Understanding the Divergence Theorem, Also Called Gauss’s Theorem

I don’t believe in rigid mathematics nor I accept these wild misconceptions about engineers being unable to do real math and they’re only deemed to apply it, as it is!  So! What is the divergence theorem?

It is the concept of calculating a flux entering a region that constitutes a volume or a surface.  A region is a volumetric body in 3D cases or a surface body in 2D cases. So why do we care about a flux going through a region of space? Well, think about the conservation of mass flow rate in a pipe. Think about the Navier Stokes equations and heat conduction! Fluxes are everywhere, they can be magnetic, fluid or in the form of heat.



What is a Flux in mathematical terms?

It is a vector with 3 components (x,y,z),for example:

\vec{F}=3.x\vec{i}+0.\vec{j}+0.\vec{k}  (1)

Vector \vec{F} is dependent on x and is horizontal at any points x in space  

Figure-1- Divergence Theorem – Vector F going through a volume body

\vec{F} magnitude is increasing horizontally in the x,y plane.

What Does a Flux Have to Do With Gauss’s Theorem?

\int \int \int div(\vec{F}). {\partial{v}}  =  \int \int \vec{F}. \vec{n}. {\partial{s}}

                      1                               2

Let’s dissect each element:

1: The first element is the summation of the divergence of \vec{F} multiplied by a differential volume (divide the region into Infinitesimal small volumes)

1:   \sum(\frac{\partial {Fi}}{\partial x} + \frac{\partial {Fj}}{\partial y} + \frac{\partial {Fk}}{\partial z}).\Delta v

2:  The second element is the summation of,\vec{F}} multiplied by a unit vector \vec{n} normal to the faces of the volumetric region.

Figure -2- Unit vector n
Figure -2- Unit vector n


2: \sum(\vec{F}.\vec{n}.\Delta.s})

  • The volumetric region in figure 1 and 2 is a cube and it has a volume of 1 m^3 and a surface area (each face) 1m^2
  • I divided the cube into 27 small cubes (figure 3).
Figure -3- differential volume
Figure -3- differential volume

\Delta v is equal to (0.3333 *0.3333*0.3333) = 0.0370259

\frac{\partial {Fi}}{\partial x} + \frac{\partial {Fj}}{\partial y} + \frac{\partial {Fk}}{\partial z} =3

1: \sum(\frac{\partial {Fi}}{\partial x} + \frac{\partial{Fj}}{\partial y} + \frac{\partial{Fk}}{\partial z}).\Delta v = \sum(3\cdot0.0370259) = 3 . 0.03702 . 27 = 3

Let’s see if element 1 is really equal to element 2.

\vec{F}.\Vec{n} \neq 0 only when \vec{F} and \vec{n} are not perpendicular, hence, \vec{F} . \vec{n} matter only at the western and eastern faces.

Figure -4- Multiplication of n and F vectors
Figure -4- Multiplication of n and F vectors





Western face: \vec{F}=3.x\vec{i}  \vec{F}.\Vec{n}=3.x\vec{i}.(-1.\vec{i})  | on the Western face, x = 1, so \vec{F}.\Vec{n}=-3

Eastern face: \vec{F}.\Vec{n}=3.x\vec{i}.(+1.\vec{i}) | on the Eastern face, x = 2 so \vec{F}.\Vec{n}=+6

There are 9 faces on the western surface and 9 faces on the eastern surface, so \sum(\vec{F}.\vec{n}.\Delta.s) = 9.(6 \cdot 0.11111)- 9.(3 \cdot 0.11111) = 3

Conclusion

What I did above is proving the Gauss’s theorem by manually doing the calculations, but you may ask, how can you relate this to a real-life problem? Well this can be applied to the Momentum equation or Continuity equation in fluids, Heat transfer and many other physical applications.

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