Tensors for Physics

Chapter 8

Integration of Fields

Abstract The integration of fields is treated in this chapter. Firstly, line integrals are
considered and the computation of potential functions from vector fields is discussed.
Secondly, surface integrals are introduced and the generalized Stokes law is derived.
Applications are the magnetic field around an electric wire and the Faraday induction.
Thirdly, volume integrals are treated and a generalized Gauss theorem is stated. The
moment of inertia tensor is defined and computed for some examples. Applications of
volume integrals in electrodynamics comprise the Gauss law and the Coulomb force,
the formulation of balance equations for energy, linear and angular momentum and
thedefinitionoftheMaxwellstresstensor.Furtherapplicationsconcernthecontinuity
equation and the flow through a pipe, the momentum balance and the force on a solid
body, the derivation of the Archimedes principle and the computation of the torque
on a rotating solid body.

The differentiation of a field provides a local information about the changes of
a function caused by small changes of the position considered. Integrals contain
a more global information since the behavior of a function over a larger region of
space is involved. These regions can be lines, surfaces or volumes, in 3D. All three
types of integrals are needed for applications in physics. They are referred to asline
integrals, surface integrals, and volume integrals.

8.1 Line Integrals

8.1.1 Definition, Parameter Representation

Letf=f(r)be a well defined, smooth function within a regionBof the 3D space.
Furthermore, letCbe a continuous, piecewise smooth curve within the regionB
with start pointr 1 and end pointr 2 .Theline integraloff(r)along the curveCis
defined by

Iμ=

∫

C

f(r)drμ. (8.1)

© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_8

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