By Roel Snieder
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By the same token, the flux through the left surface perpendicular through the x-axis is given by −vx (x, y, z)dydz, the − sign is due to the fact the component of v in the direction outward of the cube is given by −vx . ) This means that the total outward flux x through the two surfaces is given by vx (x + dx, y, z)dydz − vx (x, y, z)dydz = ∂v ∂x dxdydz. The same reasoning applies to the surfaces perpendicular to the y- and z-axes. 2. 4) where dV is the volume dxdydz of the cube and (∇ · v) is the divergence of the vector field v.
4) gives us the outward flux dΦ through an infinitesimal volume dV ; dΦ = (∇ · v)dV . We can immediately integrate this expression to find the total flux through the surface S which encloses the total volume V : S v · dS = V (∇ · v)dV . 1). This expression is called the theorem of Gauss.. 1) we did not use the dimensionality of the space, this relation holds in any number of dimensions. 1). In one dimension the vector v has only one component vx , hence (∇·v) = ∂x vx . A “volume” in one dimension is simply a line, let this line run from x = a to x = b.
The theorem of Stokes tells us how to do this. 1). 2) that we write in a slightly different form as: dS v · dr = ( × v) · n ˆ dS = ( × v) ·dS . 2) is that in the expression above we have not aligned the zaxis with the vector × v. 1). 1) holds for an infinitesimal surface area. However, this expression can immediately be integrated to give the surface integral of the curl over a finite surface S that is bounded by the curve C: C v · dr = S (∇ × v) ·dS . 2) 60 CHAPTER 7. THE THEOREM OF STOKES This result is known as the theorem of Stokes (or Stokes’ law).