A Simple Proof That the Curl Defined As Circulation Density Is a Vector-Valued Function, and an Alternative Approach to Proving Stoke’S Theorem

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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: [email protected]

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 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- The standard mathematical presentation that the curl de- sity of F at p in the direction of n”, or usually just “cir- fined as a limit of circulation density is a vector-valued culation density” it being understood that a unit normal function with the standard Cartesian expression uses vector n has been chosen. This definition has two fatal Stokes’ Theorem. Most physics books use the multidi- flaws. mensional version of Taylor’s Theorem to show this re- lationship works in the x, y plane by using linear ap- 2.1.1. Flaw 1 proximations of F, and simply assert that this special We don’t even know if the limit on the right hand side of case extends to three dimensional space, [1, pp. 71-72]. this equation exists. Indeed, it looks to be very dubious This approach requires that F have continuity in the sec- as to whether it exists. As a crude thought experiment, if ond partials, which is not necessary. Also, the assertion the loop was a circle of radius r and if the tangential that the two dimensional case extends to three dimen- component of F is 1, then in (1) we would be looking at sions is not trivial. A more elementary proof is presented 2πr 2 here, using only Green’s Theorem on a right triangle [2, lim = lim p. 1102], and the Integral Mean Value Theorem [2, p. rr00r 2 r 1071]. which, of course, does not exist. This is not a counter- example, since the tangential component of F always 2. A Criticism of Existing Methods to being 1 excludes this F from being integrable in (1), but Explain and Prove the Properties of it points out that the limit does not obviously exist. Also, the Curl the area of the loop going to zero does not force the loop 2.1. The Approach of Most Physicists to collapse about point p. In this approach, we suppose we have a vector field 2.1.2. Flaw 2 Fix,,yz = Pxyz ,, Qxyz ,, jkRxyz,, and we  Even if the limit on the right hand side of (1) exists, this are at a point p in space. The curlF p is then “de-  definition asserts that the limit produces the existence of fined” to be the vector such that for all unit vectors n, the a fixed vector curlF  p . The limit is just a scalar that following equation is true: depends on n. This produces an infinite number of equa- Frd tions, one for each n. Unless you can show linearity of C curlFn() p  =lim (1) the limit with respect to n, you can not solve this system. a0 a You can’t just assert the existence of this property. where C is a tiny loop or contour about p in the plane Nobel Prize winning physicist, Edward Purcell, points containing p with normal n, and a is the area of the inte- out this second flaw in his Berkeley Series text book

Electricity and Magnetism [1, p. 70], where he says that it can be shown that the above equation does indeed de- n fine a vector but then adds, “... we shall not do so here”. He cites no reference where it is shown that this definition proves that the vector curlF  p exists. For those thinking that the above equation defines a vector, he cre- ates a similarly defined quantity, squrlF p , by 2  Frd squrlFnp  = limC a0 a  and then asks the reader to show that squrlF p is not a vector [1, problem 2.32 in p. 85]. Figure 1. Local coordinate system.

2.2. The Approach of Most Mathematicians and YZ XZ XY Mathematicians take as their definition the standard Car- n =,,. (3) tesian formula, 222TTT RQPRQP Let T be the right-triangular patch in the xy plane curl P,, Q R = , ,. (2) xy yzzxxy with vertices  X ,0,0, 0,0,0, 0,Y ,0. Using Green’s Theorem and Integral Mean Value Theorem, we have Students find this formula mysterious and troubling.  that for some  x ,,0yT  xy , The formula should arise from the physical nature of the circulation density. It is the purpose of this paper to sup- Frd=Fxy12 ,,0d x Fxy ,,0d y TTxy xy ply the motivation for (2). FF =,,0,,0d 21xy  xy A 3. Main Result Txy xy FF Given a point p and a unit vector n, consider the plane 21  =,,0,,0x yxyT xy containing p with normal n. Form a tetrahedron by shift- xy ing the origin to the negative part of the line through p FF21  XY with direction n. The plane intersects these new coordi- =,,0,,0xy  xy xy2 nate axes at X ,0,0, 0,Y ,0 , 0,0, Z , see Figure 1.  XY Call the triangle in the plane that connects these points T =:c . (even though n has all non-zero components, the proof 3 2 below will work with minor modifications for n with Similarly for the other two right-triangular patches, some zero components). 33 Theorem 1 (curl of F) Let F : RR have con- F3  F2  YZ  Frd= 0,,yz 0,, yz tinuous partial derivatives. Given a unit vector n and a Tyz yz2  point p, let T be the triangle constructed as in Figure 1. YZ The vector curlF p can be defined by =: c1  2 Fsd T F3 F1 XZ curlF  p n =lim  Frd=xz ,0, xz ,0, T 0 T Txz xz2  XZ where T is the area of T and T is the diameter of =:c2 . the smallest ball centered at P containing T. 2

Proof. The area of T is half of the parallelogram Reversing the orientation of Txz , all the paths along formed by the vectors X ,0,Z and 0,YZ , . The the coordinate axes cancel, see Figure 2, and area of the parallelogram is the length of the cross prod- F d=sFsFsFs d  d  d uct, so that TTyz   T xz   T xy

YZ XZ XY X ,0,ZYZ 0, ,  = YZXZXY,, =.cc c 12222 3 X ,0,ZY 0, ,ZT = 2 Using (3), we then have

4. An Intuitive Proof of Stokes’ Theorem Theorem 2 (Stokes’ Theorem) Let F : RR33 have continuous partial derivatives. Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth curve. Then curlF  nFrd= S  d SS  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 integral to the surface. Refine the approximation. Figure 2. Traversing paths. 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 Fsd=ccc,, n T T 123 precise. As a practical matter, any reasonable interpreta- tion will suffice. For example, the area of the largest tri- Fsd T angular face going to zero will suffice to refine the ap- =,cc123,.c  n T proximation [5]. Taking the limit as T  0, the origin of the local coordinate system moves to p and REFERENCES ccc,,  curlF p by the continuity of the partial 123  [1] E. Purcell, “Electricity and Magnetism, Berkeley Physics derivatives. Course,” 2nd Edition, McGraw Hill, New York, 1985. By driving the radius of the ball containing the patch T [2] J. Stewart, “Calculus,” 4th Edition, Brooks/Cole, Stam- to zero, we get that the limit of the circulation density is a ford, 1999. fixed vector dot the normal, and that the expression of [3] H. M. Schey, “Div, Grad, Curl, and All That: An Infor- the fixed vector in cartesian coordinates is the standard mal Text on Vector Calculus,” 4th Edition, W. W. Norton, expression for the curlF  p . New York, 2005. The last equation in the proof makes the following two [4] E. Weisstein, “Wolfram Mathworld.” propositions immediately obvious: http://mathworld.wolfram.com/Curl.html Proposition 1. The direction of maximum circulation [5] K. Hildebrandt, K. Polthier and M. Wardetzky, “On the density is in the direction of the curl. Convergence of Metric and Geometric Properties of Po- Proposition 2. This maximum circulation density is in lyhedral Surfaces,” Geometriae Dedicata, Vol. 123, No. 1, fact just the magnitude of the curl. 2005, pp. 89-112. doi:10.1007/s10711-006-9109-5