Forward-difference Jacobian on an open domain

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, and let $F:U\to {\mathbb{R}}^n$ be $C^1$ in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms. Its Jacobian $J_F(x)$ is the matrix of first partial derivatives from The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, and the matrix of $DF(x)$ in the standard basis is $J_F(x)$ by A total derivative computes every directional derivative, and its matrix is the Jacobian. For $x\in U$, supply increments $h=(h_0,\dots ,h_{n-1})\in {\mathbb{R}}^n$. For each standard basis vector $e_i$ with $0\le i<n$, the admissible stencil conditions are $h_i\mathrm{\ne }0$ and $x+h_i{e}_i\in U$.

If any admissibility condition fails, return invalid-stencil. Otherwise form the matrix $J_{h}F(x)$ column by column: $J_{h}F(x)e_i:=\frac{F(x+h_i{e}_i)-F(x)}{h_i},0\le i<n\mathrm{.}$ Return success and this matrix. Each evaluation point is in $U$. This procedure defines a forward-difference approximation; it asserts no accuracy or convergence guarantee.