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

David McKay

California State University, Long Beach, USA

Email: dmckay@csulb.edu

Received September 24, 2011; revised November 16, 2011; accepted November 25, 2011

ABSTRACT

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

1. Introduction

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.

1071].

2. A Criticism of Existing Methods to

Explain and Prove the Properties of

the Curl

2.1. The Approach of Most Physicists

In this approach, we suppose we have a vector field

,, =,,,,

,,

yz PxyzQxyzFiRxyzjk

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:

0

d

=limC

aa

Fr

()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 [4]. 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

flaws.

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

2

00

2π2

lim= lim

rr

r

r

r

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

C

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