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}. ∇≡i∂x∂+j∂y∂+k∂z∂.
Vector operators acting on sums and products
1、 ∇ ( ϕ + ψ ) = ∇ ϕ + ∇ ψ \nabla(\phi+\psi)=\nabla\phi+\nabla\psi ∇(ϕ+ψ)=∇ϕ+∇ψ;
2、 ∇ ⋅ ( a + b ) = ∇ ⋅ a + ∇ ⋅ b \nabla \cdot (\boldsymbol{a}+\boldsymbol{b})=\nabla \cdot \boldsymbol{a}+\nabla \cdot \boldsymbol{b} ∇⋅(a+b)=∇⋅a+∇⋅b;
3、 ∇ × ( a + b ) = ∇ × a + ∇ × b \nabla \times (\boldsymbol{a}+\boldsymbol{b})=\nabla \times \boldsymbol{a}+\nabla \times \boldsymbol{b} ∇×(a+b)=∇×a+∇×b;
4、 ∇ ( ϕ ψ ) = ϕ ∇ ψ + ψ ∇ ϕ \nabla(\phi\psi)=\phi\nabla\psi+\psi\nabla\phi ∇(ϕψ)=ϕ∇ψ+ψ∇ϕ;
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} ∇(a⋅b)=a×(∇×b)+b×(∇×a)+(a⋅∇)b+(b⋅∇)a;
6、 ∇ ⋅ ( ϕ a ) = ϕ ∇ ⋅ a + a ⋅ ∇ ϕ \nabla \cdot (\phi\boldsymbol{a})=\phi \nabla \cdot \boldsymbol{a}+\boldsymbol{a} \cdot \nabla \phi ∇⋅(ϕa)=ϕ∇⋅a+a⋅∇ϕ;
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}) ∇⋅(a×b)=b⋅(∇×a)−a⋅(∇×b);
8、 ∇ × ( ϕ a ) = ∇ ϕ × a + ϕ ∇ × a \nabla\times(\phi\boldsymbol{a})=\nabla\phi\times\boldsymbol{a}+\phi\nabla\times\boldsymbol{a} ∇×(ϕa)=∇ϕ×a+ϕ∇×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} ∇×(a×b)=a(∇⋅b)−b(∇⋅a)+(b⋅∇)a−(a⋅∇)b;
where ϕ \phi ϕ and ψ \psi ψ are scalar fields, and a \boldsymbol{a} a and b \boldsymbol{b} b are vector fields.