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   Elie Cartan 
Elie Cartan   (1869-1951)

Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Differential Forms

  • Generalizing the  fundamental theorem of calculus.
  • Vectorial  surface  dotted into an observing direction gives  apparent  area.
  • Practical identities  of vector calculus.

Related articles on this site:

Related Links (Outside this Site)

Vector identities  collected by  Evans M. Harrell II  and  James V. Herod.
Electromagnetism,  in  Cartan's Corner,  by Dr. Robert M. Kiehn.
Weird Links:  Einstein-Cartan Theory (ECT)  Torsion Fields, Spin Waves...
Théorie des opérations linéaires  (Banach spaces)  by  Stefan Banach  (1932).

Student Papers :

The Poincaré Lemma and de Rham Cohomology  by  Daniel Litt  (Harvard).
An Elucidation of Vector Calculus Through Differential Forms  by
Jonathan Emberton  (Aug. 22, 2008).  University of Chicago, Class of 2011.
Wikipedia :   Differential form   |   Wedge product    |   Exterior derivative   |   Hodge dual   |   Poisson's equation   |   Newtonian potential   |   Convolution   |   Fundamental Theorem of Vector Calculus

Exterior Differential Forms

(2005-07-17)   Stokes' Theorem
The general theorem is due to  Nicolas Bourbaki... and vice-versa !

A stunning generalization of the fundamental theorem of calculus states that the integral of a form's derivative  dw  over an oriented manifold  W  is the integral of that form over the border  ¶W.  This is called  Stokes' Theorem :  Joseph-Louis Lagrange 
 (1736-1813)  Carl Friedrich Gauss 
 (1777-1855)  Lord Kelvin 
 1824-1907  Sir George Stokes 

Stokes'  Theorem
òW  dw     =     ò¶W  w

In a way, Nicolas Bourbaki and this result are  due to each other.  The urge to elucidate the latter gave birth to the former, in 1935  (as reported by Bourbaki founder André Weil,  see:  The Many Faces of Nicolas Bourbaki).

Three-Dimensional Incarnations and Relatives of  Stokes' Theorem
NameFormulaW  and ¶W
Gradient theorem òC   grad f . dr     =     f(b) - f(a)  A Path has a 2-Point Boundary
òòS   rot U . dS     =     òC   U . dr  Border of a Surface
  òòS   dS ´ grad f     =     òC   f dr  Border of a Surface
òòS   div U  dS   -   òòS   [
U/x . dS
U/y . dS
U/z . dS
      =       òC   U ´ dr
 Border of a Surface
(or Gauss) theorem
òòòV   div U  dV     =     òòS   U . dS  Boundary of a Volume
  òòòV   rot U  dV     =     òòS   dS ´ U  Boundary of a Volume
  òòòV   grad f  dV     =     òòS   f dS  Boundary of a Volume

 Mikhail Vasilevich 
 Ostrogradski (1801-1861)
Mikhail Ostrogradski

Ostrogradski's theorem  was independently discovered by Lagrange in 1762, by Gauss in 1813 and by George Green in 1825.  A  rigorous proof was given by Mikhail Vasilevich Ostrogradski (1801-1861)  in 1831.

Applied to  U  =  p grad q   (and/or  U  =  q grad p )  that theorem yields two relations known as  Green's formulas :

òòòV   ( p D q  +  grad p . grad q )   dV     =     òòS   p grad. dS
òòòV   ( p D q  -  q D p )   dV     =     òòS   ( p grad q  -  q grad p ) . dS

Applied to  U = [P,Q,0]  the third cartesian component of the  Kelvin-Stokes formula  yields the following elementary result, dubbed  Green's theorem,  which relates the counterclockwise line integral around the border  (C)  of a planar region  (S)  to a surface integral on that region:

òòS   ( Q / x - ¶P / y )  dx dy     =     òC   P dx  +  Q dy

 Lord Kelvin 

 James Clerk Maxwell 

According to E.T. Whittaker (A History of the Theories of Aether and Electricity) the Kelvin-Stokes formula (commonly called the Stokes formula) was first stated without proof by Kelvin in 1850.  George G. Stokes assigned the proof of that formula for the 1854 Smith's Prize exam, in which James Clerk Maxwell was sitting.  It was Maxwell who later traced that question back to Stokes and saw fit to give him credit for the answer...  In the end, the generalized theorem  (elucidated by the Bourbakists around 1935)  was named after Stokes too!

