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定义 梯度
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一个标量函数的梯度 f(x1, x2, x3, ..., xn) 记作 ∇f or \vec{\nabla} f 用微分算子符号 ∇ (nabla 符号) 来表示这个向量 , . 符号 "grad(f)" 也经常用来表示梯度. f的梯度定义为唯一向量场的 它是向量 v 点积 每个点xf 沿着向量 v的偏导数. 也就是,
(\nabla f(x))\cdot \mathbf{v} = D_{\mathbf v}f(x).
In a rectangular coordinate system, the gradient is the vector field whose components are the s of f:
\nabla f = \frac{\partial f}{\partial x_1 }\mathbf{e}_1 + \cdots + \frac{\partial f}{\partial x_n }\mathbf{e}_n
where the ei are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

Cartesian coordinates

In the three-dimensional , the gradient is given by:
\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}
where i, j, k are the s. For example, the gradient of the function
f(x,y,z)= \ 2x+3y^2-\sin(z)
is:
\nabla f=
\frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} = 2\mathbf{i}+ 6y\mathbf{j} -\cos(z)\mathbf{k}.
In some applications it is customary to represent the gradient as a or of its components in a rectangular coordinate system.

Cylindrical and spherical coordinates

{{main|Del in cylindrical and spherical coordinates}}
In , the gradient is given by:{{harvnb|Schey|1992|pp=139–142}}.
\nabla f(\rho, \varphi, z) =
\frac{\partial f}{\partial \rho}\mathbf{e}_\rho + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\mathbf{e}_\varphi + \frac{\partial f}{\partial z}\mathbf{e}_z
where φ is the azimuthal or angle, z is the , and eρ, eφ and ez are unit vectors pointing along the coordinate directions.
In , the gradient is given by:
\nabla f(r, \theta, \varphi) =
\frac{\partial f}{\partial r}\mathbf{e}_r+ \frac{1}{r}\frac{\partial f}{\partial \theta}\mathbf{e}_\theta+ \frac{1}{r \sin\theta}\frac{\partial f}{\partial \varphi}\mathbf{e}_\varphi
where φ is the angle and θ is the angle.
For the gradient in other s, see .

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