Chapter 3
Vector integrals and integral theorems
Syllabus covered:
1. Line, surface and volume integrals.
2. Vector and scalar forms of Divergence and Stokes’s theorems. Conservative fields: equivalence to curlfree
and existence of scalar potential. Green’s theorem in the plane.
Calculus I and II covered integrals in one, two and three dimensional Euclidean (flat) space (i.e.
R
,
R
2
and
R
3
). We are still working in
R
3
so there is no generalization to be applied to volume or triple integrals, but
one and two dimensional integration will be generalized from straight lines and planes to curves and surfaces.
integraltext
f
(
x
)
d
x
generalizes to
integraltext
C
F
.
d
r
on a curve
C
, a line integral (section 3.1).
integraltextintegraltext
f
(
x
,
y
)
d
x
d
y
generalizes to
integraltext
S
F
.
d
S
over a surface
S
, a surface integral (section 3.2).
We will then have to study the generalizations of
integraldisplay
b
a
d
f
d
x
dx
=
f
(
b
)
−
f
(
a
)
,
(3.1)
the ‘fundamental theorem of calculus’, which we use in the proofs. This theorem relates a onedimensional
integral to a zerodimensional evaluation. The higher dimensional versions do the following.
Stokes’s theorem
1
, relates the surface integral of a curl to a line integral (2 dimensions to 1): see section
3.5.
The Divergence Theorem
2
relates the volume integral of a divergence to a surface integral (3 dimensions
to 2): see section 3.3.
There is also a special case of Stokes’s theorem where the surface is a plane: this is Green’s theorem
relating the integral of a curl to a line integral (2 dimensions to 1): see section 3.4.
[Aside: All these are in fact special cases of the general Stokes’s theorem which relates an
n
−
1 dimen
sional integral of a field to the
n
dimensional integral of its derivative. Here the field is a generalization of a
vector field called an
(
n
−
1
)
form field.]
1
A linguistic comment: For a name ending in s, it used to be customary to only add an apostrophe, e.g. Stokes’. Modern use also
adds an s e.g. Stokes’s. Books, and people, vary in which they prefer.
2
First discovered by Joseph Louis Lagrange in 1762, then independently rediscovered by Carl Friedrich Gauss in 1813, by George
Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem. Thus the result may be
called Gauss’s Theorem, Green’s theorem, or Ostrogradsky’s theorem.
24
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Finally we will discuss the application to potentials, and the proofs.
Before all that, we consider the case where one wants to integrate a vector function
F
(
u
)
of one variable,
u
, with respect to
u
.
The integral can be calculated simply by integrating the components (in Cartesian
coordinates) of
F
= (
f
1
,
f
2
,
f
3
)
:
integraldisplay
F
d
u
=
parenleftbigg
integraldisplay
f
1
d
u
,
integraldisplay
f
2
d
u
,
integraldisplay
f
3
d
u
parenrightbigg
Integration of a vector in this case is just a set of three ordinary integrals.
The restriction to Cartesian
coordinates can be overcome by looking at the definition in vectorial terms: we go back to the basic definition
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 Spring '09
 Vector Calculus, Magnetism, Stokes' theorem, dx dy, Surface integral

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