Physlib.SpaceAndTime.Time.Derivatives

Given a function $f : \text{Time} \to M$, where $M$ is a topological vector space over $\mathbb{R}$ (specifically, an additive commutative group and a module over $\mathbb{R}$ with a topology), the time derivative of $f$ is a function from $\text{Time}$ to $M$. For each $t \in \text{Time}$, the value of the derivative is defined as the Fréchet derivative of $f$ at $t$ evaluated at the unit element $1 \in \text{Time}$.

The symbol $\partial_t$ is a notation for the time derivative operator `deriv`. For a function $f : \text{Time} \to M$, where $M$ is a topological vector space over $\mathbb{R}$, $\partial_t f$ denotes the derivative of $f$ with respect to time.

Let $M$ be a topological vector space over $\mathbb{R}$ (specifically, an additive commutative group and a module over $\mathbb{R}$ with a topology). For any function $f : \text{Time} \to M$ and any time $t \in \text{Time}$, the time derivative of $f$ at $t$, denoted by $\text{Time.deriv } f(t)$, is equal to the Fréchet derivative of $f$ at $t$ evaluated at the unit element $1 \in \text{Time}$.

Let $f : \text{Time} \to \mathbb{R}^d$ be a differentiable function and $k \in \mathbb{R}$ be a scalar. For any time $t \in \text{Time}$, the time derivative of the scaled function $k \cdot f$ is equal to the scalar $k$ multiplied by the time derivative of $f$, that is, $\partial_t (k \cdot f(t)) = k \cdot \partial_t f(t)$.

Let $M$ be a normed vector space over the real numbers $\mathbb{R}$. For any function $f : \text{Time} \to M$ and any time $t \in \text{Time}$, the time derivative of the negative of $f$ at $t$ is equal to the negative of the time derivative of $f$ at $t$, expressed as $\partial_t (-f)(t) = -\partial_t f(t)$.

Let $M$ be a normed vector space over the real numbers $\mathbb{R}$. For any constant element $m \in M$, the time derivative of the constant function $f(t) = m$ at any time $t$ is equal to zero, that is, $\partial_t (\lambda t, m) = 0$.

Let $M$ be a real normed space. If a function $f : \text{Time} \to M$ is of class $C^\infty$, then its time derivative $\partial_t f : \text{Time} \to M$ is differentiable.

Let $M$ be a real normed space. If a function $f : \text{Time} \to M$ is of class $C^\infty$, then its time derivative $\partial_t f$ is also of class $C^\infty$.

Let $M$ be a real normed space and $f : \text{Time} \times \text{Space } d \to M$ be a function. If the uncurried function $(t, x) \mapsto f(t, x)$ is of class $C^{n+1}$, then for any fixed $t \in \text{Time}$, the function $x \mapsto \frac{\partial f}{\partial t}(t, x)$ is of class $C^n$ on $\text{Space } d$, where $\frac{\partial f}{\partial t}$ denotes the time derivative.

Let $f : \text{Time} \to \mathbb{R}^n$ be a function from time into an $n$-dimensional Euclidean space. If for every coordinate index $i$, the component function $t \mapsto f(t)_i$ is differentiable over $\mathbb{R}$, then the function $f$ is differentiable over $\mathbb{R}$.

Let $f : \text{Time} \to \mathbb{R}^n$ be a differentiable function from time into an $n$-dimensional Euclidean space. For any coordinate index $\mu$ and any time $t \in \text{Time}$, the derivative of the $\mu$-th component function $t \mapsto f(t)_\mu$ is equal to the $\mu$-th component of the derivative of $f$. That is, $$\frac{d}{dt}(f(t)_\mu) = \left( \frac{df}{dt}(t) \right)_\mu$$

Let $f: \text{Time} \to \mathbb{R}^n$ be a differentiable function into an $n$-dimensional Euclidean space. For any coordinate index $\mu$ and time values $t, \Delta t \in \text{Time}$, the Frechet derivative of the component function $t \mapsto f(t)_\mu$ at $t$ applied to the increment $\Delta t$ is equal to the $\mu$-th component of the Frechet derivative of $f$ at $t$ applied to $\Delta t$. That is, $$D(f_\mu)(t)(\Delta t) = (Df(t)(\Delta t))_\mu$$ where $D$ denotes the Frechet derivative over the real numbers.

