Gauss Theorem, Stokes and Green's Theorem | Engineering Mathematics for Mechanical Engineering PDF Download

The Divergence Theorem Of Gauss

where ˆn is the positive (outward drawn) normal to S. Gauss diveregnce theorem can also be stated as following: The surface integral of the normal component of a vector A taken over a closed surface is equal to the integral of the divergence of A taken over the volume enclosed by the surface.
Proof: Let S be a closed surface which is such that any line parallel to the coordinate axes cuts S in at most two points. Assume the equations of the lower and upper portions, S₁ and S₂, to be z = f₁(x, y) and z = f₂(x, y) respectively. Denote the projection of the surface on the xy plane by R.Consider,

For the upper portion S₂, dydx = cos γ₂dS₂ = k.n₂dS₂ since the normal n₂ to S₂ makes an acute angle γ₂ with k. For the lower portion S₁, dydx = − cos γ₁dS₂ = k.n₁dS₁ since the normal n₁ to S₁ makes an obtuse angle γ₁ with k. Then

and
so from equations (2) and (3)

Similarly, by projecting S on the other coordinate planes,
Adding equations (4), (5) and (6),
Or
Proved.

Physical demonstration of the divergence theorem

Let A = velocity v at any point of a moving fluid. From Figure (2 a) below:
Volume of fluid crossing dS in ∆tseconds
= volume contained in cylinder of base dS and slant height v∆t
= (v∆t).ndS = v.ndS∆t
Then, volume per second of fluid crossing dS = v.dS Prom Figure (2 b)

Total volume per second of fluid emerging from closed surface S

And we know that ∇.vdV is the volume per second of fluid emerging from a volume element dV. Then Total volume per second of fluid emerging from all volume elements in S

Thus

Stokes’ Theorem

where C is traversed in the positive direction. The direction of C is called positive if an observer, walking on the boundary of S in this direction, with his head pointing in the direction of the positive normal to S, has the surface on his left.
In other words Stokes’ theorem may be stated as following: The line integral of the tangential component of a vector A taken around a simple closed curve C is equal to the surface integral of the normal component of the curl of A taken over any surface S having C as its boundary.
Proof: Let S be a surface which is such that its projections on the xy,

yz and xz planes are regions bounded by simple closed curves, as indicated in the adjoining figure. Assume S to have representation z = f(x, y) or x = g(y, z) or y = h(x, z), where f, g, h are single-valued, continuous and differentiable functions. We must show that

where Cis the boundary of S.
Consider first 
Since

If z = f(x, y) is taken as the equation of S, then the position vector to any point of S is r = xi + yj + zk = xi + yj + f(x, y)k so that . But ∂r/ ∂y is a vector tangent to S and thus perpendicular to n, so that

Substituting equation (14) into equation (12), we obtain

Now on S,₁(x, y, z) = A₁(x, y, f(x, y)) = F(x, y); hence  and equation (16) becomes

where R is the projection of S on the xy plane. By Green’s theorem for the plane the last integral equals ¸  Fdx where Γ is the boundary of R. Since at each point (x, y) of Γ the value of F is the same as the value of A₁ at each point (x, y, z) of C, and since dx is the same for both curves, we must have,

Similarly, by projections on the other coordinate planes,

 Thus by addition,Proved.
The theorem is also valid for surfaces S which may not satisfy the restrictions imposed above. For assume that S can be subdivided into surfaces S₁, S₂, .....Sk with boundaries C₁, C₂, .....C_k which do satisfy the restrictions. Then Stokes’ theorem holds for each such surface. Adding these surface integrals, the total surface integral over S is obtained. Adding the corresponding line integrals over C₁, C₂, .....C_k, the line integral over is obtained. 

Green’s Theorem in The Plane

where C is traversed in the positive (counterclockwise) direction. Unless otherwise stated we shall always assume ¸ to mean that the integral is described in the positive sense.
 Green’s theorem in the plane is a special case of Stokes’ theorem. Also, it is of interest to notice that Gauss’ divergence theorem is a generalization of Green’s theorem in the plane where the (plane) region R and its closed boundary (curve) C are replaced by a (space) region V and its closed boundary (surface) S. For this reason the divergence theorem is often called Green’s theorem in space.
Green’s theorem in the plane also holds for regions bounded by a finite number of simple closed curves which do not intersect.
Proof: Let the equations of the curves AEB and AFB (see Fig. 4) be  

y = Y₁(x) and y = Y₂(x) respectively. If R is the region bounded by C, we have

Then,

Similarly let the equations of curves EAF and EBF be x = X₁(y) and x = X₂(y) respectively. Then

Then,

Adding equations (27) and (29)

Proved.