       Gauss Theorem

• To: mathgroup at smc.vnet.net
• Subject: [mg37471] Gauss Theorem
• From: karlk at eng.umd.edu (Karl)
• Date: Thu, 31 Oct 2002 04:42:28 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```In studying fluid mechanics, I have been trying to understand Gauss
theorem in general form for a tensor of order n.  To make things
simple, I consider two-dimensional space.  Then my understanding is
that for a vector v = v1(x1,x2)i + v2(x1,x2)j:

(A)  Integral over volume of D[vi,xi] = integral over surface of vi*ni

where ni is normal in i direction.  This equation is a single
equation, which can be written explicitly (i.e. performing summation
over i) as

(B)  Integral over volume of D[v1,x1]+D[v2,x2] = integral over surface
of v1*n1 + v2*n2.

Now, this is Gauss theorem for a vector.  However, Gauss theorem for
scalar, F, states that

(C)  Integral over volume of D[F,xi] = integral over surface of F*ni

or explicitly, it yields two equations:

(D1) Integral over volume of D[F,x1] = integral over surface of F*n1
(D2) Integral over volume of D[F,x2] = integral over surface of F*n2

I am unable to understand the relation between Gauss theorem for
scalar and that for vector (not to mention 2nd order tensor), because
the following seems a contradiction:  Consider each component of the
vector, v, discussed above to be a scalar function (i.e. v1(x1,x2) is
a scalar).  Then the scalar form of Gauss theorem implies that

(E1) Integral over volume of D[v1,x1] = integral over surface of
v1*n1.

(E2) Integral over volume of D[v2,x2] = integral over surface of v2*n2

Actually, applying the scalar form of Gauss theorem completely to each
component of v, will yield two more equations in addition to just E1
and E2.  Anyway, equations E1 and E2 imply that equation (B) above, is
actually just the sum of equations E1 and E2.

However, I have been told by my profressor that equations E1 and E2
are not valid.  Where is my reasoning wrong?