Material science

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Crystallography & crystal growth

Experimental methods for x-ray diffraction

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

Material science

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Material science

Crystallography & crystal growth

Experimental methods for x-ray diffraction

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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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 – Gauss’ theorem 5.2 Green’s theorem in a plane

5.3 Fundamental theorem for curl - Stokes’ theorem 5.4 Some other important theorems

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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, Green’s theorem in a plane and the Stoke’s theorem are enunciated and proved. Some other fundamental theorems are also stated without proof.

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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 𝐹⃗ 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

𝐹⃗(𝑢) = 𝑖̂𝐹_𝑥+ 𝑗̂𝐹_𝑦+ 𝑘̂𝐹_𝑧

If the functions 𝐹_𝑥, 𝐹_𝑦, 𝐹_𝑧 are integrable functions of u, then

∫ 𝐹⃗(𝑢)𝑑𝑢 ≡ 𝑖̂ ∫ 𝐹𝑥(𝑢)𝑑𝑢 + 𝑗̂ ∫ 𝐹_𝑦(𝑢)𝑑𝑢 + 𝑘̂ ∫ 𝐹_𝑧(𝑢)𝑑𝑢

is the indefinite integral of 𝐹⃗(𝑢). If there exists a vector 𝐺⃗(𝑢), such that 𝐹⃗(𝑢) =^{𝑑𝐺⃗(𝑢)}

𝑑𝑢 , then

∫ 𝐹⃗(𝑢)𝑑𝑢 = ∫^{𝑑𝐺⃗(𝑢)}_𝑑𝑢 𝑑𝑢 = 𝐺⃗(𝑢) + 𝐾⃗⃗⃗

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

∫ 𝐹⃗(𝑢)𝑑𝑢_𝑎^𝑏 = 𝐺⃗(𝑏) − 𝐺⃗(𝑎)

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 𝐹⃗(𝑢).

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 𝐹⃗ is an integral of the form

∫^𝑏⃗⃗ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝑎⃗⃗,𝒫

Very often the notation 𝑑𝑙⃗⃗⃗⃗ is used instead of 𝑑𝑟⃗⃗⃗⃗⃗. The integral is to be carried from point in space with position vector 𝑎⃗ to one with position vector 𝑏⃗⃗, along a designated path 𝒫. In case the points 𝑎⃗ and 𝑏⃗⃗ 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:

∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

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To evaluate the line integral over an open or closed curve 𝒫, we find the scalar product of 𝐹⃗ with the displacement vector 𝑑𝑙⃗⃗⃗⃗, which is a vector along the tangent to the curve at each point.

 Remember that in general the line integral over a closed path is not zero. For example work done by a force: ∫ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗, 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 𝒫 be designated by

𝑟⃗ = 𝑥(𝑢)𝑖̂ + 𝑦(𝑢)𝑗̂ + 𝑧(𝑢)𝑘̂ (1)

Here u is a scalar parameter, and equation (1) is the equation of a curve in space. Then 𝐹⃗. 𝑑𝑙⃗⃗⃗⃗ = 𝐹_𝑥𝑑𝑥 + 𝐹_𝑦𝑑𝑦 + 𝐹_𝑧𝑑𝑧

and

∫^𝑏⃗⃗ 𝐹⃗. 𝑑𝑙⃗⃗⃗⃗

𝑎⃗⃗,𝒫 = ∫_𝑃^𝑃2(𝑢2)(𝐹_𝑥𝑑𝑥 + 𝐹_𝑦𝑑𝑦 + 𝐹_𝑧𝑑𝑧)

1(𝑢₁) (2)

Example-1

We first give an example of a curve in a plane. Evaluate the line integral of the function 𝐹⃗ = 𝑦²𝑖̂ + 2𝑦(𝑥 + 1)𝑗̂

from the point 𝑎⃗ = (0,0) to 𝑏⃗⃗ = (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

𝑥 = 𝑦; 𝑑𝑥 = 𝑑𝑦; 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = (𝑥²𝑖̂ + 2𝑦(𝑥 + 1)𝑗̂). (𝑑𝑥𝑖̂ + 𝑑𝑦𝑗̂) = 𝑥²𝑑𝑥 + 2𝑥(𝑥 + 1)𝑑𝑥

= (3𝑥²+ 2𝑥)𝑑𝑥 ⟹ ∫ (3𝑥²+ 2𝑥)𝑑𝑥

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

∫ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = ∫ 𝑥₀^{1 2}𝑑𝑥+ ∫ 4𝑦𝑑𝑦₀¹ = 8/3 Example-2

