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**Paper No. : Mathematical tools for materials**

**Module : Vector integration **
** **

** **

Prof. Vinay Gupta, Department of Physics and Astrophysics, University of Delhi, Delhi

**Development Team **

**Principal Investigator **

**Paper Coordinator **

**Content Writer **

**Content Reviewer **

Prof. P. N. Kotru ,Department of Physics, University of Jammu, Jammu-180006

Prof. V. K. Gupta, Department of Physics, University of Delhi, Delhi-110007

Prof Mahavir Singh Department of Physics, Himachal Pradesh University, Shimla

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**Description of Module ** ** **

**Subject Name ** Physics

**Paper Name ** Mathematical tools for materials
**Module Name/Title ** Vector integration

**Module Id ** VA-3

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**TABLE OF CONTENTS**

1. Ordinary integration of vectors 2. Line integral

2.1 Evaluation of line integral 2.2 Conservative fields 3. Surface integrals 4. Volume integrals

5. Fundamental theorems of vector calculus

5.1 Fundamental theorem for divergence โ Gaussโ theorem 5.2 Greenโs theorem in a plane

5.3 Fundamental theorem for curl - Stokesโ theorem 5.4 Some other important theorems

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**LEARNING OBJECTIVES**

1. In this module the topic of vector integration is taken up. First the ordinary integration of a vector is described.

2. Next the central concept of line integral is introduced. Evaluation of line integrals is described by taking up examples.

3. The special case of line integral of conservative fields is described in details with examples.

4. Next surface integral of vectors is described both for the case of open and closed surface.

5. Evaluation of volume integral is explained by an example.

6. Lastly certain fundamental theorems regarding the line, volume and surface integrals are described. The Gauss theorem, Greenโs theorem in a plane and the Stokeโs theorem are enunciated and proved. Some other fundamental theorems are also stated without proof.

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**V e c t o r I n t e g r a t i o n**

1. Ordinary integration of vectors

After having studied differentiation of vectors and vector fields, our next task is to study vector integration. The most useful concepts in this regard are the line, surface and volume integrals of vector fields. We first define the ordinary derivative of a vector quantity. If ๐นโ is a vector function of a single scalar variable u, its integral over u is defined like the integral of a function of one variable. Let

๐นโ(๐ข) = ๐ฬ๐น_{๐ฅ}+ ๐ฬ๐น_{๐ฆ}+ ๐ฬ๐น_{๐ง}

If the functions ๐น_{๐ฅ}, ๐น_{๐ฆ}, ๐น_{๐ง} are integrable functions of u, then

โซ ๐นโ(๐ข)๐๐ข โก ๐ฬ โซ ๐น๐ฅ(๐ข)๐๐ข + ๐ฬ โซ ๐น_{๐ฆ}(๐ข)๐๐ข + ๐ฬ โซ ๐น_{๐ง}(๐ข)๐๐ข

is the indefinite integral of ๐นโ(๐ข). If there exists a vector ๐บโ(๐ข), such that ๐นโ(๐ข) =^{๐๐บโ(๐ข)}

๐๐ข , then

โซ ๐นโ(๐ข)๐๐ข = โซ^{๐๐บโ(๐ข)}_{๐๐ข} ๐๐ข = ๐บโ(๐ข) + ๐พโโโ

where ๐พโโโ is a constant vector, independent of u. The definite integral between two limits u = a and u = b is then

โซ ๐นโ(๐ข)๐๐ข_{๐}^{๐} = ๐บโ(๐) โ ๐บโ(๐)

The geometrical interpretation of this definite integral follows from that of an ordinary integral, namely, its three components are the areas of the curves of the three components of ๐นโ(๐ข).

We now look on the integrals of special interest to us, viz., the line, surface and volume integrals of vector fields, and also of scalar and tensor fields. The line, surface and volume integrals refer to integral of a field over a curve, a surface or a volume in the three-dimensional space.

2. Line integral

A line integral of a vector field ๐นโ is an integral of the form

โซ^{๐}^{โโ} ๐นโ. ๐๐โโโโโ

๐โโ,๐ซ

Very often the notation ๐๐โโโโ is used instead of ๐๐โโโโโ. The integral is to be carried from point in space with position vector ๐โ to one with position vector ๐โโ, along a designated path ๐ซ. In case the points ๐โ and ๐โโ are the same, i.e., if we traverse a closed path in space, the line integral is usually denoted by a circle around the integral sign:

โฎ ๐นโ. ๐๐โโโโโ

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To evaluate the line integral over an open or closed curve ๐ซ, we find the scalar product of ๐นโ with the displacement vector ๐๐โโโโ, which is a vector along the tangent to the curve at each point.

๏ท Remember that in general the line integral over a closed path is not zero. For example work done by a force: โซ ๐นโ. ๐๐โโโโโ, may not vanish over a closed path. The most familiar everyday example is that of friction.

