20260226_vector

\begin{aligned} \nabla u & := (u_{x}, u_{y}, u_{z}) & gradient \\ Δ u & := u_{x x} + u_{y y} + u_{z z} & Laplacian \\ \nabla \cdot F & := \partial_{x} F_{1} + \partial_{y} F_{2} + \partial_{z} F_{3} & divergence \\ \nabla \times F & := (\begin{array}{c} \partial_{y} F_{3} - \partial_{z} F_{2} \\ \partial_{z} F_{1} - \partial_{x} F_{3} \\ \partial_{x} F_{2} - \partial_{y} F_{1} \end{array}) & curl \\ D^{2} u & := (\begin{array}{c} u_{x x} & u_{x y} & u_{x z} \\ u_{y x} & u_{y y} & u_{y z} \\ u_{z x} & u_{z y} & u_{z z} \end{array}) & Hessian \\ D_{v} u & := \nabla u \cdot v & directional derivative \\ D F & := (\begin{array}{c} \partial_{x} F_{1} & \partial_{y} F_{1} & \partial_{z} F_{1} \\ \partial_{x} F_{2} & \partial_{y} F_{2} & \partial_{z} F_{2} \\ \partial_{x} F_{3} & \partial_{y} F_{3} & \partial_{z} F_{3} \end{array}) & Jacobian \\ \nabla \cdot (\nabla u) & = Δ u \\ \nabla \times (\nabla u) & = 0 \\ \nabla \cdot (\nabla \times F) & = 0 \end{aligned}