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Vector Calculus

Gradient¶

Cartesian¶

\begin{matrix} \nabla V = ⟨ \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} ⟩ V \end{matrix}

cylindrical¶

\begin{matrix} \nabla V = ⟨ \frac{\partial}{\partial ρ}, \frac{1}{ρ} \frac{\partial}{\partial ϕ}, \frac{\partial}{\partial z} ⟩ V \end{matrix}

spherical¶

\begin{matrix} \nabla V = ⟨ \frac{\partial}{\partial r}, \frac{1}{r} \frac{\partial}{\partial θ}, \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} ⟩ V \end{matrix}

Divergence¶

Definition¶

hint: Stoke's Theorem

\begin{matrix} \nabla \cdot \vec{A} \equiv lim_{v \to 0} \frac{\oint_{S} \vec{A} \cdot d \vec{S}}{\int_{v} d v} \end{matrix}

Physical Interpretation¶

flux per unit volume
measure of outgoingness of vector (發散傾向)
$\nabla \cdot \vec{A} > 0$ (source)
$\nabla \cdot \vec{A} < 0$ (sink)

proof: https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

Cartesian¶

\begin{matrix} \nabla \cdot \vec{D} = \frac{\partial}{\partial x} D_{x} + \frac{\partial}{\partial y} D_{y} + \frac{\partial}{\partial z} D_{z} \end{matrix}

cylindrical¶

\begin{matrix} \nabla \cdot \vec{D} = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ D_{ρ}) + \frac{1}{ρ} \frac{\partial}{\partial ϕ} D_{ϕ} + \frac{\partial}{\partial z} D_{z} \end{matrix}

spherical¶

\begin{matrix} \nabla \cdot \vec{D} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} D_{r}) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (D_{θ} \sin θ) + \frac{1}{r \sin θ} \frac{\partial}{\partial ϕ} D_{ϕ} \end{matrix}

Laplacian¶

\begin{matrix} \nabla^{2} V \equiv \nabla \cdot (\nabla V) \\ \nabla^{2} \vec{A} \equiv \nabla (\nabla \cdot \vec{A}) - \nabla \times \nabla \times \vec{A} \end{matrix}

cylindrical

\begin{matrix} \nabla^{2} V = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ V_{ρ}) + \frac{1}{ρ^{2}} V_{ϕ ϕ} + V_{z z} \end{matrix}

spherical

\begin{matrix} \nabla^{2} V = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} V_{r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial ϕ} (\sin θ V_{θ}) + \frac{1}{r^{2} \sin^{2} θ} (V_{ϕ ϕ}) \end{matrix}

Curl¶

Definition

\begin{matrix} (\nabla \times \vec{A}) \cdot \hat{a_{n}} = lim_{Δ S \to 0} \frac{\oint_{c} \vec{A} \cdot d \vec{l}}{Δ S} \end{matrix}

Cartesian

\nabla \times \vec{A} = | \begin{matrix} \hat{a_{x}} & \hat{a_{y}} & \hat{a_{z}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{matrix} |

cylindrical

\nabla \times \vec{A} = \frac{1}{ρ} | \begin{matrix} \hat{a_{ρ}} & ρ \hat{a_{ϕ}} & \hat{a_{z}} \\ \frac{\partial}{\partial ρ} & \frac{\partial}{\partial ϕ} & \frac{\partial}{\partial z} \\ A_{ρ} & ρ A_{ϕ} & A_{z} \end{matrix} |

spherical

\nabla \times \vec{A} = \frac{1}{r^{2} \sin θ} | \begin{matrix} \hat{a_{r}} & r \hat{a_{θ}} & r \sin θ \hat{a_{ϕ}} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial ϕ} \\ A_{r} & r A_{θ} & r \sin θ A_{ϕ} \end{matrix} |

Identities¶

\begin{matrix} \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \vec{B} - (\vec{A} \cdot \vec{B}) \vec{C} \\ \nabla \cdot (\vec{A} \times \vec{B}) = (\nabla \times \vec{A}) \cdot \vec{B} - (\nabla \times \vec{B}) \cdot \vec{A} \\ \nabla \times (\vec{A} \times \vec{B}) = \vec{A} (\nabla \cdot \vec{B}) - \vec{B} (\nabla \cdot \vec{A}) + (\vec{B} \cdot \nabla) \vec{A} - (\vec{A} \cdot \nabla) \vec{B} \end{matrix}