Symbolic two-input propagation of a period–length model

Given: The function $h(T,l)=4{\pi }^{2}l{T}^{-2}$ on the open convex set $U=\{T>0\}$, a nominal input $\mu =(T_0,l_0)\in U$, and a square-integrable random vector $(T,l):\mathrm{\Omega }\to U$ with mean $\mu$ and covariance matrix as displayed.

If $h:U\to \mathbb{R}$ is Borel measurable and totally differentiable at $\mu$ and $(T,l)$ is square-integrable with mean $\mu$ and covariance matrix $\mathrm{\Sigma }$, then the linearization $x\mapsto h(\mu )+Dh(\mu )(x-\mu )$ has mean $h(\mu )$ and covariance matrix $Dh(\mu )\mathrm{\Sigma }Dh(\mu {)}^{\mathsf{T}}$. (The covariance of the linearization is $A\mathrm{\Sigma }{A}^{\mathsf{T}}$)

The covariance matrix of a square-integrable random vector is ${\mathrm{\Sigma }}_{ij}=E[(X_i-{\mu }_i)(X_j-{\mu }_j)]$ with ${\mathrm{\Sigma }}_{11}={\sigma }_T^2$, ${\mathrm{\Sigma }}_{22}={\sigma }_l^2$ and ${\mathrm{\Sigma }}_{12}={\mathrm{\Sigma }}_{21}={\sigma }_{Tl}$. (Mean vector and covariance matrix of a square-integrable random vector)

The partial derivatives of $h(T,l)=4{\pi }^{2}l{T}^{-2}$ are ${\mathrm{\partial }}_{T}h=-8{\pi }^{2}l{T}^{-3}$ and ${\mathrm{\partial }}_{l}h=4{\pi }^2{T}^{-2}$ on $U$, by the derivative of a power and the algebra of derivatives. (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case)

The Jacobian matrix of a totally differentiable map is the matrix of its total derivative in the standard bases, and matrix multiplication in the $1\times 2$ and $2\times 2$ case gives the displayed quadratic form. (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$)

On an open convex domain with $\mathrm{\parallel }{H}_h\mathrm{\parallel }\le B$, the Taylor remainder has $\mathrm{\mid }R\mathrm{\mid }\le B\mathrm{\mid }X-\mu {\mathrm{\mid }}^2\mathrm{/}2$; its mean and mean-square obey the stated bounds, and its variance differs from the linearized variance by at most $B\sqrt{s^2{M}_4}+B^2{M}_4\mathrm{/}4$ when $M_4<\mathrm{\infty }$. (A uniform Hessian bound controls the second-order Taylor remainder, Mean, mean-square and covariance bounds for second-order linearization error)

The matrix action $Hv$ and operator norm $\mathrm{\parallel }H\mathrm{\parallel }$ are those of A uniform Hessian bound controls the second-order Taylor remainder. For a $2\times 2$ matrix $H$ and vector $v$, applying Euclidean Cauchy–Schwarz to each row gives $\mathrm{\mid }Hv{\mathrm{\mid }}^2\le ({\sum }_{0\le i,j<2}{H}_{ij}^2)\mathrm{\mid }v{\mathrm{\mid }}^2$, so its operator norm is at most the square root of the sum of squared entries. (The Hessian matrix and critical points of a scalar field, Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs)

If $h$ is continuous, it has Borel preimages of Borel sets; if every first partial derivative exists on a neighbourhood of $\mu$ and is continuous at $\mu$, then $h$ is totally differentiable there with Jacobian derivative. The $C^k$ convention is continuity of the iterated coordinate derivatives through order $k$. (A continuous map has Borel preimages of Borel sets, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^k$ Euclidean maps and diffeomorphisms)

Define $c_0:=4{\pi }^2$ and $c_{k+1}:=-(k+2)c_k$ for integers $k\ge 0$. For each $k$, any iterated partial with $k$ derivatives in $T$ and no $l$ derivative is $c_{k}l{T}^{-k-2}$; with one $l$ derivative in any position it is $c_k{T}^{-k-2}$; with at least two $l$ derivatives it is zero. These formulas follow by repeated differentiation of powers and the product rule and are continuous on $T>0$, so $h$ is $C^{\mathrm{\infty }}$ under the convention of [L7]. (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, $C^k$ Euclidean maps and diffeomorphisms)