A tensor product of two positive Gaussian rules has a derivative error bound

Given: Two finite positive order-$2n$ Gaussian moment data, their $n$-node rules, and a function with the stated continuous pure partial derivatives.

Positive-measure Gaussian quadrature moment data and The m-node Gaussian quadrature rule for a positive measure: order-$2n$ data require finite absolute moments through $2n$; discarding the last moment condition gives the same measure's order-$2n-1$ data used to define the Gaussian rules.

Gaussian quadrature is exact through degree 2m−1 and has positive weights: each rule has positive weights summing to its measure's total mass.

Gaussian quadrature has the Hermite mean-value remainder: for $u\in C^{2n}$, each one-coordinate error has absolute value at most $K_{\mu }\mathrm{\parallel }{u}^{(2n)}{\mathrm{\parallel }}_{\mathrm{\infty }}$ or $K_{\nu }\mathrm{\parallel }{u}^{(2n)}{\mathrm{\parallel }}_{\mathrm{\infty }}$, respectively.

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: increments of a primitive are integrals of its continuous derivative.

Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous: each continuous partial derivative on the compact rectangle is uniformly continuous.

The Borel sigma-algebra of a topological space and Integrable real and complex functions, and their integrals: continuous bounded functions on the compact intervals are Borel measurable and integrable for the finite measures.

Monotonicity and nonnegative homogeneity of the nonnegative integral and The Lebesgue integral is linear on $L^1(\mu )$: $\mathrm{\mid }\int vd\nu \mathrm{\mid }\le M_{\nu }\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$, bounded integrands are integrable, and finite integral differences distribute linearly.

The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$ and Laws of finite sums and products in $\mathbb{N}$, and $\iota \left({\sum }_{k<n}{a}_k\right)={\sum }_{k<n}\iota (a_k)$: $(2n){!}_{\mathbb{R}}$ is the positive real image of ${\prod }_{j=1}^{2n}j$.

For a nonempty subset of ${\mathbb{R}}^n$ with $n\ge 1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: the rectangle is closed and bounded in ${\mathbb{R}}^2$, hence compact, and its continuous real-valued partial derivatives attain finite extrema, so their supremum norms are finite.