Physlib.SpaceAndTime.Space.Derivatives.Curl

For a vector field $\mathbf{f} : \text{Space } 3 \to \mathbb{R}^3$, the curl operator $\nabla \times \mathbf{f}$ is defined as the function mapping each point $x$ to a vector in $\mathbb{R}^3$ whose components are: $$(\nabla \times \mathbf{f})_0 = \frac{\partial f_2}{\partial x_1} - \frac{\partial f_1}{\partial x_2}$$ $$(\nabla \times \mathbf{f})_1 = \frac{\partial f_0}{\partial x_2} - \frac{\partial f_2}{\partial x_0}$$ $$(\nabla \times \mathbf{f})_2 = \frac{\partial f_1}{\partial x_0} - \frac{\partial f_0}{\partial x_1}$$ where $f_i$ denotes the $i$-th component of the vector field $\mathbf{f}$, and $\frac{\partial}{\partial x_j}$ denotes the partial derivative with respect to the $j$-th coordinate (using 0-based indexing for $i, j \in \{0, 1, 2\}$).

The notation $\nabla \times \mathbf{f}$ is defined to represent the curl of a function $\mathbf{f}$, where $\nabla \times$ denotes the curl operator `curl`.

The curl of the zero vector field $\mathbf{f} : \text{Space } 3 \to \mathbb{R}^3$, where $\mathbf{f}(\mathbf{x}) = \mathbf{0}$ for all $\mathbf{x}$, is equal to the zero vector field. This is denoted as $\nabla \times \mathbf{0} = \mathbf{0}$.

For any constant vector $\mathbf{v} \in \mathbb{R}^3$, let $\mathbf{f}: \text{Space } 3 \to \mathbb{R}^3$ be the constant vector field defined by $\mathbf{f}(\mathbf{x}) = \mathbf{v}$ for all $\mathbf{x} \in \text{Space } 3$. Then the curl of $\mathbf{f}$ is zero, denoted as $\nabla \times \mathbf{f} = 0$.

Let $\mathbf{f}_1, \mathbf{f}_2: \text{Space } 3 \to \mathbb{R}^3$ be two differentiable vector fields. The curl of the sum of these vector fields is equal to the sum of their individual curls: $$\nabla \times (\mathbf{f}_1 + \mathbf{f}_2) = \nabla \times \mathbf{f}_1 + \nabla \times \mathbf{f}_2$$

Let $\mathbf{f} : \text{Space } 3 \to \mathbb{R}^3$ be a differentiable vector field and $k \in \mathbb{R}$ be a scalar constant. The curl of the vector field scaled by $k$ is equal to $k$ times the curl of $\mathbf{f}$: $$\nabla \times (k \mathbf{f}) = k (\nabla \times \mathbf{f})$$ where $\nabla \times$ denotes the curl operator and $k \mathbf{f}$ is the scalar multiplication of the vector field.

Let $\mathbf{f} : \text{Space } 3 \to \mathbb{R}^3$ be a differentiable vector field. The curl of the negative of the vector field is equal to the negative of the curl of $\mathbf{f}$: $$\nabla \times (-\mathbf{f}) = -(\nabla \times \mathbf{f})$$ where $\nabla \times$ denotes the curl operator.

Let $\mathbf{f}_1, \mathbf{f}_2: \text{Space } 3 \to \mathbb{R}^3$ be two differentiable vector fields. The curl of the difference of these vector fields is equal to the difference of their individual curls: $$\nabla \times (\mathbf{f}_1 - \mathbf{f}_2) = \nabla \times \mathbf{f}_1 - \nabla \times \mathbf{f}_2$$ where $\nabla \times$ denotes the curl operator.

Let $W$ be a vector space over $\mathbb{R}$. Suppose $\mathbf{f} : W \to (\text{Space } 3 \to \mathbb{R}^3)$ is a linear map such that for every $w \in W$, the vector field $\mathbf{f}(w)$ is differentiable. Then the mapping $w \mapsto \nabla \times \mathbf{f}(w)$, which assigns each $w$ to the curl of the vector field $\mathbf{f}(w)$, is also a linear map over $\mathbb{R}$.

Let $f: \text{Space } 3 \to \mathbb{R}^3$ be a twice continuously differentiable ($C^2$) function. Let $f_k$ denote the $k$-th component of the function $f$ and $\partial_k$ denote the partial derivative with respect to the $k$-th coordinate. For any indices $i, u, v, w \in \{0, 1, 2\}$, the second-order partial derivatives distribute over the sum of the coordinate-wise derivatives: $$\partial_i \left( \partial_u f_u + \partial_v f_v + \partial_w f_w \right) = \partial_i \partial_u f_u + \partial_i \partial_v f_v + \partial_i \partial_w f_w$$

Let $f: \text{Space } 3 \to \mathbb{R}^3$ be a twice continuously differentiable ($C^2$) vector field. For any coordinate indices $u, v, w \in \{0, 1, 2\}$, the partial derivative with respect to the $u$-th coordinate of the difference between the partial derivatives $\partial_v f_w$ and $\partial_w f_v$ is equal to the difference of the second-order partial derivatives: $$\partial_u (\partial_v f_w - \partial_w f_v) = \partial_u \partial_v f_w - \partial_u \partial_w f_v$$ where $\partial_i$ denotes the spatial derivative in the direction of the $i$-th standard basis vector and $f_i$ denotes the $i$-th component of the vector field $f$. This identity demonstrates that second-order partial derivatives distribute coordinate-wise over the subtraction of terms typically found in the components of a curl.

