Physlib.SpaceAndTime.Space.Derivatives.Basic

Given a function $f: \text{Space } d \to M$, where $M$ is a real topological vector space, and a basis index $\mu \in \{0, \dots, d-1\}$, the spatial derivative is the function that maps each point $x \in \text{Space } d$ to the Fréchet derivative of $f$ at $x$ evaluated in the direction of the $\mu$-th standard basis vector $e_\mu$. This definition corresponds to the partial derivative of $f$ with respect to the $\mu$-th coordinate, denoted as $\frac{\partial f}{\partial x_\mu}$ or $\partial_\mu f$.

This definition introduces the mathematical notation $\partial[i]$ as a shorthand for the spatial derivative operator `Space.deriv` in the direction of the $i$-th basis vector. Given a function $f: \text{Space } d \to M$ and an index $i \in \{0, \dots, d-1\}$, the expression $\partial[i] f$ denotes the partial derivative of $f$ with respect to the $i$-th coordinate direction.

Let $M$ be a real topological vector space and $f: \text{Space } d \to M$ be a function. For any basis index $\mu \in \{0, \dots, d-1\}$ and any point $x \in \text{Space } d$, the spatial derivative of $f$ at $x$ in the direction of the $\mu$-th standard basis vector is equal to the Fréchet derivative of $f$ at $x$ evaluated at the $\mu$-th basis vector $e_\mu$: \[ \text{deriv } \mu \, f(x) = \text{D}f(x)(e_\mu) \] where $\text{D}f(x)$ denotes the Fréchet derivative of $f$ at $x$, and $e_\mu$ is the $\mu$-th vector of the standard orthonormal basis of $\text{Space } d$.

Let $M$ be a real topological vector space and $f: \text{Space } d \to M$ be a function. For any index $\mu \in \{0, \dots, d-1\}$, the spatial derivative $\partial_\mu f$ is the function that maps each point $x \in \text{Space } d$ to the Fréchet derivative of $f$ at $x$ evaluated in the direction of the $\mu$-th standard basis vector $e_\mu$. Mathematically, $\partial_\mu f = \lambda x, Df(x) \cdot e_\mu$, where $Df(x)$ denotes the Fréchet derivative of $f$ at $x$ and $e_\mu$ is the $\mu$-th standard orthonormal basis vector of $\text{Space } d$.

Let $M$ be a real topological vector space and $f: \text{Space } d \to M$ be a function. For any point $x \in \text{Space } d$ and any basis index $\mu \in \{0, \dots, d-1\}$, the spatial derivative of $f$ in the direction $\mu$ at $x$, denoted $\partial_\mu f(x)$, is equal to the Fréchet derivative of $f$ at $x$ evaluated at the $\mu$-th standard basis vector $e_\mu$.

Let $M$ be a real topological vector space and $f: \text{Space } d \to M$ be a function. For any points $x, y \in \text{Space } d$, the Fréchet derivative of $f$ at $x$ evaluated in the direction $y$, denoted $Df(x)(y)$, is equal to the sum over all basis indices $i \in \{0, \dots, d-1\}$ of the $i$-th coordinate of $y$ scaled by the spatial derivative of $f$ at $x$ in the $i$-th direction: \[ Df(x)(y) = \sum_{i=0}^{d-1} y_i \cdot \partial_i f(x) \] where $y_i$ is the $i$-th component of $y$ and $\partial_i f(x)$ is the partial derivative of $f$ at $x$ with respect to the $i$-th standard basis vector.

Let $M$ be a real normed space and $f: \text{Space } d \to M$ be a function. For any point $x \in \text{Space } d$ and any basis index $\mu \in \{0, \dots, d-1\}$, the spatial derivative of $f$ in the direction $\mu$ at $x$, denoted $\partial_\mu f(x)$, is equal to the manifold derivative of $f$ at $x$ (with respect to the standard model spaces $\mathbb{R}^d$ and $M$) evaluated at the $\mu$-th standard basis vector $e_\mu$.