Those special forces for which the line integral over a closed path is indeed zero are known as
*conservative forces. *

2.1 Evaluation of line integral

To evaluate the line integral defined above, a simple method is to first choose a coordinate system. In Cartesian coordinates, let the points along the curve ๐ซ be designated by

๐โ = ๐ฅ(๐ข)๐ฬ + ๐ฆ(๐ข)๐ฬ + ๐ง(๐ข)๐ฬ (1)

Here u is a scalar parameter, and equation (1) is the equation of a curve in space. Then
๐นโ. ๐๐โโโโ = ๐น_{๐ฅ}๐๐ฅ + ๐น_{๐ฆ}๐๐ฆ + ๐น_{๐ง}๐๐ง

and

โซ^{๐}^{โโ} ๐นโ. ๐๐โโโโ

๐โโ,๐ซ = โซ_{๐}^{๐}^{2}^{(๐ข}^{2}^{)}(๐น_{๐ฅ}๐๐ฅ + ๐น_{๐ฆ}๐๐ฆ + ๐น_{๐ง}๐๐ง)

1(๐ข_{1}) (2)

Example-1

We first give an example of a curve in a plane. Evaluate the line integral of the function
๐นโ = ๐ฆ^{2}๐ฬ + 2๐ฆ(๐ฅ + 1)๐ฬ

from the point ๐โ = (0,0) to ๐โโ = (1,1) over the two paths shown: (i) straight line between the two points and (ii) along the path shown by arrows.

(1,1) (i)

(ii)

(0,0) (ii)

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(a) Along (i) x = y

๐ฅ = ๐ฆ; ๐๐ฅ = ๐๐ฆ; ๐นโ. ๐๐โโโโโ = (๐ฅ^{2}๐ฬ + 2๐ฆ(๐ฅ + 1)๐ฬ). (๐๐ฅ๐ฬ + ๐๐ฆ๐ฬ) = ๐ฅ^{2}๐๐ฅ + 2๐ฅ(๐ฅ + 1)๐๐ฅ

= (3๐ฅ^{2}+ 2๐ฅ)๐๐ฅ โน โซ (3๐ฅ^{2}+ 2๐ฅ)๐๐ฅ

1 0

= 2

(b) For this curve there are two parts. In the first part *dy = 0 from (0,0) to (1,0). In the second part dx=0 *
from (1,0) to (1,1). Hence

โซ ๐นโ. ๐๐โโโโโ = โซ ๐ฅ_{0}^{1} ^{2}๐๐ฅ+ โซ 4๐ฆ๐๐ฆ_{0}^{1} = 8/3
Example-2

Find the total work done in moving a particle in a force field given by ๐นโ = ๐ฅ๐ฆ๐ฬ + ๐ง๐ฬ โ 10๐ฅ๐ฆ๐ฬ

along a path whose parametric equation is given by

๐ฅ = 1 + ๐ก^{2}; ๐ฆ = 2๐ก^{2}; ๐ง = ๐ก^{3}; from ๐ก = 0 to ๐ก = 1

โซ ๐นโ. ๐๐โโโโโ = โซ(๐ฅ๐ฆ๐ฬ + ๐ง๐ฬ โ 10๐ฅ๐ฆ๐ฬ). (๐๐ฅ๐ฬ + ๐๐ฆ๐ฬ + ๐๐ง๐ฬ) = โซ(๐ฅ๐ฆ๐๐ฅ + ๐ง๐๐ฆ โ 10๐ฅ๐ฆ๐๐ง)

= โซ [(1 + ๐ก_{0}^{1} ^{2})8๐ก^{3}๐๐ก + 4๐ก^{4}๐๐ก โ 60(1 + ๐ก^{2})๐ก^{4}๐๐ก]= โซ (8๐ก_{0}^{1} ^{3}โ 56๐ก^{4}+ 8๐ก^{5}โ 60๐ก^{6})๐๐ก= 1726/105

2.2 Conservative fields

In analogy with the force, any vector field whose line integral is independent of the chosen path and depends only on the two end points is called a conservative field. Both the above examples are of non-conservative fields. In fact there is a simple and well known criterion to decide whether a given field is conservative or not. The result is given by the following theorems:

Theorem-1

The line integral of the gradient of a scalar function ๐(๐ฅ, ๐ฆ, ๐ง) along any curve from the point ๐โโโโ to the point ๐โ is _{0}
independent of the path taken and equals the difference between the values of the function at the two points. That
is,

โซ^{๐โ} โโโโ๐. ๐๐โโโโโ

๐_{0}

โโโโโ ,๐ซ = ๐(๐โ) โ ๐(๐โโโโ) _{0} (3)

independent of the path ๐ซ chosen.