For any twice continuously differentiable ($C^2$) vector field $\mathbf{f} : \text{Space } 3 \to \mathbb{R}^3$, the divergence of its curl is identically zero, i.e., $\nabla \cdot (\nabla \times \mathbf{f}) = 0$.

Let $f: \text{Space } 3 \to \mathbb{R}$ be a scalar field that is twice continuously differentiable ($C^2$). The curl of its gradient is the zero vector field: $$\nabla \times (\nabla f) = 0$$ where $\nabla f$ denotes the gradient of $f$ and $\nabla \times$ denotes the curl operator.

Let $\mathbf{f} : \mathbb{R}^3 \to \mathbb{R}^3$ be a twice continuously differentiable ($C^2$) vector field. The curl of the curl of $\mathbf{f}$ is equal to the gradient of the divergence of $\mathbf{f}$ minus the vector Laplacian of $\mathbf{f}$: $$\nabla \times (\nabla \times \mathbf{f}) = \nabla(\nabla \cdot \mathbf{f}) - \Delta \mathbf{f}$$ where $\nabla \times$ denotes the curl operator, $\nabla \cdot$ denotes the divergence, $\nabla$ denotes the gradient operator, and $\Delta$ denotes the vector Laplacian.

The linear map `distCurl` computes the curl of a distribution $f \in \text{Space } 3 \to d[\mathbb{R}] \mathbb{R}^3$. It is defined as the composition of the distributional Fréchet derivative $\mathcal{D}f$ with a linear operator that extracts the curl components. For a distribution $f = (f_0, f_1, f_2)$ mapping the 3-dimensional Euclidean space to $\mathbb{R}^3$, the components of the resulting distribution $\nabla \times f$ are given by: \begin{align*} (\nabla \times f)_0 &= \partial_2 f_1 - \partial_1 f_2 \\ (\nabla \times f)_1 &= \partial_0 f_2 - \partial_2 f_0 \\ (\nabla \times f)_2 &= \partial_1 f_0 - \partial_0 f_1 \end{align*} where $\partial_i$ denotes the distributional partial derivative with respect to the $i$-th coordinate of $\mathbb{R}^3$.

Let $f$ be an $\mathbb{R}^3$-valued distribution on the 3-dimensional Euclidean space $\mathbb{R}^3$, and let $\eta \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$ be a Schwartz test function. Let $\{e_0, e_1, e_2\}$ be the standard orthonormal basis of $\mathbb{R}^3$. The evaluation of the $0$-th component (index 0) of the distributional curl $\nabla \times f$ at the test function $\eta$ is given by $$((\nabla \times f) \eta)_0 = -f(\partial_{e_2} \eta)_1 + f(\partial_{e_1} \eta)_2$$ where $\partial_{e_i} \eta$ denotes the directional derivative of the test function $\eta$ in the direction of the basis vector $e_i$, and $f(\phi)_j$ represents the $j$-th component of the vector obtained by applying the distribution $f$ to a test function $\phi$.

Let $f$ be an $\mathbb{R}^3$-valued distribution on $\mathbb{R}^3$ and $\eta \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$ be a Schwartz test function. The evaluation of the $1$-st component (index 1) of the distributional curl $\nabla \times f$ at $\eta$ is given by $$((\nabla \times f) \eta)_1 = -f(\partial_0 \eta)_2 + f(\partial_2 \eta)_0$$ where $\partial_i \eta$ denotes the partial derivative of $\eta$ with respect to the $i$-th coordinate, and $f(\phi)_j$ represents the $j$-th component of the vector result obtained by applying the distribution $f$ to a test function $\phi$.

Let $f: \mathbb{R}^3 \to d[\mathbb{R}] \mathbb{R}^3$ be a distribution on the 3-dimensional Euclidean space and $\eta \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$ be a Schwartz test function. Let $e_0, e_1, e_2$ be the standard orthonormal basis of $\mathbb{R}^3$. The third component (index 2) of the distributional curl $\nabla \times f$ applied to the test function $\eta$ is given by $$((\nabla \times f) \eta)_2 = -f(\partial_{e_1} \eta)_0 + f(\partial_{e_0} \eta)_1$$ where $\partial_{e_i} \eta$ denotes the directional derivative of $\eta$ in the direction of the basis vector $e_i$, and $f(\cdot)_i$ denotes the $i$-th component of the vector-valued distribution applied to a test function.

Let $f$ be an $\mathbb{R}^3$-valued distribution on the 3-dimensional Euclidean space $\mathbb{R}^3$, and let $\eta \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$ be a Schwartz test function. The value of the distributional curl $(\nabla \times f)$ applied to $\eta$ is a vector in $\mathbb{R}^3$ whose components are given by: \begin{align*} ((\nabla \times f) \eta)_0 &= -f(\partial_{2} \eta)_1 + f(\partial_{1} \eta)_2 \\ ((\nabla \times f) \eta)_1 &= -f(\partial_{0} \eta)_2 + f(\partial_{2} \eta)_0 \\ ((\nabla \times f) \eta)_2 &= -f(\partial_{1} \eta)_0 + f(\partial_{0} \eta)_1 \end{align*} where $\partial_i \eta$ denotes the partial derivative of the test function $\eta$ with respect to the $i$-th coordinate, and $f(\phi)_j$ denotes the $j$-th component of the vector obtained by applying the distribution $f$ to a test function $\phi$.

Let $f$ be a real-valued distribution on the 3-dimensional Euclidean space $\mathbb{R}^3$. The distributional curl of the distributional gradient of $f$ is the zero distribution, denoted by: $$\nabla \times (\nabla f) = 0$$ where $\nabla f$ is the distributional gradient and $\nabla \times$ is the distributional curl operator.