Find the total work done in moving a particle in a force field given by 𝐹⃗ = 𝑥𝑦𝑖̂ + 𝑧𝑗̂ − 10𝑥𝑦𝑘̂

along a path whose parametric equation is given by

𝑥 = 1 + 𝑡²; 𝑦 = 2𝑡²; 𝑧 = 𝑡³; from 𝑡 = 0 to 𝑡 = 1

∫ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = ∫(𝑥𝑦𝑖̂ + 𝑧𝑗̂ − 10𝑥𝑦𝑘̂). (𝑑𝑥𝑖̂ + 𝑑𝑦𝑗̂ + 𝑑𝑧𝑘̂) = ∫(𝑥𝑦𝑑𝑥 + 𝑧𝑑𝑦 − 10𝑥𝑦𝑑𝑧)

= ∫ [(1 + 𝑡₀^{1 2})8𝑡³𝑑𝑡 + 4𝑡⁴𝑑𝑡 − 60(1 + 𝑡²)𝑡⁴𝑑𝑡]= ∫ (8𝑡₀^{1 3}− 56𝑡⁴+ 8𝑡⁵− 60𝑡⁶)𝑑𝑡= 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 𝑉(𝑥, 𝑦, 𝑧) along any curve from the point 𝑟⃗⃗⃗⃗ to the point 𝑟⃗ is ₀ independent of the path taken and equals the difference between the values of the function at the two points. That is,

∫^𝑟⃗ ∇⃗⃗⃗𝑉. 𝑑𝑟⃗⃗⃗⃗⃗

𝑟₀

⃗⃗⃗⃗⃗ ,𝒫 = 𝑉(𝑟⃗) − 𝑉(𝑟⃗⃗⃗⃗) ₀ (3)

independent of the path 𝒫 chosen.

Proof: By definition 𝑑𝑟⃗⃗⃗⃗⃗. ∇⃗⃗⃗𝑉 = 𝑑𝑉

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Hence

∫^𝑟⃗ ∇⃗⃗⃗𝑉. 𝑑𝑟⃗⃗⃗⃗⃗

𝑟₀

⃗⃗⃗⃗⃗ ,𝒫 = ∫_𝑟^𝑟⃗ 𝑑𝑉

⃗⃗⃗⃗⃗ ,𝒫0 = 𝑉(𝑟⃗) − 𝑉(𝑟⃗⃗⃗⃗) 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 𝐹⃗ about every closed curve vanishes, then it can be written as the gradient of some scalar function 𝜑(𝑟⃗).

Proof: We are given that

∮ 𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = 0 (5)

for every closed curve in a certain region. Let 𝑎⃗ be a fixed point in space and 𝑟⃗ a variable point. Draw any two paths, C and 𝐶′ between these two points. Then C and −𝐶′ together form a closed curve and hence

∫^𝑟⃗ 𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝑎⃗⃗,𝐶 + ∫^𝑟⃗ 𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝑎⃗⃗,−𝐶′ = 0 ⟹ ∫ 𝑓⃗. 𝑑𝑟^𝑟⃗ ⃗⃗⃗⃗⃗

𝑎⃗⃗,𝐶 = ∫^𝑟⃗ 𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝑎⃗⃗,𝐶^′

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 𝜑 of the coordinates:

∫ 𝑓⃗. 𝑑𝑟^𝑟⃗ ⃗⃗⃗⃗⃗

𝑎⃗⃗ = 𝜑(𝑟⃗) (6)

Let the two points 𝑟⃗ and 𝑎⃗ be chosen to be infinitely close to each other. Then

𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = 𝑑𝜑 = ∇⃗⃗⃗𝜑. 𝑑𝑟⃗⃗⃗⃗⃗ ⟹ 𝑓⃗ = ∇⃗⃗⃗𝜑 (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:

𝑓⃗ = ∇⃗⃗⃗𝜑 ⟺ ∇⃗⃗⃗𝘹𝑓⃗ = 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

∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = 0

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where 𝐹⃗ = (2𝑥𝑦 + 𝑦𝑧)𝑖̂ + (𝑥²+ 𝑧 + 𝑥𝑧)𝑗̂ + (𝑦 + 𝑥𝑦)𝑘̂

