∇ = [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] ， ∇ × = [ 0 − ∂ ∂ z ∂ ∂ y ∂ ∂ z 0 − ∂ ∂ x − ∂ ∂ y ∂ ∂ x 0 ] ， ∇ 2 = [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] T [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] = ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 ) \nabla= \begin{bmatrix} \frac{\partial}{\partial x} \\\\ \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} \end{bmatrix}， \nabla \times = \begin{bmatrix} 0 & -\frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} & 0 & -\frac{\partial}{\partial x} \\\\ -\frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{bmatrix}， \nabla^2= \begin{bmatrix} \frac{\partial}{\partial x} \\\\ \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} \end{bmatrix}^T \begin{bmatrix} \frac{\partial}{\partial x} \\\\ \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} \end{bmatrix} =(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}) ∇=⎣⎢⎢⎢⎢⎡∂x∂∂y∂∂z∂⎦⎥⎥⎥⎥⎤，∇×=⎣⎢⎢⎢⎢⎡0∂z∂−∂y∂−∂z∂0∂x∂∂y∂−∂x∂0⎦⎥⎥⎥⎥⎤，∇2=⎣⎢⎢⎢⎢⎡∂x∂∂y∂∂z∂⎦⎥⎥⎥⎥⎤T⎣⎢⎢⎢⎢⎡∂x∂∂y∂∂z∂⎦⎥⎥⎥⎥⎤=(∂x2∂2+∂y2∂2+∂z2∂2)
(1) ∇ ϕ = [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] ϕ = [ ∂ ϕ ∂ x ∂ ϕ ∂ y ∂ ϕ ∂ z ] \nabla \phi=\begin{bmatrix} \frac{\partial}{\partial x} \\\\ \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} \end{bmatrix}\phi=\begin{bmatrix} \frac{\partial \phi}{\partial x} \\\\ \frac{\partial \phi}{\partial y} \\\\ \frac{\partial \phi}{\partial z} \end{bmatrix}\tag 1 ∇ϕ=⎣⎢⎢⎢⎢⎡∂x∂∂y∂∂z∂⎦⎥⎥⎥⎥⎤ϕ=⎣⎢⎢⎢⎢⎡∂x∂ϕ∂y∂ϕ∂z∂ϕ⎦⎥⎥⎥⎥⎤(1) (2) ∇ ⋅ A ⃗ = [ ∂ ∂ x ∂ ∂ y ∂ ∂ z ] T [ A x A y A z ] \nabla \cdot \vec A =\begin{bmatrix} \frac{\partial}{\partial x} \\\\ \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} \end{bmatrix}^T\begin{bmatrix} A_x \\\\ A_y \\\\ A_z \end{bmatrix}\tag 2 ∇⋅A=⎣⎢⎢⎢⎢⎡∂x∂∂y∂∂z∂⎦⎥⎥⎥⎥⎤T⎣⎢⎢⎢⎢⎡AxAyAz⎦⎥⎥⎥⎥⎤(2) (3) ∇ × A ⃗ = [ 0 − ∂ ∂ z ∂ ∂ y ∂ ∂ z 0 − ∂ ∂ x − ∂ ∂ y ∂ ∂ x 0 ] [ A x A y A z ] \nabla \times \vec A = \begin{bmatrix} 0 & -\frac{\partial }{\partial z} & \frac{\partial}{\partial y} \\\\ \frac{\partial}{\partial z} & 0 & -\frac{\partial}{\partial x} \\\\ -\frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{bmatrix}\begin{bmatrix} A_x \\\\ A_y \\\\ A_z \end{bmatrix}\tag 3 ∇×A=⎣⎢⎢⎢⎢⎡0∂z∂−∂y∂−∂z∂0∂x∂∂y∂−∂x∂0⎦⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎡AxAyAz⎦⎥⎥⎥⎥⎤(3) (4) ∇ 2 ϕ = ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 ) ϕ \nabla^2\phi=(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})\phi\tag 4 ∇2ϕ=(∂x2∂2+∂y2∂2+∂z2∂2)ϕ(4)
扫一扫
188
被折叠的 条评论
为什么被折叠?
到【灌水乐园】发言