Proof: By definition ๐๐โโโโโ. โโโโ๐ = ๐๐

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Hence

โซ^{๐โ} โโโโ๐. ๐๐โโโโโ

๐_{0}

โโโโโ ,๐ซ = โซ_{๐}^{๐โ} ๐๐

โโโโโ ,๐ซ0 = ๐(๐โ) โ ๐(๐โโโโ) _{0} (4)

without any reference to the path taken.

Theorem-2

This is the converse of the above theorem: If the line integral of a vector function ๐นโ about every closed curve vanishes, then it can be written as the gradient of some scalar function ๐(๐โ).

Proof: We are given that

โฎ ๐โ. ๐๐โโโโโ = 0 (5)

for every closed curve in a certain region. Let ๐โ be a fixed point in space and ๐โ a variable point. Draw any two paths, C and ๐ถโฒ between these two points. Then C and โ๐ถโฒ together form a closed curve and hence

โซ^{๐โ} ๐โ. ๐๐โโโโโ

๐โโ,๐ถ + โซ^{๐โ} ๐โ. ๐๐โโโโโ

๐โโ,โ๐ถโฒ = 0 โน โซ ๐โ. ๐๐^{๐โ} โโโโโ

๐โโ,๐ถ = โซ^{๐โ} ๐โ. ๐๐โโโโโ

๐โโ,๐ถ^{โฒ}

This implies that the value of the integral is independent of the path of integration and depends only on the end point. Hence it is some function ๐ of the coordinates:

โซ ๐โ. ๐๐^{๐โ} โโโโโ

๐โโ = ๐(๐โ) (6)

Let the two points ๐โ and ๐โ be chosen to be infinitely close to each other. Then

๐โ. ๐๐โโโโโ = ๐๐ = โโโโ๐. ๐๐โโโโโ โน ๐โ = โโโโ๐ (7)

Now we know from the theorem of vector differentiation that a vector field can be written as gradient of a scalar, if and only if, its curl vanishes:

๐โ = โโโโ๐ โบ โโโโ๐น๐โ = 0 (8)

Thus we have a simple criterion:

Theorem-3

The line integral of a vector field over a closed curve is zero, if and only if, the field is curl free.

Example

Prove that the integral

โฎ ๐นโ. ๐๐โโโโโ = 0

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where ๐นโ = (2๐ฅ๐ฆ + ๐ฆ๐ง)๐ฬ + (๐ฅ^{2}+ ๐ง + ๐ฅ๐ง)๐ฬ + (๐ฆ + ๐ฅ๐ฆ)๐ฬ

โโโโ๐น๐นโ = [^{๐}

๐๐ฆ(๐ฆ + ๐ฅ๐ฆ) โ ^{๐}

๐๐ง(๐ฅ^{2}+ ๐ง + ๐ฅ๐ง)] ๐ฬ + [^{๐}

๐๐ง(2๐ฅ๐ฆ + ๐ฆ๐ง) โ ^{๐}

๐๐ฅ(๐ฆ + ๐ฅ๐ฆ)] + [^{๐}

๐๐ฅ(๐ฅ^{2}+ ๐ง + ๐ฅ๐ง) โ

๐

๐๐ฆ(2๐ฅ๐ฆ + ๐ฆ๐ง)] = 0 Hence

โฎ ๐โ. ๐๐โโโโโ = 0

3. Surface integrals

Let S be a two-sided surface. If the surface is a closed surface, the outer surface is taken as the positive and the
direction of the outward drawn normal is taken as positive. If the surface is an open one, then any side is
arbitrarily taken as positive. Given such a surface and an element of this surface *dS, we can associate with it a *
vector ๐๐โโโโโ = ๐ฬ๐๐, with magnitude dS and direction given by ๐ฬ. The integral

โฌ ๐นโ. ๐๐โโโโโ

๐ = โฌ ๐นโ. ๐ฬ๐๐_{๐}

is example of a surface integral. This is called the *flux of *๐นโ over S. We can define other surface integrals
involving a scalar field or a vector field as well. Some other useful surface integrals are

โฌ ๐นโ๐น๐๐โโโโโ = โฌ ๐นโ๐น๐ฬ๐๐ ; โฌ ๐๐๐; โฌ ๐๐ฬ๐๐ = โฌ ๐๐๐โโโโโ

๐ฬ

dS

S

If the integration is over a closed surface, it is usually denoted by โฏ . In general we would expect the integral over a surface to depend on the boundary as well as the actual surface with that boundary. However there is a class of functions for which the integral depends only on the boundary and not the actual surface. For such functions the integral over a closed surface is zero.

