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
- $laplace = (diff ^ 2) / (diff r ^ 2) + 1 / r diff / (diff r) + 1 / (r ^ 2) (diff ^ 2) / (diff theta ^ 2) + (diff ^ 2) / (diff z ^ 2)$