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。

Vector Calculus

矢量微积分

Many quantities which are of interest in physics are both directed quantities (vectors) and can take on a continuous range of values, making calculus methods necessary. Several operations from the mathematical field of vector calculus are of particular importance in solving physical problems.

Basic vector calculus operations
向量微积分基本运算
Vector calculus identities
矢量微积分恒等式
Divergence theorem
散度定理
Stokes' theorem
斯托克斯定理

物理学中有很多量既是方向量（矢量），又能取连续范围的值，因此需要微积分方法。从向量微积分中几个操作在解决物理问题中特别重要。
 Index

Vector concepts
索引矢量概念
 
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Vector Calculus Operations

矢量微积分运算

Three vector calculus operations which find many applications in physics are:

1. The divergence of a vector function

2. The curl of a vector function
3. The Gradient of a scalar function
These examples of vector calculus operations are expressed in Cartesian coordinates, but they can be expressed in terms of any orthogonal coordinate system, aiding in the solution of physical problems which have other than rectangular symmetries.
在物理中有很多应用的三个向量微积分运算是：1. 向量函数的散度 2. 向量函数的旋度 3. 标量函数的梯度。这些向量微积分运算的例子是在笛卡尔坐标系中表达的，但也可以用任何正交坐标系来表达，有助于解决具有非矩形对称性的物理问题。
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Vector calculus
索引 矢量微积分
 
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The Del Operator

Del 运算符

The collection of partial derivative operators

is commonly called the del operator. It shows up in important vector calculus operations.

Gradient
梯度
Divergence
散度
Curl
旋度
LaPlacian
拉普拉斯算子
偏导数算子的集合通常被称为del算子。它出现在重要的矢量微积分运算中。
 Index

Vector calculus
索引 矢量微积分
 
HyperPhysics*****HyperMath*****Calculus
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