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For a nonconstant rational map between round unit Riemann spheres, $R^{\ast }d\Omega =J_{R}d\Omega$. Counting inverse images with multiplicity gives $\int J_{R}d\Omega =4\pi \mathrm{deg}R$. The Jacobian determinant vanishes at ramification points of a holomorphic map, but the integral still records the global degree of a holomorphic map.

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.

If $X$ and $Y$ are independent random variables with Gamma distributions of shapes $a,b>0$ and the same rate $\theta >0$, then $U=X/(X+Y)$ has the Beta distribution with parameters $(a,b)$, and $V=X+Y$ has the Gamma distribution of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse change of variables $(x,y)=(uv,(1-u)v)$ has absolute Jacobian determinant $v$, which makes the joint probability density factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.

For planar Brownian motion started at $x+iy$ in $0<y<\pi$, this is its unconditional exit density on the bottom boundary. Its mass is $1-y/\pi$; the remaining mass $y/\pi$ lies on the top boundary. The conformal bijection $z\mapsto e^z$, the Poisson kernel for the upper half-plane and the boundary Jacobian determinant derive the formula. Conditional bottom-boundary density requires division by its mass.

For the blowup of a complex manifold at a point in complex dimension $n$, the differential gives a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$ whose divisor on a complex manifold is $(n-1)E$. Indeed, a local chart has $x_i=u$ and $x_j=u{t}_j$ for $j\ne i$, so its Jacobian determinant is $\pm u^{n-1}$. The holomorphic line bundle associated to a divisor then gives the formula for the canonical bundle.

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change $X_0/X_i=(X_i/X_0{)}^{-1}$ and $X_j/X_i=(X_j/X_0)/(X_i/X_0)$ gives the displayed Jacobian determinant. Multiplying the $i$th local generator by $(-1{)}^i$ removes this sign and yields the transition functions of the twisting sheaf on projective space with degree $-n-1$.

A double integral integrates a function over a two-dimensional region. A change of variables formula can replace the region and area element using a Jacobian determinant.

For unnormalized model posterior densities $h_j$, model-selection probabilities $s_{jk}$ and parameter-proposal densities $q_{jk}$, accept a proposal $(j,\theta )\to (k,{\theta }^{\prime })$ with the minimum of one and $h_k({\theta }^{\prime })s_{kj}({\theta }^{\prime })q_{kj}(\theta \mid {\theta }^{\prime })/[h_j(\theta )s_{jk}(\theta )q_{jk}({\theta }^{\prime }\mid \theta )]$. A deterministic matching-map formulation instead includes its explicit Jacobian determinant; it must not be counted again when already absorbed into a direct proposal density.

For independent Grassmann variables and an invertible ordinary matrix $S$, the linear change $\eta =S\xi$ transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by $\det S$, and coefficient extraction must undo that factor. For paired variables $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$ the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.

The density of expected Brownian occupation time before exit from a planar domain. For planar Brownian motion with infinitesimal generator $\frac{1}{2}\Delta$, it obeys $-\frac{1}{2}{\Delta }_w{G}_D(z,w)={\delta }_z(w)$ and has singularity $-{\pi }^{-1}\log |w-z|$. A conformal bijection preserves this Dirichlet Green function: the squared derivative in the conformal Brownian clock cancels the area Jacobian determinant.

A hierarchical Bayesian model assigns distributions to local latent variables conditional on shared hyperparameters, then distributions to observations conditional on those latent variables. Its probabilistic graphical model records this conditional factorization, often using a plate for repeated observations. Integrating out latent variables gives the marginal likelihood of the hyperparameters. Priors described as flat on logarithms require a Jacobian determinant when densities are written in the original positive variables.

An involutive Metropolis proposal uses a deterministic bijection $S$ satisfying $S^2=I$. For a volume-preserving $S$ and target probability density function $\rho$, accept $S(z)$ with probability $\min (1,\rho (S(z))/\rho (z))$. The identity $\rho (z)\alpha (z)=\min (\rho (z),\rho (S(z)))$ and a change of variables under $S$ prove detailed balance. A non-unit Jacobian determinant must be included when volume is not preserved.

For a smooth flow map ${\phi }_t$ generated by a vector field $F$, the chain rule and $\det (I+tDF)=1+t\mathrm{tr}(DF)+o(t)$ give ${\partial }_{t}J[{\phi }_t](x)=(\nabla \cdot F)({\phi }_t(x))J[{\phi }_t](x)$. With initial value one, its solution is the exponential of the time integral of this divergence. Thus a divergence-free generator preserves Euclidean volume and orientation. This connects the Jacobian determinant with a volume-preserving vector field.

The displayed smooth map from ${\mathbb{R}}^2$ to the punctured plane has Jacobian determinant one: in polar coordinates its radius is $e^s$ and its angle is $r{e}^{-2s}$, so the determinant is $e^{2s}{e}^{-2s}=1$. Nevertheless $a(0,2\pi k)=(1,0)$ for every integer $k$. Thus a local volume-preserving map need not be injective or have bounded weighted inverse multiplicity. Compactly supported data distributed over many inverse branches disprove a determinant-only dispersion with a nonlinear velocity map estimate.