物理引擎基础

genius
2021-11-22 / 0 评论 / 23 阅读 / 正在检测是否收录...

基础理论

数学基础

  • ∇:向量微分算子、哈密尔顿算子、Nabla算子、劈形算子,倒三角算子是一个微分算子: $\nabla = {\frac{\partial }{\partial x}}\mathbf{i}+ {\frac{\partial }{\partial y}}\mathbf{j}+ {\frac{\partial }{\partial z}}\mathbf{k}$
  • grad F=▽F,梯度(gradient),梯度指向标量场增长最快的方向,梯度的长度是这个最大的变化率: $▽f=\frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}$
  • div F=▽·F,散度(divergence),描述了通量源的密度,可用表征空间各点矢量场发散的强弱程度: $\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial U}{\partial x} +\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z }$
  • Δ、$∇^2$ or ∇·∇:拉普拉斯算子(Laplace operator),定义为梯度(▽f)的散度(▽·▽f): $\Delta f = \nabla^2 f = \nabla \cdot \nabla f$, $\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$
  • rot F 或curl F=∇ × F,旋度(curl,rotation),是向量场沿法向量的平均旋转强度: $\operatorname{curl}\,\mathbf{F} = \nabla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y}- \frac{\partial F_2}{\partial z}\right)\mathbf{e}_1 - \left(\frac{\partial F_3}{\partial x}- \frac{\partial F_1}{\partial z}\right)\mathbf{e}_2 + \left(\frac{\partial F_2}{\partial x}- \frac{\partial F_1}{\partial y}\right)\mathbf{e}_3$

    基本关系:

    一个标量场f梯度场是无旋场,也就是说它的旋度处处为零:$\nabla\times (\nabla f) = 0$

    一个矢量场F的旋度场是无源场,也就是说它的散度处处为零:$\nabla\cdot (\nabla \times \mathbf{F} ) = 0$

    F的旋度场的旋度场是: $\nabla \times (\nabla \times \mathbf{F} ) = \nabla(\nabla\cdot \mathbf{F}) - \nabla^2 \mathbf{F}$

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