Visual proof of Gauss Theorem?

Question

I saw some proofs about Gauss Theorem here but I could not understand everything about it.

Is it possible to have a visual proof of Gauss Theorem, as it can be very interesting to understand and see.

Answer 1

Surface integral of vectorial quantity is the net flux & Divergence of vectorial quantity is total vectorial quantity produce or sink otherwords, total sources or sinks of vector quantity. So physically we can see,

“Total vectorial quantity produce or sink inside closed surface throughout the volume is equal to net flex of this vectorial quantity across the volume boundary.”

Let’s consider, S is a closed surface such that no line parallel to co ordinate axes cuts it in more than two points. Thus, S is a double valued surface over its projection on the region D in XY plane. S be consist of 2 sub surfaces lower surface S1 & upper surface S2.

S1 : $z = f1 (x,y)$ consisting of points $(x,y,f1)$

S2 : $z = f2 (x,y)$ consisting of points $(x,y,f2)$

Now, assuming a vectorial quantity, $F = Fx i +Fy j +Fz k$

Total vector quantity produce or sink inside S throughout the volume V is

Taking a part,

Since Upper surface S2 where unit normal $η2$ makes an angle with Z axis then $η2.k=cosθ2$

Lower surface S1 where unit normal $η1$ makes an angle with Z axis then $η1.k=-cosθ1$

Thus, The projections are,
$S2 : dxdy = cosθ2*dS2$ & $S1 : dxdy = -cosθ1*dS1$

Similarly,

Therefore,

QED.