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梯度,散度,旋度,Laplace, nabla

这里 $x_i^->$ 表示一个方向的单位向量

nabla

可当做一个向量看待

$$limits(nabla)^-> eq.def sum_i x_i^-> diff / (diff x_i) $$

梯度

$$"grad"(f) eq.def limits(nabla)^-> f = sum_i x_i^-> (diff f) / (diff x_i)$$

  • 输入 $f$: $RR^n -> R$ 标量场,输出向量场.
  • $nabla f$ 对于一个向量输入,输出一个向量

散度

$$"div"limits(F)^-> eq.def limits(nabla)^-> dot limits(F)^-> = sum_i (diff F_x_i) / (diff x_i) $$ - 输入向量场,输出标量场

旋度

$$"curl" limits(F)^-> eq.def nabla times limits(F)^->$$

Laplace

$$laplace eq.def sum_i (diff ^ 2) / (diff x_i^2)$$

  • 输入标量场,输出标量场(先做梯度运算,再做散度运算)
  • Laplace 算子的极坐标,柱坐标和球坐标表示: ref: https://zhuanlan.zhihu.com/p/193094897
    • $laplace = (diff ^ 2) / (diff r ^ 2) + 1 / r diff / (diff r) + 1 / (r ^ 2) (diff ^ 2) / (diff theta ^ 2) + (diff ^ 2) / (diff z ^ 2)$