Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

The Gradient

梯度

The gradient is a vector operation which operates on a scalar function to produce a vector whose magnitude is the maximum rate of change of the function at the point of the gradient and which is pointed in the direction of that maximum rate of change. In rectangular coordinates the gradient of function f(x,y,z) is:

If S is a surface of constant value for the function f(x,y,z) then the gradient on the surface defines a vector which is normal to the surface.

Applications
应用
Conduction heat transfer
传导热传递

The divergence of the gradient is called the LaPlacian. It is widely used in physics.

梯度的散度称为拉普拉斯算子。它在物理学中被广泛应用。


Gradient in other coordinate systems

其他坐标系中的梯度
梯度是一种向量运算，它对一个标量函数作用，产生一个向量，其大小是该函数在梯度点的最大变化率，方向即为该最大变化率的方向。在直角坐标系中，函数f(x,y,z)的梯度为：如果S是函数f(x,y,z)的常值表面，那么梯度在该表面上定义了一个垂直于该表面的向量。
 Index

Vector calculus
索引 矢量微积分
 
HyperPhysics*****HyperMath*****Calculus
HyperPhysics ***** HyperMath ***** 微积分
R Nave
 Go Back
返回





Gradient, Various Coordinates

梯度，不同坐标系

Compared to the gradient in rectangular coordinates:

In cylindrical polar coordinates:
and in spherical polar coordinates:
与直角坐标系中的梯度相比：在圆柱坐标系中：以及在球面坐标系中：
 Index

Vector calculus
索引 矢量微积分
 
HyperPhysics*****HyperMath*****Calculus
HyperPhysics ***** HyperMath ***** 微积分
R Nave
 Go Back
返回





Divergence, Various Coordinates

发散，各种坐标

Compared to the divergence in rectangular coordinates:

In cylindrical polar coordinates:
and in spherical polar coordinates:
与直角坐标系中的散度相比：在柱坐标系中：以及在球坐标系中：
 Index

Vector calculus
索引 矢量微积分
 
HyperPhysics*****HyperMath*****Calculus
HyperPhysics ***** HyperMath ***** 微积分
R Nave
 Go Back
返回