11.05.2012
Vector Analysis and Geometry
Subject Expert: Bijumon R
Content Editor: Nandakumar
PRESENTER:RASHMI.M
E- Content in Mathematics
9
Volume integrals and Gauss’s Divergence Theorem
Objectives
From this unit a learner is expected to achieve the following
- Familiarize gauss divergence theorem
- Learn the definition of surface integral of a vector field.
- Learn the method of finding surface integral of a vector field.
Sections
1. Introduction
2. Volume Integral
3. Vector Volume Integral
4. The Divergence Theorem
In vector calculus, the divergence theorem (also known as Gauss Divergence Theorem)is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface. Before entering into the details of divergence theorem, we consider the fundamentals of volume integrals.
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Volume Integral
Let V be the volume bounded by a surface S. Let F(x, y, z) be a valued function (may or may not be a vector function) defined over V. Now subdivide the volume V into n elements of volumes V1, V2, . . ., Vn. Let P1(x1, y1 , z1), P2(x2, y2 , z2), . . ., Pn(xn, yn, zn) be arbitrary points in V1, V2, . . ., Vn respectively. Consider the finite sum
.
The limit of the above sum when n and Vr 0, if it exists, is called the volume integral of F(x, y, z) over V and is denoted by
That is,
If we subdivide the volume V into small cuboids by drawing lines parallel to the thee co-ordinate axes, then dV = dx dy dz and consequently, the above volume integral becomes
Example1 Let and let V denote the closed region bounded by the planes Evaluate
Solution
The region enclosed by the planes can be found to be given by the following system of inequalities:
Hence
= 128.
Example 2 Let F = 2xzi x j + y2k . Evaluate , where V is the region bounded by the surfaces x = 0, y = 0, y = 6, z = x2, z = 4.
Solution
We know that
The region V is the set of all points (x, y, z) for which 0 x 2; 0 y 6; x2 z 4.
For the integration over V we first integrate with respect to z (from z = x2 to z = 4) considering x and y as constant. Then we integrate with respect to y (from y = 0 to y = 6)keeping x fixed. Finally, integrate from x = 0 to x = 2. The procedure can be seen below:
= 128 i 24 j + 384k , on simplification.
3. Vector Volume Integral
Instead of the scalar function F(x, y, z) , if we consider a vector function F (x, y, z), then the volume integral is called vector volume integral and is denoted by
If F =F1i + F2 j + F3k, then
4. The Divergence Theorem
Theorem(Gauss’s divergence theorem)The volume integral of the divergence of F, taken throughout a bounded domain V , equals the surface integral of the normal component of F taken over S,the boundary of V.
i.e., . . . (1)
Remarks
- In other words, theorem says that, the total divergence within D equals the net flux emerging from D.
- This theorem enables us to convert a surface integral into a volume integral and thus gives another method for the evaluation of surface integral.
- The Gauss divergence theorem can be extended to surfaces where a line parallel to any coordinate axis meets them in even number of times.
Example 3 Consider the function F =x2i + z j + yz k and the volume enclosed by the cube given by 0 x 1, 0 y 1 and 0 z 1. Verify Gauss divergence theorem for F
Solution
Now
on simplification. (2)
The surface integral is to be evaluated as the sum of six integrals corresponding to the 6 faces of the cube (Ref figure) .
(i) For the face on the XY plane ; z = 0 and the out ward drawn normal is k and hence we have
(ii) For the face on the z = 1 plane, the out ward drawn normal is k and hence we have
(iii) For the face on the ZX plane; y = 0 and the outward direction unit normal is j and hence we have
(iv) For the face on the y = 1 plane the outward direction unit normal is j and hence we have
(v) For the face on the YZ plane; x = 0 and the outward direction unit normal is i and hence we have
(vi) For the face on the x = 1, the outward direction unit normal is i and hence we have
Adding the results in (i) to (vi) we have
(3)
From (1) and (2) Gauss theorem is verified.
Example4 Let F = xy2 i + yz2 j + zx2 k . Using Divergence Theorem, evaluate over the sphere given by x2 + y2 + z2 =1.
Solution
By Divergence theorem, we have
,
where V is the region bounded by the sphere S : x2 + y2 + z2 =1.
Now divF = .F = .( xy2 i + yz2 j + zx2 k) = y2 + z2 + x2 .
by using the spherical polar coordinates
, on simplification.
Example5 If F = and .F = 0, show (using divergence theorem) that
where V is the volume enclosed by S.
Solution
We know that .(F) = .F + .F
= . + 0 , since F = and .F = 0
= . = = F2 = F2
by Gauss’s divergence theorem, we have
Example6 Show that , where V is the volume enclosed by the surfaces and r is the position vector.
Solution
(In the solution above we have used the fact that for
)
Example7 Show that , where S is a sphere, center origin and radius r and .
Solution
By Gauss’s Divergence theorem, we have
,
where V is the volume enclosed by the surface S.
Now
by changing to polar spherical co-ordinates where r varies from 0 to r, varies from 0 to 2 and from to 0.
Summarynow its time to summarise the class we have discussed the volume integrals and vector volume integrals. Explained the famous theorem gauss divergence theorem with suitable examples
Assignments
1.Find the integral of f (x,y,z)=y over the volume of the sphere contained inside x2+y2+z2 = 1
- Show that where V is the volume enclosed by the surfaces and r is the position vector.
3By the divergence theorem, evaluate , where F = 4 x i 2y2 j + z2 k and S is the surface bounded by x2 y2 = 4, z = 0 and z = 3.
4Prove that for any closed surface S.
FAQ
1. What is the physical meaning of Gauss Divergence Theorem?
Ans. If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sink there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is true. The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Quiz:
1. Divergence theorem states that
(a) outward flux of a vector fieldis equal to the volume integral of the region inside the surface
(b) outward flux of a vector field is greater than the volume integral of the region inside the surface
(c) outward flux of a vector field is less than the volume integral of the region inside the surface
(d) there is no relation between outward flux and the volume integral
Ans.(a)outward flux of a vector field is equal to the volume integral of the region inside the surface
2. gauss theorem gives the method for the evaluation of
(a) surface integral (b)volume integral(c)surface area(d)none of these
Ans: (a) surface integral
GLOSSARY
Volume integral:
Gauss’s divergence theorem:The volume integral of the divergence of F, taken throughout a bounded domain V , equals the surface integral of the normal component of F taken over S,the boundary of V.
i.e.,
REFERENCES
Books
- Murray R. Spiegel, Vector Analysis, Schaum Publishing Company, New York.
- N. Saran and S. N. Nigam, Introduction to Vector Analysis, Pothishala Pvt. Ltd., Allahabad.
- Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1999.
- Shanti Narayan, A Text Book of Vector Calculus, S. Chand & Co., New Delhi.
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