Let $f : \text{Time} \to \text{Vector}_d$ be a differentiable function from the time domain to the space of $d$-dimensional Lorentz vectors. For any time $t \in \text{Time}$ and any coordinate index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the time derivative of the $i$-th component function $t \mapsto f(t)_i$ is equal to the $i$-th component of the time derivative of $f$ at $t$. That is, $$\frac{d}{dt} (f(t)_i) = \left( \frac{df}{dt}(t) \right)_i$$ where $\frac{d}{dt}$ denotes the time derivative `Time.deriv`.

Let $c, g : \text{Time} \to \mathbb{R}$ be real-valued functions of time. If $c$ and $g$ are differentiable at $t \in \text{Time}$ and $g(t) \neq 0$, then the time derivative of their quotient at $t$ is given by: $$\partial_t \left( \frac{c}{g} \right)(t) = \frac{(\partial_t c)(t) \cdot g(t) - c(t) \cdot (\partial_t g)(t)}{(g(t))^2}$$ where $\partial_t$ denotes the time derivative operator `Time.deriv`.

Let $M$ be a real normed vector space and $s$ be a finite set of indices. If for every $i \in s$, the function $a_i : \text{Time} \to M$ is differentiable at a time $t$, then the time derivative of the sum of these functions at $t$ is equal to the sum of their individual time derivatives at $t$: $$\frac{d}{dt} \left( \sum_{i \in s} a_i(t) \right) = \sum_{i \in s} \frac{d}{dt} a_i(t)$$ where $\frac{d}{dt}$ denotes the time derivative (defined as the Fréchet derivative at $t$ evaluated at the unit time).

The function $\text{Time.val} : \text{Time} \to \mathbb{R}$, which maps a time value to its underlying real-number representation, is $n$-times continuously differentiable for any $n \in \mathbb{N} \cup \{\infty\}$.

Let $P$ be an affine space (torsor) modeled on a real normed vector space $V$. For any function $f : \text{Time} \to P$, any reference point $p \in P$, and any time $t \in \text{Time}$, the time derivative $\partial_{t,v} f(t)$ is equal to the Fréchet derivative of the displacement function $s \mapsto f(s) - p$ at $t$ evaluated at the unit time $1$: $$\partial_{t,v} f(t) = \text{D}(s \mapsto f(s) - p)(t) \cdot 1$$ This identity shows that the derivative of a trajectory into a torsor is independent of the reference point $p$ used to define the displacement.

Let $V$ be a real topological vector space and $P$ be an affine space modeled on $V$. For any function $f : \text{Time} \to P$ and any time $t \in \text{Time}$, the derivative vector $\text{Time.derivVec} f t$ is equal to the Fréchet derivative of the displacement function $s \mapsto f(s) - f(t)$ evaluated at $t$ and applied to the unit scalar $1$.

Let $M$ be a real normed space and let $t \in \text{Time}$. If the functions $f, g : \text{Time} \to M$ are differentiable at $t$, then the time derivative of their sum at $t$ is equal to the sum of their time derivatives at $t$: $$\text{deriv}(f + g)(t) = \text{deriv } f(t) + \text{deriv } g(t)$$

For a function $f : \text{Time} \to \mathbb{R}$ and a constant $c \in \mathbb{R}$, if $f$ is differentiable at a time $t$, then the time derivative of the function $s \mapsto f(s) \cdot c$ at $t$ is equal to the time derivative of $f$ at $t$ multiplied by $c$. That is, $$\frac{d}{dt}(f(t) \cdot c) = \left( \frac{d}{dt} f(t) \right) \cdot c$$ where $\frac{d}{dt}$ denotes the time derivative operator `Time.deriv`.

