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

The Divergence Theorem Of Gauss

The divergence theorem of Gauss states that if V is the volume bounded by a closed surface S and A is a vector function of position with continuous derivatives, then
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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.Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringConsider,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringFor 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
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering
and
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineeringso from equations (2) and (3)
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Similarly, by projecting S on the other coordinate planes,
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringAdding equations (4), (5) and (6),
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringOr
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringProved.

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)
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Total volume per second of fluid emerging from closed surface S

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Thus

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Stokes’ Theorem

Stokes’ theorem states that if S is an open, two-sided surface bounded by a closed, nonintersecting curve C (simple closed curve) then if A has continuous derivatives

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineeringyz 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
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

where Cis the boundary of S.
Consider firstGauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering 
Since

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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 Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering. But ∂r/ ∂y is a vector tangent to S and thus perpendicular to n, so that

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Now on S,1(x, y, z) = A1(x, y, f(x, y)) = F(x, y); hence Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering and equation (16) becomes

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

where R is the projection of S on the xy plane. By Green’s theorem for the plane the last integral equals ¸ Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering 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,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Similarly, by projections on the other coordinate planes,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

 Thus by addition,Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical EngineeringProved.
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, .....Ck 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, .....Ck, the line integral over is obtained. 

Green’s Theorem in The Plane

If R is a closed region of the xy plane bounded by a simple closed curve C and if M and N are continuous functions of x and y having continuous derivatives in R, then

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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  
Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Then,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

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

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Then,

Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Adding equations (27) and (29)Gauss Theorem, Stokes and Green`s Theorem | Engineering Mathematics for Mechanical Engineering

Proved.

The document Gauss Theorem, Stokes and Green's Theorem | Engineering Mathematics for Mechanical Engineering is a part of the Mechanical Engineering Course Engineering Mathematics for Mechanical Engineering.
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FAQs on Gauss Theorem, Stokes and Green's Theorem - Engineering Mathematics for Mechanical Engineering

1. What is the Divergence Theorem of Gauss?
Ans. The Divergence Theorem of Gauss, also known as Gauss's Theorem, states that the total outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface.
2. How is Stokes' Theorem related to the Divergence Theorem of Gauss?
Ans. Stokes' Theorem is a generalization of the Fundamental Theorem of Calculus and is related to the Divergence Theorem of Gauss through the concept of curl. While the Divergence Theorem relates the flux of a vector field to its divergence, Stokes' Theorem relates the circulation of a vector field to its curl.
3. What is Green's Theorem in the Plane?
Ans. Green's Theorem in the Plane relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It establishes a connection between line integrals and double integrals, similar to how the Divergence Theorem and Stokes' Theorem connect surface and line integrals.
4. How are the Divergence Theorem, Stokes' Theorem, and Green's Theorem applied in Mechanical Engineering?
Ans. In Mechanical Engineering, these theorems are used to analyze and solve problems related to fluid flow, heat transfer, stress analysis, and other physical phenomena. They provide a mathematical framework for understanding and predicting the behavior of vector fields in different engineering applications.
5. What are some practical examples where the Divergence Theorem, Stokes' Theorem, and Green's Theorem are used in Mechanical Engineering?
Ans. Some practical examples include analyzing fluid flow in pipes, calculating heat transfer in thermal systems, determining stress distribution in structural components, and predicting electromagnetic field behavior in electronic devices. These theorems play a crucial role in the design and optimization of various mechanical systems.
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