Evaluation of surface integrals is usually simplified by considering the projection of the given surface over one of the coordinate planes. Normal to the x-y plane is ๐ฬ. The projection of a surface element dS with normal ๐ฬ on x-y plane is given by

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|๐๐๐ฬ. ๐ฬ| = ๐๐ฅ. ๐๐ฆ โน ๐๐ =^{๐๐ฅ๐๐ฆ}

|๐ฬ.๐ฬ |

If the projection of the surface S on the x-y plane is the region R, then the given surface integral takes the form

โฌ ๐นโ. ๐๐โโโโโ

๐ = โฌ ๐นโ. ๐ฬ๐๐_{๐} = โฌ (๐นโ. ๐ฬ)^{๐๐ฅ๐๐ฆ}

|๐ฬ.๐ฬ |

๐ (9)

Example

Evaluate the surface integral

โฌ ๐นโ. ๐๐โโโโโ

๐ = โฌ ๐นโ. ๐ฬ๐๐_{๐}
where

๐นโ = ๐ง๐ฬ + ๐ฅ๐ฬ + 3๐ฆ๐ฬ

over part of the surface 2๐ฅ + 2๐ฆ + ๐ง = 8 in the first octant.

Gradient to the above surface is given by

โโโโ(3๐ฅ + 2๐ฆ + ๐ง) = 2๐ฬ + 2๐ฬ + ๐ฬ โน ๐ฬ = ^{2๐ฬ+2๐ฬ+๐}^{ฬ}

|2๐ฬ+2๐ฬ+๐ฬ |=^{2๐ฬ+2๐ฬ+๐}^{ฬ}

3 โน ๐ฬ. ๐ฬ = 1/3 Further

๐นโ. ๐ฬ = (๐ง๐ฬ + ๐ฅ๐ฬ + 3๐ฆ๐ฬ).^{2๐ฬ+2๐ฬ+๐}^{ฬ}

3 =^{2๐ง+2๐ฅ+3๐ฆ}

3 =^{16โ2๐ฅโ๐ฆ}

3

For the projected surface in the x-y plane 0 โค ๐ฅ โค 4; 0 โค ๐ฆ โค 4 โ ๐ฅ.

Hence the required surface integral is

โฌ ๐นโ. ๐๐โโโโโ

๐ = โฌ (๐นโ. ๐ฬ)^{๐๐ฅ๐๐ฆ}

|๐ฬ.๐ฬ |

๐
= โซ ๐๐ฅ_{0}^{4} โซ_{0}^{4โ๐ฅ}๐๐ฆ(16 โ 2๐ฅ โ ๐ฆ)= 320/3
4. Volume integrals

Consider a closed surface in space enclosing a volume V. Then volume integral is an expression of the form

โญ ๐(๐โ)๐๐_{๐} or โญ ๐นโ(๐โ)๐๐_{๐}

For example if ๐ is the charge density of a body, โญ ๐(๐โ)๐๐_{๐} is the total charge contained in the body.

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Example

Calculate the volume integral of the scalar function ๐ = ๐ฅ^{2}๐ฆ๐ง over the volume of the prism shown in the
diagram.

z

2

O * 1 * * y *
1 * x+y=1 *

x

In this case the integration can be performed in any order. So we write

โญ ๐(๐โ)๐๐_{๐} = โญ ๐ฅ_{๐} ^{2}๐ฆ๐ง๐๐= โซ ๐ฅ_{0}^{1} ^{2}๐๐ฅโซ_{0}^{1โ๐ฅ}๐ฆ๐๐ฆโซ ๐ง๐๐ง_{0}^{2} = 2 โซ ๐ฅ_{0}^{1} ^{2}(1 โ ๐ฅ)^{2}/2๐๐ฅ= 1/30

5. Fundamental theorems of vector calculus The fundamental theorem of calculus states that

โซ_{๐}^{๐}^{๐๐(๐ฅ)}_{๐๐ฅ} ๐๐ฅ = ๐(๐) โ ๐(๐) (10)

The result can be interpreted as follows: ๐๐ =^{๐๐(๐ฅ)}_{๐๐ฅ} ๐๐ฅ is the infinitesimal change in *f as we go from point x to *
*x+dx. If the interval from a to b is subdivided into infinitesimal parts, the total change in the function is the sum *
of changes in each step, so that at the end it equals the value of the function at the end point minus its value at the
initial point. In other words, the integral over a boundary is related to value at the end points.

There are similar theorem for vector calculus as well, one each for gradient, divergence and the curl. If we move from a point ๐โ to a point ๐โโ along a certain path, then the change in the function over an infinitesimal interval ๐๐โโโโโ

will be

๐๐ =^{๐๐}

๐๐ฅ๐๐ฅ +^{๐๐}

๐๐ฆ๐๐ฆ +^{๐๐}

๐๐ง๐๐ง = โโโโ๐. ๐๐โโโโโ

The total change in the function will be

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โซ^{๐}^{โโ} โโโโ๐. ๐๐โโโโโ

๐โโ,๐ซ = ๐(๐โโ) โ ๐(๐โ) (11)

This may be regarded as the fundamental theorem for gradients.

