Area Integral

An area integral of a vector function E can be defined as the integral on a surface of the scalar product of E with area element dA. The direction of the area element is defined to be perpendicular to the area at that point on the surface. 矢量函数 E 的面积积分可以定义为在表面上对 E 与面积元 dA 的标量积的积分。面积元的方向定义为在该表面点处垂直于面积的方向。

The outward directed surface integral over an entire closed surface is denoted It is appropriate for such physical applications as Gauss' law. 整个闭合曲面的向外指向的曲面积分表示为，这在如高斯定律这样的物理应用中是合适的。

Line Integral

Vector functions such as electric field and magnetic field occur in physical applications, and scalar products of these vector functions with another vector such as distance or path length appear with regularity. When such a product is summed over a path length where the magnitudes and directions change, that sum becomes an integral called a line integral. A line integral is also used for the general definition of work in mechanics. Applications 应用 矢量函数如电场和磁场在物理应用中出现，这些矢量函数与另一个矢量（如距离或路径长度）的标量积经常出现。当这样的积沿着路径长度进行求和，而其中的大小和方向发生变化时，这个和就变成了一个称为线积分的积分。线积分也用于力学中对功的一般定义。

Applications of Line Integrals

The line integral of electric field around a closed loop is equal to the voltage generated in that loop (Faraday's law):

Such an integral is also used for the calculation of voltage difference since voltage is work per unit charge. Calculating the voltage difference near a point charge is a good example.

The line integral of a force over a path is equal to the work done by that force on the path.