Let $M$ be a normed real vector space and $f : \text{Time} \to M$ be a function. If $f$ is differentiable at $-t$ for some $t \in \text{Time}$, then the time derivative of the time-reversed function $s \mapsto f(-s)$ at time $t$ is equal to the negative of the time derivative of $f$ at time $-t$. That is, $$\partial_t (f(-s)) \big|_{s=t} = - \partial_t f(-t)$$ where $\partial_t$ denotes the time derivative operator `Time.deriv`.

Let $d$ be a natural number and $f: \text{Time} \to \text{Space } d$ be a differentiable function. For any $t \in \text{Time}$ and index $i \in \{0, \dots, d-1\}$, the derivative of the $i$-th component function $s \mapsto f(s)_i$ at time $t$ is equal to the $i$-th component of the derivative of $f$ at $t$. That is: $$\text{deriv}\left(\lambda s, f(s)_i\right)(t) = \left(\text{deriv } f(t)\right)_i$$

Let $V$ be a topological vector space over $\mathbb{R}$ and $P$ be an affine space (torsor) modeled on $V$. For any point $p \in P$, the derivative of the constant trajectory $s \mapsto p$ at any time $t$ is the zero vector in $V$. That is: $$\partial_{t} p = 0$$

Let $M$ be a real normed space and $f : \text{Time} \to M$ be a function. If $f$ is $C^{n+1}$ for some $n \in \mathbb{N} \cup \{\infty\}$, then its time derivative $\partial_t f$ is a $C^n$ function.

Let $M$ be a real normed space and $f : \text{Time} \to M$ be a function. If $f$ is twice continuously differentiable ($C^2$), then its time derivative $\partial_t f$ is differentiable.

Let $M$ be a real normed space and $f : \text{Time} \to M$ be a smooth ($C^\infty$) function. For any $t \in \text{Time}$, the second time derivative of the time-reversed curve $s \mapsto f(-s)$ evaluated at $t$ is equal to the second time derivative of $f$ evaluated at $-t$. Mathematically, this is expressed as: $$\partial_t (\partial_t (s \mapsto f(-s)))(t) = \partial_t (\partial_t f)(-t)$$ where $\partial_t$ denotes the time derivative operator.

Let $\iota$ be a finite index set. Let $r : \text{Time} \to \mathbb{R}^\iota$ be a curve in Euclidean space and $f : \mathbb{R} \to \mathbb{R}$ be a real-valued function. For a given time $t \in \text{Time}$ and a coordinate index $i \in \iota$, suppose that $r$ is differentiable at $t$ and $f$ has a derivative $f'$ at the $i$-th coordinate of the curve at that time, $r(t)_i$. Then the time derivative of the composite function $s \mapsto f(r(s)_i)$ at time $t$ is given by: $$\partial_t (f(r(s)_i))|_{s=t} = f' \cdot (\partial_t r(t))_i$$ where $\partial_t r(t)_i$ denotes the $i$-th component of the velocity (time derivative) of the curve $r$ at time $t$.

Let $d$ be a natural number and $f : \text{Time} \to \text{Space } d$ be a differentiable trajectory. For any time $t \in \text{Time}$ and any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the torsor derivative of $f$ at $t$ is equal to the time derivative of the $i$-th coordinate function of $f$ evaluated at $t$. This can be expressed as: $$(\partial_{tv} f(t))_i = \frac{d}{dt} (f(t)_i)$$ where $\partial_{tv}$ denotes the torsor derivative (the derivative of a map into an affine space), and $f(t)_i$ represents the $i$-th coordinate of the position in $\text{Space } d$.

Let $d$ be a natural number and $f: \text{Time} \to \text{Space } d$ be a differentiable function. For any time $t \in \text{Time}$, the torsor derivative $\partial_t^v f(t)$ (the derivative of $f$ viewed as a trajectory in an affine space) is equal to the displacement vector from the origin $0 \in \text{Space } d$ to the vector space derivative $\partial_t f(t)$ (the derivative of $f$ viewed as a trajectory in an additive commutative group). That is, $$\partial_t^v f(t) = \partial_t f(t) \ominus 0$$ where $\ominus$ (or $-_v$) denotes the vector subtraction operation that produces a displacement vector from two points in $\text{Space } d$.