5.1 Fundamental theorem for divergence โ Gaussโ theorem The fundamental theorem for divergence states that

โซ โ_{๐}โโโ. ๐นโ๐๐= โฏ ๐นโ. ๐๐โโโโโ

๐ (12)

This theorem is most often referred to as Gauss theorem and sometimes as Greens theorem. The โboundaryโ of a curve is its end points, that of an open surface is its perimeter and that of volume is the enclosing surface. This theorem is also in the spirit of the fundamental theorem of calculus in that it relates the integral of the derivative of a function over a volume to the function at its boundary, that is, the bounding surface.

*z *

๐ฬ

S ๐_{1}

๐_{2}
๐ฬ

y

โ

*Projection of S on x-y plane *
x

Proof

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Let S be a convex closed surface. Any line parallel to one of the axes will cut the surface in at most two points. In
the case of line being parallel to the *z-axis, such points will divide the surface into two parts, the lower and the *
upper part. Let the equations of the upper and lower parts be respectively

๐ง = ๐_{1}(๐ฅ, ๐ฆ); ๐ง = ๐_{2}(๐ฅ, ๐ฆ)

Let the projection of the surface on the x-y plane be โ. Consider first the z- component of the vector ๐นโ. For this component we have

โญ_{๐}^{๐๐น}_{๐๐ง}^{3}๐๐= โญ ^{๐๐น}^{3}

๐๐ง ๐๐ฅ๐๐ฆ๐๐ง

๐ = โฌ ๐๐ฅ๐๐ฆ[โซ ^{๐๐น}^{3}

๐๐ง
๐_{1}(๐ฅ,๐ฆ)
๐_{2}(๐ฅ,๐ฆ)

โ. ๐๐ง] = โฌ [๐น_{โ} _{3}(๐ฅ, ๐ฆ, ๐_{1}) โ๐น_{3}(๐ฅ, ๐ฆ, ๐_{2})]๐๐ฅ๐๐ฆ

For the upper portion of the surface, ๐_{1}, let ๐ผ_{1} be the angle between the normal and the z-axis. This angle being
acute, we have

๐๐ฅ๐๐ฆ = cos ๐ผ_{1}๐๐_{1}= ๐ฬ. ๐ฬ๐๐_{1}

Similarly, the lower portion of the surface makes an obtuse angle ๐ผ_{2} with the z-axis and we have
๐๐ฅ๐๐ฆ = cos ๐ผ_{2}๐๐_{2}= โ๐ฬ. ๐ฬ๐๐_{2}

As a result

โฌ [๐น_{โ} 3(๐ฅ, ๐ฆ, ๐_{1})= โฌ ๐น_{๐} _{3}๐ฬ. ๐ฬ๐๐_{1}

1 ; โฌ [๐น_{โ} _{3}(๐ฅ, ๐ฆ, ๐_{2})= โ โฌ ๐น_{๐} _{3}๐ฬ. ๐ฬ๐๐_{1}

2 ;

โฌ [๐น_{โ} 3(๐ฅ, ๐ฆ, ๐_{1}) โ๐น_{3}(๐ฅ, ๐ฆ, ๐_{2})]๐๐ฅ๐๐ฆ = โฌ ๐น_{๐} 3๐ฬ. ๐ฬ๐๐_{1}

1 + โฌ ๐น_{๐} _{3}๐ฬ. ๐ฬ๐๐_{1}

2 = โฌ ๐น_{๐} _{3}๐ฬ. ๐ฬ๐๐
So that

โญ_{๐}^{๐๐น}_{๐๐ง}^{3}๐๐= โฌ ๐น_{๐} _{3}๐ฬ. ๐ฬ๐๐

Similarly on projecting the give surface on the other two coordinate planes and adding all the contributions together, we obtain

โญ (^{๐๐น}_{๐๐ฅ}^{1}+^{๐๐น}^{2}

๐๐ฆ +^{๐๐น}^{3}

๐๐ง)๐๐

๐ = โฌ (๐น_{๐} _{1}๐ฬ + ๐น_{2}๐ฬ + ๐น_{3}๐ฬ). ๐ฬ๐๐
Or

โซ โ_{๐}โโโ. ๐นโ๐๐= โฏ ๐นโ. ๐๐โโโโโ

๐ Q.E.D.

Physically the meaning of the theorem is clear. Let ๐นโ refer to the velocity of a fluid (it may even refer to an electric field). As we have already seen โโโโ. ๐นโ๐๐ is the volume of the fluid coming out of the volume element dV per second, and โญ โโโโ. ๐นโ๐๐ is the total fluid emerging out of the volume. On the other hand, volume of fluid

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crossing surface element dS per second is ๐นโ. ๐ฬ๐๐ so that the total fluid crossing the closed surface enveloping the volume is โฌ ๐นโ. ๐ฬ๐๐. Gauss theorem thus simply expresses the obvious equality of the two expressions.

