Combinations of grad, div and curl_div grad curl-优快云博客

本文链接：https://blog.youkuaiyun.com/justidle/article/details/112893154

We now consider the action of two vector operators in succession on a scalar or vector field. We can immediately discard four of the nine obvious combinations of grad, div and curl. where $ϕ\phi$ is a scalar field and $a\boldsymbol{a}$ is a vector field.

$4$ meaningless

these four combinations are grad(grad $ϕ\phi$ ), div(div $a\boldsymbol{a}$ ), curl(div $a\boldsymbol{a}$ ) and grad(curl $a\boldsymbol{a}$ ).
In each case the second (outer) vector operator is acting on the wrong type of field, i.e. scalar instead of vector or vice versa. In grad(grad $ϕ\phi$ ), for example, grad acts on grad $ϕ\phi$ , which is a vector field, but we know that grad only acts on scalar fields.

$2$ always zero

curl grad $ϕ\phi$ = $∇×∇ϕ=0\nabla\times\nabla\phi=0$ ,
div curl $a\boldsymbol{a}$ = $∇⋅(∇×a)=0\nabla\cdot(\nabla \times \boldsymbol{a})=0$ .

$3$ Meaningful

div grad $ϕ=∇⋅∇ϕ=∇2ϕ=∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2\phi=\nabla\cdot\nabla\phi=\nabla^2\phi=\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}$ ,
grad div $a\boldsymbol{a}$ = $∇(∇⋅a)=(∂2ax∂x2+∂2ay∂x∂y+∂2az∂x∂z)i+(∂2ax∂y∂y+∂2ay∂2y+∂2az∂y∂z)j+(∂2ax∂z∂x+∂2ay∂z∂y+∂2az∂2z)k\nabla(\nabla\cdot\boldsymbol{a})\\=(\frac{\partial^2a_x}{\partial x^2}+\frac{\partial^2a_y}{\partial x \partial y}+\frac{\partial^2a_z}{\partial x \partial z})\boldsymbol{i}+(\frac{\partial^2a_x}{\partial y\partial y}+\frac{\partial^2a_y}{\partial^2 y}+\frac{\partial^2a_z}{\partial y \partial z})\boldsymbol{j}+(\frac{\partial^2a_x}{\partial z\partial x}+\frac{\partial^2a_y}{\partial z\partial y}+\frac{\partial^2a_z}{\partial ^2z})\boldsymbol{k}$ ,
curl curl $a\boldsymbol{a}$ = $∇×(∇×a)=∇(∇⋅a)−∇2a\nabla\times(\nabla\times\boldsymbol{a})=\nabla(\nabla\cdot\boldsymbol{a})-\nabla^2\boldsymbol{a}$