Gauss Divergence Therom

Made By Vikash

1. Gauss Divergence Therom
Prepared By:-
Vikash Maurya
Guided :By-
SUBJECT:- Gauss Divergence Theorem

2. The Divergence Theorem
In this section, we will learn about:
The Divergence Theorem & Gauss Divergence Theorem.

3. Divergence of a vector Field
The divergence of a vector field ar a point is a scalar
quantity of magnitude equal to flux of that vector field
diverging out per unit volume through that point in
mathematical from, the dot product of del operator and
the vector field A(x,y,z,) gives the divergence of a vector field
A. i.e. ,
div A. = .. A

4. But = Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z and A = ÎAX + ĵ AY +
AZ k

. A = (Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z).(ÎAX + ĵ AY + AZ k)
. A = ( ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)
div A = ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z

5. The divergence of a vector field can bev
positive nigetive or zero.
(1) if the divergance of a vector field at a point is
positive (div A = +ve) it means that the flux
entering through the surface is less then the flux
comeing out through that surface. In other words ,
there is a source of that vector field in the region.

6. (2) if the divergance of a vector field at a
point is nigetive (div A = -ve) it means
that the flux entering through the surface
is more then the flux comeing out
through that surface. In other words ,
there is sink of that vector field in the
region.

7.  if the divergance of a vector field is
zero. it implies that the flux entering
through the surface is equal to the flux
leaving that surface. In other words ,
there is neither the source nor sink in
that region.

8. According to this theorem the volume intrigle of
divergence of a vector field A over a volume V is
equal to the surface integral of that vector field A
taken over the surface S which encloses that
volume V. I,.e.
 ∫∫∫(divA)dV = ∫∫A. da
thus this theorem is used to convert the volume
integral into the surface integral or to convert
surface integral into the volume intrgral.

9. Proof : In cortesian coordinates,
div A = .A = (Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z).(ÎAX + ĵ AY + AZ
k)
= ( ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)
And dV = dxdydz
While A.da = (ÎAX + ĵ AY + AZ k).(Îdax + ĵday + kdaz)
= Axdax + Ayday + Azdaz
= Axdax + Ayday + Azdxdy
y

10. According to Gauss Theorem,
∫∫∫ (∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)dxdydz = ∫∫ (Axdydz + Aydxdz + Azdxdy)