Example

Gauss theorem can very often be used to simplify the evaluation of surface or volume integrals by converting one

into the other. We evaluate the surface integral โฌ ๐นโ. ๐ฬ๐๐ over the surface of the cube described by 0 โค ๐ฅ, ๐ฆ, ๐ง โค 2 for the function ๐นโ = 4๐ฅ๐ฆ๐ฬ + ๐ฆ๐ง๐ฬ + ๐ฅ๐ฆ๐ง๐ฬ.

Using Gauss theorem, the required integral is

โญ โโโโ. ( 4๐ฅ๐ฆ๐ฬ + ๐ฆ๐ง๐ฬ + ๐ฅ๐ฆ๐ง๐ฬ)๐๐ = โญ( 4๐ฆ + ๐ง + ๐ฅ๐ฆ)๐๐ = โญ( 4๐ฆ + ๐ง + ๐ฅ๐ฆ)๐๐ฅ๐๐ฆ๐๐ง

= โซ ๐๐ฅ

2 0

โซ ๐๐ฆ โซ ๐๐ง(4๐ฆ + ๐ง + ๐ฅ๐ฆ) = 48

2 0 2 0

5.2 Greenโs theorem in a plane

Greenโs theorem in the plane can be regarded as the special case of the more general Stokesโ theorem, which we
shall take up immediately after this. Given two differentiable functions M(x,y) and N(x,y), a closed region in the
*x-y plane, R, bounded by the simple curve C, Greenโs theorem in a plane states that *

โฎ (๐๐๐ฅ + ๐๐๐ฆ)_{๐ถ} = โฌ (^{๐๐}

๐๐ฅโ^{๐๐}

๐๐ฆ) ๐๐ฅ๐๐ฆ

๐

The curve C is to be traversed in the counterclockwise direction.

y

D

C

A * B *

E

O a * b * x

Proof

Let the equation of the part AEB of the closed curve be ๐ฆ = ๐น_{1}(๐ฅ) and that of ADB be ๐ฆ = ๐น_{2}(๐ฅ). Then

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โฌ_{๐
}^{๐๐}_{๐๐ฆ}๐๐ฅ๐๐ฆ= โซ [โซ ^{๐๐}

๐๐ฆ๐๐ฆ

๐น_{2}(๐ฅ)

๐น_{1}(๐ฅ) ] ๐๐ฅ

๐

๐ = โซ [๐(๐ฅ, ๐น_{๐}^{๐} _{2}) โ ๐(๐ฅ, ๐น_{1})]๐๐ฅ

= โ โซ ๐(๐ฅ, ๐น_{1})๐๐ฅ

๐ ๐

โ โซ ๐(๐ฅ, ๐น_{2})๐๐ฅ

๐ ๐

= โ โฎ ๐๐๐ฅ

๐ถ

Or

โฎ ๐๐๐ฅ_{๐ถ} = โ โฌ ^{๐๐}

๐๐ฆ๐๐ฅ๐๐ฆ

๐

Similarly, let the equation of the parts *EAD and EBD of the curve be *๐ฅ = ๐บ_{1}(๐ฆ) and ๐ฅ = ๐บ_{2}(๐ฆ), respectively.

Following exactly the same procedure but with the order of the x and y integrations interchanged, we obtain

โฎ ๐๐๐ฆ_{๐ถ} = โฌ ^{๐๐}

๐๐ฅ๐๐ฅ๐๐ฆ

๐

On adding the two results we obtain the desired result:

โฎ (๐๐๐ฅ + ๐๐๐ฆ)_{๐ถ} = โฌ (^{๐๐}

๐๐ฅโ^{๐๐}

๐๐ฆ) ๐๐ฅ๐๐ฆ

๐

5.3 Fundamental theorem for curl - Stokesโ theorem

We now take up the Stokesโ theorem. This theorem is also in the mould of the fundamental theorem of calculus.

It relates the surface integral of the derivative of a function to the value of the function along the boundary of the surface, i.e., its periphery. The theorem states that

โฌ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐= โฌ (โโโโ๐น๐นโ). ๐๐โโโโโ

๐ = โฎ ๐นโ. ๐๐โโโโโ

๐ถ (13)

Here S is an open two-sided surface, bounded by a closed non-intersecting curve ๐ถ. Since the surface is an open
surface, direction of the normal ๐ฬ is not defined. Nor is the direction along which the curve C is to be traversed is
defined. There is flexibility in the definition of one of these. However, once the positive sense of one of them is
defined, that of other is fixed by the relation. The consistency of the Stokesโ theorem demands the *right hand *
*rule: If the fingers point along the direction of the line integral then the position of the thumb gives the direction *
of the normal to the surface.