Let $d$ be a dimension and $M$ be a real normed vector space. For a function $f : \text{Space } d \to M$ and a point $x \in \text{Space } d$, $f$ is manifold-differentiable at $x$ with respect to the specific manifold structure `manifoldStructure d` on the domain and the canonical manifold structure $\mathcal{I}(\mathbb{R}, M)$ on the codomain if and only if $f$ is Fréchet differentiable at $x$ in the standard sense of real normed spaces.

Let $M$ be a real normed space and $f: \text{Space } d \to M$ be a function. For any point $x \in \text{Space } d$ and any coordinate index $\mu \in \{0, \dots, d-1\}$, the spatial derivative of $f$ at $x$ in the direction $\mu$ is equal to the manifold Fréchet derivative of $f$ at $x$ applied to the $\mu$-th standard basis vector $e_\mu \in \mathbb{R}^d$. This derivative is taken with respect to the specific manifold structure `Space.manifoldStructure d` on the domain and the canonical manifold structure $\mathcal{I}(\mathbb{R}, M)$ on the codomain. Mathematically: \[ \text{deriv } \mu \, f(x) = (df)_x(e_\mu) \] where $(df)_x$ is the manifold derivative at $x$, and $e_\mu$ is the vector in the Euclidean space $\mathbb{R}^d$ with $1$ at the $\mu$-th coordinate and $0$ elsewhere.

Let $M$ be a real normed space and $d$ be a natural number representing the dimension of the domain $\text{Space } d$. For any constant value $m \in M$ and any basis index $\mu \in \{0, \dots, d-1\}$, let $f: \text{Space } d \to M$ be the constant function $f(x) = m$. The spatial derivative of $f$ in the direction of the $\mu$-th basis vector at any point $t \in \text{Space } d$ is zero, denoted as $\partial_\mu f(t) = 0$.

Let $M$ be a real normed space and $d$ be a natural number. Let $f_1, f_2: \text{Space } d \to M$ be two functions that are differentiable on $\text{Space } d$. For any basis index $u \in \{0, \dots, d-1\}$, the spatial derivative of the sum of the functions is equal to the sum of their individual spatial derivatives: \[ \partial_u (f_1 + f_2) = \partial_u f_1 + \partial_u f_2 \] where $\partial_u$ denotes the partial derivative with respect to the $u$-th coordinate (the Fréchet derivative evaluated in the direction of the $u$-th standard basis vector).

Let $f_1, f_2: \text{Space } d \to \mathbb{R}^d$ be two differentiable functions. For any basis index $u \in \{0, \dots, d-1\}$ of the domain and any coordinate index $i$ of the codomain $\mathbb{R}^d$, the spatial derivative $\partial_u$ of the sum of the $i$-th components of $f_1$ and $f_2$ is equal to the sum of the spatial derivatives of the individual $i$-th components: \[ \partial_u (f_1(x)_i + f_2(x)_i) = \partial_u (f_1(x)_i) + \partial_u (f_2(x)_i) \] where $f(x)_i$ denotes the $i$-th component of the vector $f(x)$.

Let $M$ be a real normed space and $\mathbb{k}$ be a normed algebra over $\mathbb{R}$. For functions $c: \text{Space } d \to \mathbb{k}$ and $f: \text{Space } d \to M$ that are differentiable at a point $x \in \text{Space } d$, the spatial derivative in direction $u \in \{0, \dots, d-1\}$ of their scalar product satisfies the Leibniz rule: \[ \partial_u (c \cdot f)(x) = c(x) \cdot \partial_u f(x) + (\partial_u c(x)) \cdot f(x) \] where $\partial_u$ denotes the partial derivative with respect to the $u$-th standard basis vector.

Let $M$ be a real normed space and $R$ be a semiring acting on $M$ (such that the action commutes with the real scalar multiplication and constant scalar multiplication is continuous). Let $f: \text{Space } d \to M$ be a differentiable function and $c \in R$ be a constant. For any index $u \in \{0, \dots, d-1\}$, the spatial derivative in direction $u$ (denoted by $\partial_u$) satisfies: \[ \partial_u (c \cdot f) = c \cdot \partial_u f \] where $\partial_u f$ is the derivative of $f$ in the direction of the $u$-th standard orthonormal basis vector of $\text{Space } d$.

