.

A cardioid is an epicycloid with one cusp; a standard polar equation is $r=a(1+\cos \theta )$.

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

Spherical coordinates $(r,\theta ,\varphi )$ in ${\mathbb{R}}^3$ use

If orthogonal coordinates $q_i$ satisfythen $h_i=|\partial \mathbf{x}/\partial q_i|$ are their scale factors.

For $\mathbf{A}=A_r{\mathbf{e}}_r+A_{\theta }{\mathbf{e}}_{\theta }+A_{\varphi }{\mathbf{e}}_{\varphi }$, the spherical-coordinate curl follows from the orthogonal-coordinate scale factors $1,r,r\sin \theta$.

The radius-$R$ ball in ${\mathbb{R}}^n$ has volume ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

The boundary of the radius-$R$ ball has area $2{\pi }^{n/2}{R}^{n-1}/\Gamma (n/2)$.