Now there are an infinite number of surfaces which have the given closed curve as its periphery. Stokesโ theorem states that the surface integral of (โโโโ๐น๐นโ). ๐ฬ is the same for all such surfaces, since they are all equal to the line integral on the right hand side. Ordinarily the flux integral depends on the surface over which it is evaluated, but for curl of a vector it is independent of the surface and depends only on the boundary of the surface. Thus for the curl of a vector function swe have the result

๏ The integral โฌ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐ depends only on the boundary of the surface.

๏ The integral โฏ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐ over a closed surface vanishes, since the boundary line shrinks to a point.

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Proof of Stokesโ theorem

We now come to the proof of Stokesโ theorem, which as we have mentioned can be regarded as the generalization of the above proved Greenโs theorem to surfaces in three dimensions. Let S be a surface whose projections in the three coordinate planes are regions bounded by simple closed curves. The equation of the surface can be written in any of the three given forms

๐ฅ = ๐(๐ฆ, ๐ง); ๐ฆ = ๐(๐ฅ, ๐ง); ๐ง = โ(๐ฅ, ๐ฆ)

We have to demonstrate that

โฌ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐= โฌ [โ_{๐} โโโ๐น(๐น_{๐ฅ}๐ฬ + ๐น_{๐ฆ}๐ฬ + ๐น_{๐ง}๐ฬ)]. ๐ฬ๐๐= โฎ ๐นโ. ๐๐โโโโโ

๐ถ

*z *

dS

๐ฬ

dSd

* S dS *

SS

C

y

dxdy

R

*Cโ *

x

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Consider the first term

โฌ [โ_{๐} โโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐

Now

[โโโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐ = (^{๐๐น}^{๐ฅ}

๐๐ง ๐ฬ. ๐ฬ โ^{๐๐น}^{๐ฅ}

๐๐ฆ๐ฬ. ๐ฬ)๐๐ (14)

If we take the equation of the surface as ๐ง = โ(๐ฅ, ๐ฆ), the position vector of any point in S is ๐โ = ๐ฅ๐ฬ + ๐ฆ๐ฬ + ๐ง๐ฬ = ๐ฅ๐ฬ + ๐ฆ๐ฬ + โ(๐ฅ, ๐ฆ)๐ฬ

Regarding this as the parametric form of the equation of a surface, ^{๐๐โ}

๐๐ฅ and ^{๐๐โ}

๐๐ฆ are vectors tangent to the surface.

Hence
๐ฬ.^{๐๐โ}

๐๐ฆ= ๐ฬ. ๐ฬ +^{๐๐ง}

๐๐ฆ๐ฬ. ๐ฬ = 0 โน ๐ฬ. ๐ฬ = โ^{๐๐ง}

๐๐ฆ๐ฬ. ๐ฬ

Substituting in (14) we have
(^{๐๐น}^{๐ฅ}

๐๐ง ๐ฬ. ๐ฬ โ^{๐๐น}^{๐ฅ}

๐๐ฆ ๐ฬ. ๐ฬ) = โ^{๐๐น}^{๐ฅ}

๐๐ง

๐๐ง

๐๐ฆ๐ฬ. ๐ฬ โ^{๐๐น}^{๐ฅ}

๐๐ฆ๐ฬ. ๐ฬ = โ(^{๐๐น}^{๐ฅ}

๐๐ง

๐๐ง

๐๐ฆ+^{๐๐น}^{๐ฅ}

๐๐ฆ)๐ฬ. ๐ฬ

Or

[โโโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐ = โ(^{๐๐น}^{๐ฅ}

๐๐ง

๐๐ง

๐๐ฆ+^{๐๐น}^{๐ฅ}

๐๐ฆ)๐ฬ. ๐ฬ๐๐ (15)

Now on the given surface S, ๐น_{๐ฅ}(๐ฅ, ๐ฆ, ๐ง) = ๐น_{๐ฅ}(๐ฅ, ๐ฆ, โ(๐ฅ, ๐ฆ)) = ๐บ(๐ฅ, ๐ฆ) โน (^{๐๐น}^{๐ฅ}

๐๐ง

๐๐ง

๐๐ฆ+^{๐๐น}^{๐ฅ}

๐๐ฆ) =^{๐๐บ}

๐๐ฆ

Hence equation (15) becomes
[โโโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐ = โ^{๐๐บ}

