Integral Calculus on Manifolds

  • Bjørn Felsager
Part of the Graduate Texts in Contemporary Physics book series (GTCP)


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 ℝ2.Then we divide this regionUinto cells △ i with area ∈2and in each cell, △ i we choose a point(x i y i ). (See Figure 8.1.) We can now form the Riemann sum:
$$\sum\limits_i {f\left( {{x_i},{y_i}} \right){ \in ^2}} $$


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

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Bjørn Felsager
    • 1
    • 2
  1. 1.Mathematics DepartmentOdense UniversityDenmark
  2. 2.The Niels Bohr InstituteThe University of CopenhagenHaslevDenmark

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