Vector operators acting on sums and products_a× ×b+a b+b a= a b-优快云博客

本文链接：https://blog.youkuaiyun.com/justidle/article/details/112887156

这篇博客详细介绍了向量算子∇，也称为梯度，及其在笛卡尔坐标系中的定义。文章列举了∇在不同数学运算中的行为，包括加法、点乘、叉乘等，并给出了相应的运算规则，如∇(ϕ+ψ)=∇ϕ+∇ψ，∇×(a+b)=∇×a+∇×b等，这些规则对于理解矢量场的性质和计算至关重要。

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定义

Central to all these differential operations is the vector operator $∇\nabla$ , which is called del (or sometimes nabla) and in Cartesian coordinates is defined by
$∇≡i∂∂x+j∂∂y+k∂∂z.\nabla\equiv \boldsymbol{i} \frac{\partial}{\partial x}+\boldsymbol{j}\frac{\partial}{\partial y}+\boldsymbol{k}\frac{\partial}{\partial z}.$

Vector operators acting on sums and products

1、 $∇(ϕ+ψ)=∇ϕ+∇ψ\nabla(\phi+\psi)=\nabla\phi+\nabla\psi$ ;
2、 $∇⋅(a+b)=∇⋅a+∇⋅b\nabla \cdot (\boldsymbol{a}+\boldsymbol{b})=\nabla \cdot \boldsymbol{a}+\nabla \cdot \boldsymbol{b}$ ;
3、 $∇×(a+b)=∇×a+∇×b\nabla \times (\boldsymbol{a}+\boldsymbol{b})=\nabla \times \boldsymbol{a}+\nabla \times \boldsymbol{b}$ ;
4、 $∇(ϕψ)=ϕ∇ψ+ψ∇ϕ\nabla(\phi\psi)=\phi\nabla\psi+\psi\nabla\phi$ ;
5、 $∇(a⋅b)=a×(∇×b)+b×(∇×a)+(a⋅∇)b+(b⋅∇)a\nabla (\boldsymbol{a} \cdot \boldsymbol{b})=\boldsymbol{a} \times (\nabla \times \boldsymbol{b})+\boldsymbol{b} \times (\nabla \times \boldsymbol{a})+(\boldsymbol{a} \cdot\nabla)\boldsymbol{b}+(\boldsymbol{b} \cdot\nabla)\boldsymbol{a}$ ;
6、 $∇⋅(ϕa)=ϕ∇⋅a+a⋅∇ϕ\nabla \cdot (\phi\boldsymbol{a})=\phi \nabla \cdot \boldsymbol{a}+\boldsymbol{a} \cdot \nabla \phi$ ;
7、 $∇⋅(a×b)=b⋅(∇×a)−a⋅(∇×b)\nabla\cdot(\boldsymbol{a} \times \boldsymbol{b})=\boldsymbol{b}\cdot(\nabla \times\boldsymbol{a} ) -\boldsymbol{a}\cdot(\nabla \times\boldsymbol{b})$ ;
8、 $∇×(ϕa)=∇ϕ×a+ϕ∇×a\nabla\times(\phi\boldsymbol{a})=\nabla\phi\times\boldsymbol{a}+\phi\nabla\times\boldsymbol{a}$ ;
9、 $∇×(a×b)=a(∇⋅b)−b(∇⋅a)+(b⋅∇)a−(a⋅∇)b\nabla \times (\boldsymbol{a}\times\boldsymbol{b})=\boldsymbol{a}(\nabla\cdot\boldsymbol{b})-\boldsymbol{b}(\nabla\cdot\boldsymbol{a})+(\boldsymbol{b}\cdot\nabla)\boldsymbol{a}-(\boldsymbol{a}\cdot\nabla)\boldsymbol{b}$ ;
where $ϕ\phi$ and $ψ\psi$ are scalar fields, and $a\boldsymbol{a}$ and $b\boldsymbol{b}$ are vector fields.