Let $f: \text{Space } d \to \mathbb{R}^d$ be a differentiable function and $k \in \mathbb{R}$ be a scalar constant. For any point $x \in \text{Space } d$, coordinate index $i \in \{0, \dots, d-1\}$, and basis index $u \in \{0, \dots, d-1\}$, the spatial derivative in direction $u$ (denoted by $\partial_u$) of the $i$-th component of $f$ scaled by $k$ satisfies: \[ \partial_u (x \mapsto k \cdot f_i(x)) = k \cdot \partial_u f_i(x) \] where $f_i(x)$ denotes the $i$-th coordinate of the vector $f(x)$.

Let $M$ be a real normed space and $f: \text{Space } d \to M$ be a function. If $f$ is twice continuously differentiable (of class $C^2$), then for any indices $u, v \in \{0, \dots, d-1\}$, the spatial derivatives in those directions commute: \[ \partial_u (\partial_v f) = \partial_v (\partial_u f) \] where $\partial_\mu f$ denotes the derivative of $f$ in the direction of the $\mu$-th standard orthonormal basis vector of $\text{Space } d$.

For any dimension $d$, given an index $\mu \in \{0, \dots, d-1\}$ and a point $x$ in the space $\text{Space } d$, the spatial derivative of the $\mu$-th coordinate function $f(x) = x_\mu$ in the direction of the $\mu$-th standard basis vector is equal to 1: \[ \partial_\mu (x \mapsto x_\mu) = 1 \] where $x_\mu$ denotes the $\mu$-th component of the vector $x$.

For a given dimension $d$, let $\mu$ and $\nu$ be distinct indices in $\{0, 1, \dots, d-1\}$ (i.e., $\mu \neq \nu$). For any point $x$ in $\text{Space } d$, the spatial derivative of the $\nu$-th coordinate function with respect to the $\mu$-th direction is zero: \[ \partial_\mu (x \mapsto x_\nu) = 0 \] where $x_\nu$ denotes the $\nu$-th component of $x$, and $\partial_\mu$ denotes the derivative in the direction of the $\mu$-th standard orthonormal basis vector $e_\mu$.

For a given dimension $d$, let $\mu$ and $\nu$ be indices in $\{0, 1, \dots, d-1\}$. For any point $x$ in $\text{Space } d$, the spatial derivative of the $\mu$-th coordinate function with respect to the $\nu$-th direction is given by: \[ \partial_\nu (x \mapsto x_\mu) = \begin{cases} 1 & \text{if } \nu = \mu \\ 0 & \text{if } \nu \neq \mu \end{cases} \] where $x_\mu$ denotes the $\mu$-th component of the vector $x$, and $\partial_\nu$ denotes the derivative in the direction of the $\nu$-th standard basis vector.

In a $d$-dimensional space $\text{Space } d$, let $\mu$ and $\nu$ be indices in $\{0, 1, \dots, d-1\}$. For any point $x \in \text{Space } d$, the spatial derivative of the square of the $\mu$-th coordinate function in the direction of the $\nu$-th standard basis vector is given by: \[ \partial_\nu (x \mapsto x_\mu^2) = \begin{cases} 2x_\mu & \text{if } \nu = \mu \\ 0 & \text{if } \nu \neq \mu \end{cases} \] where $x_\mu$ denotes the $\mu$-th component of the vector $x$, and $\partial_\nu$ denotes the derivative evaluated in the direction of the $\nu$-th standard basis vector $e_\nu$.

Let $f: \text{Space } d \to \mathbb{R}^n$ be a differentiable function, where the codomain is the $n$-dimensional Euclidean space. For any point $x \in \text{Space } d$, any domain basis index $\nu \in \{0, \dots, d-1\}$, and any codomain component index $\mu \in \{0, \dots, n-1\}$, the spatial derivative of the $\mu$-th component of $f$ in the direction $\nu$ is equal to the $\mu$-th component of the spatial derivative of $f$ in the direction $\nu$. That is, \[ \partial_\nu (x \mapsto f(x)_\mu) = (\partial_\nu f)_\mu \]

Let $f: \text{Space } d \to \text{Lorentz.Vector } d$ be a differentiable function. For any spatial coordinate index $\nu \in \{0, \dots, d-1\}$ and any Lorentz vector component index $\mu$, the spatial derivative of the $\mu$-th component of $f$ at a point $x$ is equal to the $\mu$-th component of the spatial derivative of $f$ at $x$. Mathematically, $$\partial_\nu (f_\mu)(x) = (\partial_\nu f(x))_\mu$$ where $\partial_\nu$ denotes the derivative with respect to the $\nu$-th standard basis vector of $\text{Space } d$.

