本文深入探讨了矢量微积分的基本概念，包括梯度、散度、旋度和拉普拉斯算子的定义及其在三维空间中的表达。通过详细的数学公式，解释了这些概念如何应用于物理和其他科学领域。
∇=[∂∂x∂∂y∂∂z]，∇×=[0−∂∂z∂∂y∂∂z0−∂∂x−∂∂y∂∂x0]，∇2=[∂∂x∂∂y∂∂z]T[∂∂x∂∂y∂∂z]=(∂2∂x2+∂2∂y2+∂2∂z2)\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)∇⋅A⃗=[∂∂x∂∂y∂∂z]T[AxAyAz] \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)∇×A⃗=[0−∂∂z∂∂y∂∂z0−∂∂x−∂∂y∂∂x0][AxAyAz] \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)∇2ϕ=(∂2∂x2+∂2∂y2+∂2∂z2)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)
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