Physlib.Mathematics.Calculus.ParametricIntegration

Let $M$ and $N$ be real normed spaces. Let $F: M \to \mathbb{R} \to N$ be a function such that the uncurried function $(x, t) \mapsto F(x, t)$ is of class $C^1$. For any $x_0 \in M$, the parametric integral $x \mapsto \int_0^1 F(x, t) \, dt$ is Fréchet differentiable at $x_0$, and its Fréchet derivative is given by $\int_0^1 \text{D}_x F(x_0, t) \, dt$, where $\text{D}_x F(x_0, t)$ denotes the Fréchet derivative of the map $x \mapsto F(x, t)$ at $x_0$.

Let $M$ and $N$ be normed spaces and $F : M \to \mathbb{R} \to N$ be a function such that the uncurried map $(x, t) \mapsto F(x, t)$ is continuously differentiable ($C^1$). For any point $x_0 \in M$ and any vector $v \in M$, the Fréchet derivative of the parametric integral $x \mapsto \int_0^1 F(x, t) \, dt$ at $x_0$ applied to the vector $v$ is equal to the integral of the Fréchet derivative of $x \mapsto F(x, t)$ at $x_0$ applied to $v$: $$D \left( \int_0^1 F(x, t) \, dt \right) (x_0) v = \int_0^1 \left( D_x F(x_0, t) v \right) \, dt$$ where $D$ denotes the Fréchet derivative.

Let $M$ and $N$ be real normed spaces. Let $F : M \times \mathbb{R} \to N$ be a function such that the map $(x, t) \mapsto F(x, t)$ is continuously differentiable ($C^1$). Then the Fréchet derivative of the function mapping $x \in M$ to the integral $\int_0^1 F(x, t) \, dt$ is equal to the function mapping $x$ to the integral of the partial Fréchet derivative $\int_0^1 \text{D}_x F(x, t) \, dt$.

Let $M$ and $N$ be normed spaces over $\mathbb{R}$. If a function $F: M \times \mathbb{R} \to N$ is $C^1$ (first-order continuously differentiable) as a function of its joint arguments, then the function $g: M \to N$ defined by the parametric interval integral $g(x) = \int_{0}^{1} F(x, t) \, dt$ is also $C^1$.

Let $M$ be a finite-dimensional real vector space and $N$ be a normed space. For any natural number $n$, if a function $F: M \times \mathbb{R} \to N$ is such that the uncurried function $(x, t) \mapsto F(x, t)$ is of class $C^{n+1}$, then the function $x \mapsto \int_{0}^{1} F(x, t) \, dt$ is also of class $C^{n+1}$ on $M$.

Let $M$ be a finite-dimensional, proper, real normed space, and let $n$ be a natural number. If a function $F: M \times \mathbb{R} \to N$ (represented here in its uncurried form $\text{↿}F$) is $C^n$ smooth, then the parametric integral function $x \mapsto \int_0^1 F(x, t) \, dt$ is also $C^n$ smooth.