After recalling three theorems that we have encountered in the previous chapters (The Fundamental Theorem of Calculus, The classical Stokes’ Theorem and Gauss’ divergence theorem), in the first section we will point out that they follow a common pattern and then, in the following sections, we will pin down how to express this pattern in general mathematical terms.

Contents

• Introduction
• Integration domains
• Differential forms
• Integration
• Stokes' formula
• Corollaries
• Summary

The integral calculus of the previous chapter provides a convenient bridge between the physical laws of electromagnetism and the compact mathematical expressions of the same laws known as Maxwell's equations. In this chapter we provide the details of how this works.

Contents

• Principle of continuity
• Gauss' laws

The goal of the material provided here is to explain Stokes' theorem in a way that is suitable for students in Engineering, Physics and Mathematics. It follows the outline that was discussed in the JEM Lisbon workshop

The learner will find:

• What it means and what is it useful for.
• Its significance in Mathematics, in Physics, and in other application areas.
• Insights about why is it true, and in particular the realization that the main root for its truth is the fundamental theorem of calculus.

To properly understand Stokes' theorem, it is useful to recall a few basic notations and results from calculus.

These notations and results refer to the gradient of a function, to the curl and divergence of a vector field, and to a few fundamental relations among them.

Contents

• Gradient
• Curl
• Divergence

In this chapter the learner will find the basic facts about line, surface and volume integrals that we will need.

Contents

• Line Integrals

• Surface Integrals

• Volume Integrals

• Summary

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