In the $d$-dimensional real vector space $\text{Space } d$, the function mapping each element $x$ to the square of its Euclidean norm, $x \mapsto \|x\|^2$, is differentiable over $\mathbb{R}$.

In the $d$-dimensional real inner product space $\text{Space } d$, for any point $x \in \text{Space } d$ and any basis index $i \in \{0, 1, \dots, d-1\}$, the spatial derivative of the squared Euclidean norm function $x \mapsto \|x\|^2$ in the direction of the $i$-th basis vector at $x$ is equal to $2x_i$. Mathematically, \[ \partial_i (\|x\|^2) = 2x_i \] where $x_i$ denotes the $i$-th component of the vector $x$, and $\partial_i$ denotes the spatial derivative in the direction of the $i$-th standard basis vector $e_i$.

For any dimension $d \in \mathbb{N}$, the function $y \mapsto \langle y, y \rangle$ mapping an element of the $d$-dimensional real inner product space $\text{Space } d$ to its inner product with itself is differentiable over the real numbers $\mathbb{R}$.

For any dimension $d \in \mathbb{N}$ and any element $x$ of the $d$-dimensional real inner product space $\text{Space } d$, the function mapping $y \in \text{Space } d$ to the inner product $\langle y, y \rangle$ is differentiable at $x$ over the real numbers $\mathbb{R}$.

Let $M$ be a real normed space and $d$ be a natural number. Suppose $f, g: M \to \text{Space } d$ are functions. If $f$ and $g$ are differentiable at a point $x \in M$, then the scalar-valued function mapping $y \in M$ to the inner product $\langle f(y), g(y) \rangle$ is differentiable at $x$.

Let $M$ be a real normed space and $d$ be a natural number. If the functions $f: M \to \text{Space } d$ and $g: M \to \text{Space } d$ are differentiable, then the function mapping each $y \in M$ to the inner product $\langle f(y), g(y) \rangle$ is also differentiable.

For any dimension $d$ and any $n \in \mathbb{N} \cup \{\infty\}$, the function on the $d$-dimensional space $\text{Space } d$ that maps a vector $y$ to its inner product with itself, $\langle y, y \rangle = \sum_{i=1}^d y_i^2$, is $n$-times continuously differentiable ($C^n$) over the real numbers $\mathbb{R}$.

Let $M$ be a real normed vector space and $d$ be a natural number. Let $\text{Space } d$ denote the $d$-dimensional real inner product space where the inner product of two vectors $p, q$ is given by $\langle p, q \rangle = \sum_{i} p_i q_i$. If $f: M \to \text{Space } d$ and $g: M \to \text{Space } d$ are $n$-times continuously differentiable functions (of class $C^n$) for $n \in \mathbb{N} \cup \{\infty\}$, then the mapping $y \mapsto \langle f(y), g(y) \rangle$ is also an $n$-times continuously differentiable function from $M$ to $\mathbb{R}$.

In the $d$-dimensional real inner product space $\text{Space } d$, for any point $x$ and any basis index $i \in \{0, 1, \dots, d-1\}$, the spatial derivative of the function $x \mapsto \langle x, x \rangle$ in the direction of the $i$-th standard basis vector at $x$ is equal to $2x_i$. Mathematically, \[ \partial_i \langle x, x \rangle = 2x_i \] where $\langle \cdot, \cdot \rangle$ denotes the real inner product and $x_i$ denotes the $i$-th component of the vector $x$.

