<aside>
</aside>
The nabla symbol, represented by the upside-down delta symbol (∇), is a mathematical operator used in vector calculus and differential geometry. It is used to denote the gradient of a scalar field, the divergence of a vector field, and the curl of a vector field. The nabla symbol is a vector operator that can be applied to various types of functions, including scalar functions, vector functions, and tensor functions.
The gradient of a scalar field is a vector that points in the direction of the maximum rate of change of the scalar field. It is defined using the nabla operator as follows:
∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k 最快增加方向【梯度下降法对应负梯度】
where f is a scalar function, and i, j, and k are the unit vectors in the x, y, and z directions, respectively.
The divergence of a vector field is a scalar that represents the amount of "source" or "sink" at a given point in the field. It is defined using the nabla operator as follows:
∇ · F = (∂Fx/∂x) + (∂Fy/∂y) + (∂Fz/∂z) 面内有发射源【点电荷，高斯定理】
where F is a vector field.
The curl of a vector field is a vector that represents the amount of rotation at a given point in the field. It is defined using the nabla operator as follows:
∇ x F = ( ∂Fz/∂y - ∂Fy/∂z )i + ( ∂Fx/∂z - ∂Fz/∂x )j + ( ∂Fy/∂x - ∂Fx/∂y )k 漩涡中心，大小及方向【一个小棒在视为速度场的场中随波旋转】
where F is a vector field.
In conclusion, the nabla symbol is a powerful mathematical tool that plays an important role in many areas of physics and engineering. It allows us to analyze and manipulate scalar, vector, and tensor fields in a concise and elegant manner.
混合积
$$ \boldsymbol a\cdot(\boldsymbol b\times \boldsymbol c)=\boldsymbol b\cdot(\boldsymbol c\times \boldsymbol a)=\boldsymbol c\cdot(\boldsymbol a\times \boldsymbol b) $$
矢积(AB)C=(CA)B-(CB)A末尾两次作用于前两位
$$
(\boldsymbol a\times \boldsymbol b)\times \boldsymbol c=(\boldsymbol c\cdot \boldsymbol a)\boldsymbol b-(\boldsymbol c\cdot \boldsymbol b)\boldsymbol a $$