Among the above formulae, the least popular is surely the one involving an irreducible nabla  (as tabulated above in terms of cartesian coordinates).  I believe it has never been given a name:

òòS   [ div U  dS  -   (U.Ñ) dS  ]       =       òC   U ´ dr

The next section features this formula and demonstrates what's involved in elementary proofs of such, when more elegant general reasoning is shunned.

Highbrow Poincaré duality is a loosely related topic.

(2005-08-01)   Area Vector  S  and  Apparent Area  u.S
The apparent area bounded by an oriented loop is the scalar product of its vectorial surface  S  by a unit vector  u  pointing to a distant observer.

 A twisted loop, with  
 color-coded apparent area. This apparent area is a signed quantity which is positive for an observer looking at the  north side  of the loop  (readily identified if the loop is not  too  twisted).

The  area vector, or  surface vector  is an axial vector, defined by a contour integral around the  oriented  loop:

S     =     ½   òC+   r ´ dr

For a closed loop  C+  this defining integral does not depend on the choice of origin for the position vector  r.  Anyone encountering this for the first time is encouraged to work out  S  explicitely for a circle of radius  R,  with the following parametric equations  (0 < q < 2p).

x   =   a  +  R cos q     ;     y   =   b  +  R sin q     ;     z   =   c

For simple planar loops, the magnitude of  S  is simply the usual  surface area enclosed by the loop  (the vector is perpendicular to the plane and points to whichever direction is implied by the orientation of the loop).  This definition is consistent with the general integration formulas tabulated above.  (HINT:  With  U = r  you'll end subtracting two very simple integrands:  3 dS  and  dS.)

An Elementary Argument :

The apparent area surrounded by a simple planar curve is proportional to the cosine of its tilt to the observer.  This establishes the advertised property in terms of scalar products, for all simple oriented planar curves, including triangles.

The same is true of a non-planar quadrilateral, because such a polygon always has the same apparent area as two triangles sharing an edge  (one of the quadrilateral's diagonal)  if they are oriented in such a way that this hinge is traveled in opposite directions.  The quadrilateral contour is effectively equivalent to that of the triangles  (as the hinge would be counted once positively and once negatively if the triangles were considered individually).

This argument may be extended by induction to any polygon and, by continuity, to any smooth enough curve.  So, the above formula for  S  and its relevance to apparent areas hold for all  rectifiable  3D curves.  QED

An  elementary  proof of the Kelvin-Stokes formula could proceed similarly.

Note that the  surface area  of a surface bounded by a given loop has little to do with the above vectorial area.  For example, an hemisphere of radius  R  has a surface area of  2pR2,  which is twice the magnitude of the equator's vectorial area...  The vectorial area of an  entire  sphere is zero !

(2005-07-31)   Formulas of Vector Calculus
Differential identities for three-dimensional fields.

All our operators are additive  (e.g.,  div (A+B)  =  div A + div B )  and they  commute  with the  Laplacian  operator  D, defined in the second line below.

div rot A   =   0 rot grad p   =   0
Dp   =   div grad p DA   =   grad div A - rot rot A
div (p A)   =   p div A  +  A . grad p rot (p A)   =   p rot A + grad p ´ A
div (A ´ B)   =   B . rot A - A . rot B grad (p q)   =   p grad q  +  q grad p
D(pq)  =  pDq + qDp + 2 grad p . grad q D(pA) =  p DA + ADp + 2 (grad p.Ñ) A
rot (A ´ B)     =     A div B  -  B div A  +  (B.Ñ)A  -  (A.Ñ)B
grad (A . B) = A ´ rot B + B ´ rot A + (B.Ñ)A + (A.Ñ)B
 Hermann von Helmholtz 

Helmholtz Decomposition

The  Fundamental Theorem of Vector Calculus  states that any  smooth  vector field  (decaying rapidly enough at large distances)  is the sum of an  irrotational  field  (of zero rotational)  and a  solenoidal  one  (of zero divergence).  Such a sum is called a  Helmholtz decomposition.

Hodge decomposition is the generalization of this to all differential forms.

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