For any dimension $d$, let $x_1$ and $x_2$ be vectors in the $d$-dimensional real inner product space $\text{Space } d$. For any basis index $i \in \{0, \dots, d-1\}$, the spatial derivative of the function $f(x) = \langle x, x_2 \rangle$ in the direction of the $i$-th standard basis vector, evaluated at the point $x_1$, is equal to the $i$-th component of $x_2$: $$\partial_i (\langle x, x_2 \rangle)|_{x=x_1} = (x_2)_i$$

Let $\text{Space } d$ be a $d$-dimensional real inner product space. For any vectors $x_1, x_2 \in \text{Space } d$ and any basis index $i \in \{0, \dots, d-1\}$, let $f: \text{Space } d \to \mathbb{R}$ be the function defined by $f(x) = \langle x_1, x \rangle$. The spatial derivative of $f$ in the direction of the $i$-th standard basis vector at the point $x_2$ is equal to the $i$-th component of $x_1$: $$\partial_i (\langle x_1, x \rangle)|_{x=x_2} = (x_1)_i$$

Let $M$ be a real normed space and $f: \text{Space } d \to M$ be a function. If $f$ is twice continuously differentiable ($C^2$), then for any index $i \in \{0, \dots, d-1\}$, the spatial derivative $\partial_i f$ is differentiable. Here, $\partial_i f$ denotes the partial derivative of $f$ in the direction of the $i$-th standard basis vector of $\text{Space } d$.

Let $f: \text{Space } d \to \mathbb{R}$ be a function. If $f$ is continuously differentiable of order $n+1$ ($C^{n+1}$), then the function that maps each point $x \in \text{Space } d$ to the collection of its spatial partial derivatives $(\partial_i f(x))_{i=0}^{d-1}$ is continuously differentiable of order $n$ ($C^n$). Here, $\text{Space } d$ is a $d$-dimensional real inner product space, and $\partial_i f$ denotes the derivative of $f$ in the direction of the $i$-th vector of the standard orthonormal basis.

Let $\text{Space } d$ be a $d$-dimensional real inner product space and $M$ be a real normed vector space. Given an index $\mu \in \{0, 1, \dots, d-1\}$, the operator $\text{distDeriv } \mu$ is an $\mathbb{R}$-linear map that sends a distribution $f: \text{Space } d \to d[\mathbb{R}] M$ to its partial derivative in the direction of the $\mu$-th standard basis vector $e_\mu$. This derivative is defined by composing the distributional Fréchet derivative $Df$ with the evaluation map at $e_\mu$, representing the directional derivative $\partial_\mu f$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space and $M$ be a real normed vector space. Let $\{e_0, e_1, \dots, e_{d-1}\}$ be the standard orthonormal basis of $\text{Space } d$. For a distribution $f : \text{Space } d \to d[\mathbb{R}] M$, an index $\mu \in \{0, 1, \dots, d-1\}$, and a Schwartz test function $\epsilon \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the evaluation of the distributional partial derivative $\partial_\mu f$ at $\epsilon$ is equal to the distributional Fréchet derivative $Df$ evaluated at $\epsilon$ and applied to the basis vector $e_\mu$: \[ (\partial_\mu f)(\epsilon) = (Df)(\epsilon)(e_\mu) \]

For a Schwartz function $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$ on a $d$-dimensional real inner product space, let $\{e_i\}$ be the standard orthonormal basis of $\text{Space } d$. For any point $x \in \text{Space } d$ and any indices $\mu, \nu \in \{0, \dots, d-1\}$, the mixed partial derivatives of $\eta$ commute: \[ \partial_\mu (\partial_\nu \eta)(x) = \partial_\nu (\partial_\mu \eta)(x) \] where $\partial_i$ denotes the directional derivative in the direction of the $i$-th basis vector $e_i$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space and $M$ be a real normed vector space. For any $M$-valued distribution $f: \text{Space } d \to d[\mathbb{R}] M$ and any indices $\mu, \nu \in \{0, 1, \dots, d-1\}$, the distributional partial derivatives with respect to the standard basis vectors commute: \[ \partial_\nu (\partial_\mu f) = \partial_\mu (\partial_\nu f) \] where $\partial_i$ denotes the operator `distDeriv i` acting on the space of distributions.