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 1869-1951

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© 2000-2005 Gérard P. Michon, Ph.D.

Differential Forms

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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 the form  w  over the border  ¶W.  This is called  Stokes' Theorem :

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 prominent Bourbaki founder André Weil,  see:  The Many Faces of Nicolas Bourbaki).

Three-Dimensional Incarnations and Relatives of  Stokes' Theorem
NameFormulaW  and ¶W
Conservativity òC   grad f . dr     =     f(b) - f(a)  A Path has a 2-Point Boundary
Kelvin-Stokes 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   é
ê
ë
U/x . dS
U/y . dS
U/z . dS
ù
ú
û
      =       òC   U ´ dr
 Border of a Surface
Ostrogradski
(Gauss)
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) Otrogradski's formula was independently discovered by Lagrange in 1762, by Gauss in 1813 and by Green in 1825.  It was proved by Mikhail Vasilevich Ostrogradski in 1831.

Special cases of Ostrogradski's theorem, known as  Green's formulas  are obtained with   U   =   p grad q :

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

Among the above formulae, the least popular is probably the one involving an irreducible nabla  (as tabulated above in terms of cartesian coordinates).

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.


(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 arbitrary origin chosen 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, a 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)   Vector Calculus
Differential identities for three-dimensional fields.

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

ScalarVector
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

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