Geometry, Particles, and Fields pp 402-483 | Cite as

# Integral Calculus on Manifolds

Chapter

## Abstract

The next problem we must attack is how we can extend the integral calculus to differential forms. Before we proceed to give precise definitions, we would like to give an

*intuitive*feeling of the integral concept we are going to construct. In the familiar theory of Riemann integrals we consider a function of, say, two variables*f(x*,y) defined on a “nice” subset*U*of ℝ^{2}.Then we divide this region*U*into cells △_{ i }with area ∈^{2}and in each cell, △_{ i }we choose a point*(x*_{ i }y_{ i }). (See Figure 8.1.) We can now form the Riemann sum:$$\sum\limits_i {f\left( {{x_i},{y_i}} \right){ \in ^2}}
$$

## Keywords

Scalar Field Differential Form Open Covering Regular Domain Integral Calculus
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## Copyright information

© Springer Science+Business Media New York 1998