๐๐ฆ๐ฬ. ๐ฬ๐๐ = โ^{๐๐บ}

๐๐ฆ๐๐ฅ๐๐ฆ Using this expression in the surface integral, we have

โฌ [โ_{๐} โโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐= โ โฌ ^{๐๐บ}

๐ ๐๐ฆ๐๐ฅ๐๐ฆ

Here R is the projection of S on the x-y plane. Now we use the Greenโs theorem for a plane which we have just proved above and obtain

โฌ [โ_{๐} โโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐= โ โฌ ^{๐๐บ}

๐
๐๐ฆ๐๐ฅ๐๐ฆ = โฎ ๐บ๐๐ฅ_{๐ถ}_{โฒ}

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where ๐ถ^{โฒ} is the boundary of the planar region R. Since at each point (x,y) of ๐ถ^{โฒ} the value of G is the same as that
of ๐น_{๐ฅ} at each point (x,y,z) of C, and since dx is the same for both the curves, it follows that

โฎ ๐บ๐๐ฅ_{๐ถ}โฒ = โฎ ๐ด_{๐ถ} _{๐ฅ}๐๐ฅโน โฌ [โ_{๐} โโโ๐น(๐น_{๐ฅ}๐ฬ)]. ๐ฬ๐๐= โฎ ๐ด_{๐ถ} _{๐ฅ}๐๐ฅ

On making similar projections on the other two planes and adding the results together we obtain the desired result

โฌ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐= โฎ ๐นโ. ๐๐โโโโโ

๐ถ Q.E.D.

Example

Verify Stokeโs theorem for the vector field ๐นโ = (4๐ฅ โ ๐ฆ)๐ฬ + ๐ฆ^{2}๐ง^{3}๐ฬ + ๐ฆ^{3}๐ง^{2}๐ฬ for the upper surface of the sphere
๐ฅ^{2}+ ๐ฆ^{2}+ ๐ง^{2}= 1.

The surface in this case is the upper half sphere and the boundary is the circle ๐ฅ^{2}+ ๐ฆ^{2}= 1. If the path is
traversed along the counter clockwise direction, the normal to the surface will be โupwardsโ. Write the equation
of the circle in the parametric form

๐ฅ = sin ๐ก ; ๐ฆ = โ cos ๐ก ; 0 โค ๐ก โค 2๐

Then

โฎ ๐นโ. ๐๐โโโโโ

๐ถ = โซ (4 sin ๐ก + cos ๐ก) cos ๐ก ๐๐ก_{0}^{2๐} = ๐
On the other hand

โโโโ๐น๐นโ = ๐ฬ

Hence

โฌ (โ_{๐} โโโ๐น๐นโ). ๐ฬ๐๐= โฌ ๐ฬ. ๐ฬ๐๐_{๐} = โฌ ๐๐ฅ๐๐ฆ_{๐
} = ๐

Here R is the projection of the hemisphere on the x-x plane, i.e., ๐ฅ^{2}+ ๐ฆ^{2}= 1. Hence the rightmost integral is
nothing but the area of a unit circle which equals ฯ.

5.4 Some other important theorems

There are quite a few other useful and related theorems which either follow from the above theorems or can be proved in very similar ways. We simply list these theorems without offering any proof.

๏ Greenโs first identity

Given scalars ฮฆ and ฮจ, Greenโs first identity states that
โญ [ฮฆโ_{๐} ^{2}ฮจ + (โโโโฮจ). (โโโโฮฆ)]dV= โฌ (ฮฆโโโโ

๐ ฮจ). ๐๐โโโโโ

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๏ Greenโs second identity

Given scalars ฮฆ and ฮจ, Greenโs first identity states that

โญ [ฮฆโ_{๐} ^{2}ฮจ โ ฮจโ^{2}ฮฆ)]dV= โฌ (ฮฆโโโโ

๐ ฮจ โ ฮจโโโโฮฆ). ๐๐โโโโโ

๏ Divergence like theorem for curl
โซ โ_{๐}โโโ๐น๐นโ๐๐= โฏ ๐๐_{๐}โโโโโ๐น๐นโ

๏ Stokeโs like theorem for scalars

โฎ ฮฆdrโโโโโ

๐ถ = โฌ ๐๐_{๐}โโโโโ๐นโโโโฮฆ

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**SUMMARY**

๏ท In this module we take up the topic of vector integration. First we describe the ordinary integration of a vector.

๏ท Next we introduce the central concept of line integral and describe the evaluation of line integrals by examples.

๏ท We discuss in detail the special case of line integral of conservative fields and prove theorems on the condition for a field to be conservative.

๏ท After line integral we describe surface integrals of vectors both for the case of open and closed surfaces.

๏ท Then we explain evaluation of volume integral by an example.

๏ท We next state and prove the very important fundamental theorems regarding the line, volume and surface integrals; viz., the Gauss theorem, Greenโs theorem in a plane and the Stokeโs theorem.

๏ท Finally we state some other useful theorems without proof.