Vector Newton remainder under a Lipschitz Jacobian

Given: The open set $U$, the $C^1$ map $F$, the convex set $K$, the points $x_{\ast },x\in K$, the Jacobian $J$, and the Lipschitz bound with constant $L$.

Continuously differentiable maps, local inverses, and local diffeomorphisms: a $C^1$ map is totally differentiable at every point, and its derivative-matrix entries are continuous.

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the Jacobian matrix has entries ${\mathrm{\partial }}_j{F}_i$ in the standard basis; A total derivative computes every directional derivative, and its matrix is the Jacobian identifies this matrix with the matrix of $DF$.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: the derivative of a composition is the composition of the derivatives.

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: vector derivatives and integrals are componentwise; a vector-valued function is integrable when its components are integrable, and its integral is their vector of integrals.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: every continuous real-valued function on a compact interval is Riemann integrable.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: if a vector-valued function is differentiable on the closed interval and its vector derivative is integrable, then the integral of the derivative is the endpoint increment.

Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$ and [F4]: the vector-valued integral is linear, by componentwise scalar linearity.

For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: for an integrable vector-valued $g$ on $[a,b]$, $\mathrm{\parallel }{\int }_a^{b}g{\mathrm{\parallel }}_2\le {\int }_a^b\mathrm{\parallel }g{\mathrm{\parallel }}_2$.

If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$: pointwise order between integrable real functions on $[a,b]$ implies the same order between their integrals.

Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative: if $H^{\mathrm{\prime }}=h$ on a closed interval and $h$ is integrable, then $\int h=H(1)-H(0)$ on $[0,1]$; in particular ${\int }_0^{1}cdt=c$ for a constant $c$ and ${\int }_0^1(1-t)dt=1\mathrm{/}2$.