矢量的梯度与Jacobi矩阵

• 梯度符号：
  \(\nabla = \left[\frac{\partial}{\partial x} \quad \frac{\partial}{\partial y} \quad \frac{\partial}{\partial z} \right]^T\)

• 矢量的梯度：
  \(\nabla \mathbf{v} =\left[\begin{array}{l}\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array}\right]\left[\begin{array}{lll}u & v & w\end{array}\right]=\left[\begin{array}{lll} \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} \\ \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \\ \frac{\partial u}{\partial z}& \frac{\partial v}{\partial z}& \frac{\partial w}{\partial z}\end{array}\right]=\frac{\partial u_{j}}{\partial x_i}\)
  https://zhuanlan.zhihu.com/p/139105732

• 矢量的Jacobi矩阵：
  \(\mathbf{v}\nabla^T=\left(\nabla \mathbf{v}\right)^T\)
  https://www.zhihu.com/question/374466553/answer/1043618492
  梯度之上： Jacobian矩阵 和 Hessian矩阵

• 综合：
  10935 梯度、散度、旋度、Jacobian、Hessian、Laplacian 的关系图
  导数、梯度、 Jacobian、Hessian
  Vector, Matrix, and Tensor Derivatives
  The Matrix Calculus You Need For Deep Learning