Advances in Pure Mathematics, 2012, 2, 33-35
http://dx.doi.org/10.4236/apm.2012.21007 Published Online January 2012 (http://www.SciRP.org/journal/apm)
A Simple Proof That the Curl Defined as Circulation
Density Is a Vector-Valued Function, and an Alternative
Approach to Proving Stoke’s Theorem
California State University, Long Beach, USA
Received September 24, 2011; revised November 16, 2011; accepted November 25, 2011
This article offers a simple but rigorous proof that the curl defined as a limit of circulation density is a vector-valued
function with the standard Cartesian expression.
Keywords: Curl; Circulation Density
The standard mathematical presentation that the curl de-
fined as a limit of circulation density is a vector-valued
function with the standard Cartesian expression uses
Stokes’ Theorem. Most physics books use the multidi-
mensional version of Taylor’s Theorem to show this re-
lationship works in the x, y plane by using linear ap-
proximations of F, and simply assert that this special
case extends to three dimensional space, [1, pp. 71-72].
This approach requires that F have continuity in the sec-
ond partials, which is not necessary. Also, the assertion
that the two dimensional case extends to three dimen-
sions is not trivial. A more elementary proof is presented
here, using only Green’s Theorem on a right triangle [2,
p. 1102], and the Integral Mean Value Theorem [2, p.
2. A Criticism of Existing Methods to
Explain and Prove the Properties of
2.1. The Approach of Most Physicists
In this approach, we suppose we have a vector field
curl pF and we
are at a point p in space. The is then “de-
fined” to be the vector such that for all unit vectors n, the
following equation is true:
()curlp Fn (1)
where C is a tiny loop or contour about p in the plane
containing p with normal n, and a is the area of the inte-
rior of the loop, see [3, p. 81] and . The limit on the
right hand side of (1) is given the name “circulation den-
sity of F at p in the direction of n”, or usually just “cir-
culation density” it being understood that a unit normal
vector n has been chosen. This definition has two fatal
2.1.1. Fl aw 1
We don’t even know if the limit on the right hand side of
this equation exists. Indeed, it looks to be very dubious
as to whether it exists. As a crude thought experiment, if
the loop was a circle of radius r and if the tangential
component of F is 1, then in (1) we would be looking at
which, of course, does not exist. This is not a counter-
example, since the tangential component of F always
being 1 excludes this F from being integrable in (1), but
it points out that the limit does not obviously exist. Also,
the area of the loop going to zero does not force the loop
to collapse about point p.
2.1.2. Fl aw 2
Even if the limit on the right hand side of (1) exists, this
definition asserts that the limit produces the existence of
a fixed vector
curl pF. The limit is just a scalar that
depends on n. This produces an infinite number of equa-
tions, one for each n. Unless you can show linearity of
the limit with respect to n, you can not solve this system.
You can’t just assert the existence of this property.
Nobel Prize winning physicist, Edward Purcell, points
out this second flaw in his Berkeley Series text book
opyright © 2012 SciRes. APM
Electricity and Magnetism [1, p. 70], where he says that
it can be shown that the above equation does indeed de-
fine a vector but then adds, “... we shall not do so here”.
He cites no reference where it is shown that this defini-
tion proves that the vector exists. For those
thinking that the above equation defines a vector, he cre-
ates a similarly defined quantity, , by
squrl = li
and then asks the reader to show that is not
a vector [1, problem 2.32 in p. 85].
2.2. The Approach of Most Mathematicians
Mathematicians take as their definition the standard Car-
Students find this formula mysterious and troubling.
The formula should arise from the physical nature of the
circulation density. It is the purpose of this paper to sup-
ply the motivation for (2).
3. Main Result
Given a point p and a unit vector n, consider the plane
containing p with normal n. Form a tetrahedron by shift-
ing the origin to the negative part of the line through p
with direction n. The plane intersects these new coordi-
nate axes at
,0 , 0, ,Y,0X
see Figure 1.
Call the triangle in the plane that connects these points T
(even though n has all non-zero components, the proof
below will work with minor modifications for n with
some zero components).
Theorem 1 (curl of F) Let have con-
tinuous partial derivatives. Given a unit vector n and a
point p, let T be the triangle constructed as in Figure 1.
The vector can be defined by
where T is the area of T and T is the diameter of
the smallest ball centered at P containing T.
Proof. The area of T is half of the parallelogram
formed by the vectors ,0,
and 0, ,YZ
area of the parallelogram is the length of the cross prod-
uct, so that
ZY Z YZXZXY
Figure 1. Local coordinate system.
y be the right-triangular patch in the xy plane
. Using Green’s
Theorem and Integral Mean Value Theorem, we have
that for some
Similarly for the other two right-triangular patches,
d=0, ,0, ,2
Reversing the orientation of
z, all the paths along
the coordinate axes cancel, see Figure 2, and
TT T T
yz xz xy
YZ XZ XY
Using (3), we then have
Copyright © 2012 SciRes. APM
Copyright © 2012 SciRes. APM
4. An Intuitive Proof of Stokes’ Theorem
Figure 2. Traversing paths.
Taking the limit as 0T, the origin of the local co-
ordinate system moves to p and
123 by the continuity of the partial
Theorem 2 (Stokes’ Theorem) Let have
continuous partial derivatives. Let S be an oriented
piecewise-smooth surface that is bounded by a simple,
closed, piecewise-smooth curve. Then
By driving the radius of the ball containing the patch T
to zero, we get that the limit of the circulation density is a
fixed vector dot the normal, and that the expression of
the fixed vector in cartesian coordinates is the standard
expression for the .
The last equation in the proof makes the following two
propositions immediately obvious:
Proposition 1. The direction of maximum circulation
density is in the direction of the curl.
Proposition 2. This maximum circulation density is in
fact just the magnitude of the curl.
Proof. Triangulate the surface S. Apply Theorem 1 to
each of the triangular faces approximating S, and all inte-
rior paths cancel leaving an approximate boundary inte-
gral to the surface. Refine the approximation.
This idea can be made into a rigorous proof, but there
is quite a bit of mathematical machinery that is necessary
for the meaning of “refine the approximation” to be made
precise. As a practical matter, any reasonable interpreta-
tion will suffice. For example, the area of the largest tri-
angular face going to zero will suffice to refine the ap-
 E. Purcell, “Electricity and Magnetism, Berkeley Physics
Course,” 2nd Edition, McGraw Hill, New York, 1985.
 J. Stewart, “Calculus,” 4th Edition, Brooks/Cole, Stam-
 H. M. Schey, “Div, Grad, Curl, and All That: An Infor-
mal Text on Vector Calculus,” 4th Edition, W. W. Norton,
New York, 2005.
 E. Weisstein, “Wolfram Mathworld.”
 K. Hildebrandt, K. Polthier and M. Wardetzky, “On the
Convergence of Metric and Geometric Properties of Po-
lyhedral Surfaces,” Geometriae Dedicata, Vol. 123, No. 1,
2005, pp. 89-112. doi:10.1007/s10711-006-9109-5