∇⃗⃗⃗𝘹𝐹⃗ = [^𝜕

𝜕𝑦(𝑦 + 𝑥𝑦) − ^𝜕

𝜕𝑧(𝑥²+ 𝑧 + 𝑥𝑧)] 𝑖̂ + [^𝜕

𝜕𝑧(2𝑥𝑦 + 𝑦𝑧) − ^𝜕

𝜕𝑥(𝑦 + 𝑥𝑦)] + [^𝜕

𝜕𝑥(𝑥²+ 𝑧 + 𝑥𝑧) −

𝜕

𝜕𝑦(2𝑥𝑦 + 𝑦𝑧)] = 0 Hence

∮ 𝑓⃗. 𝑑𝑟⃗⃗⃗⃗⃗ = 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 𝑑𝑆⃗⃗⃗⃗⃗ = 𝑛̂𝑑𝑆, with magnitude dS and direction given by 𝑛̂. The integral

∬ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 = ∬ 𝐹⃗. 𝑛̂𝑑𝑆_𝑆

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

∬ 𝐹⃗𝘹𝑑𝑆⃗⃗⃗⃗⃗ = ∬ 𝐹⃗𝘹𝑛̂𝑑𝑆 ; ∬ 𝜑𝑑𝑆; ∬ 𝜑𝑛̂𝑑𝑆 = ∬ 𝜑𝑑𝑆⃗⃗⃗⃗⃗

𝑛̂

dS

S

If the integration is over a closed surface, it is usually denoted by ∯ . 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 𝑘̂. The projection of a surface element dS with normal 𝑛̂ on x-y plane is given by

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|𝑑𝑆𝑛̂. 𝑘̂| = 𝑑𝑥. 𝑑𝑦 ⟹ 𝑑𝑆 =^{𝑑𝑥𝑑𝑦}

|𝑛̂.𝑘̂ |

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

∬ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 = ∬ 𝐹⃗. 𝑛̂𝑑𝑆_𝑆 = ∬ (𝐹⃗. 𝑛̂)^{𝑑𝑥𝑑𝑦}

|𝑛̂.𝑘̂ |

𝑅 (9)

Example

Evaluate the surface integral

∬ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 = ∬ 𝐹⃗. 𝑛̂𝑑𝑆_𝑆 where

𝐹⃗ = 𝑧𝑖̂ + 𝑥𝑘̂ + 3𝑦𝑘̂

over part of the surface 2𝑥 + 2𝑦 + 𝑧 = 8 in the first octant.

Gradient to the above surface is given by

∇⃗⃗⃗(3𝑥 + 2𝑦 + 𝑧) = 2𝑖̂ + 2𝑗̂ + 𝑘̂ ⟹ 𝑛̂ = ^{2𝑖̂+2𝑗̂+𝑘̂}

|2𝑖̂+2𝑗̂+𝑘̂ |=^{2𝑖̂+2𝑗̂+𝑘̂}

3 ⟹ 𝑛̂. 𝑘̂ = 1/3 Further

𝐹⃗. 𝑛̂ = (𝑧𝑖̂ + 𝑥𝑘̂ + 3𝑦𝑘̂).^{2𝑖̂+2𝑗̂+𝑘̂}

3 =^{2𝑧+2𝑥+3𝑦}

3 =^{16−2𝑥−𝑦}

3

For the projected surface in the x-y plane 0 ≤ 𝑥 ≤ 4; 0 ≤ 𝑦 ≤ 4 − 𝑥.

Hence the required surface integral is

∬ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 = ∬ (𝐹⃗. 𝑛̂)^{𝑑𝑥𝑑𝑦}

|𝑛̂.𝑘̂ |

𝑅 = ∫ 𝑑𝑥₀⁴ ∫₀^4−𝑥𝑑𝑦(16 − 2𝑥 − 𝑦)= 320/3 4. Volume integrals

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

∭ 𝜑(𝑟⃗)𝑑𝑉_𝑉 or ∭ 𝐹⃗(𝑟⃗)𝑑𝑉_𝑉

For example if 𝜑 is the charge density of a body, ∭ 𝜑(𝑟⃗)𝑑𝑉_𝑉 is the total charge contained in the body.

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Example

Calculate the volume integral of the scalar function 𝜑 = 𝑥²𝑦𝑧 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

∭ 𝜑(𝑟⃗)𝑑𝑉_𝑉 = ∭ 𝑥_𝑉 ²𝑦𝑧𝑑𝑉= ∫ 𝑥₀^{1 2}𝑑𝑥∫₀^1−𝑥𝑦𝑑𝑦∫ 𝑧𝑑𝑧₀² = 2 ∫ 𝑥₀^{1 2}(1 − 𝑥)²/2𝑑𝑥= 1/30

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

∫_𝑎^{𝑏𝑑𝑓(𝑥)}_𝑑𝑥 𝑑𝑥 = 𝑓(𝑏) − 𝑓(𝑎) (10)

The result can be interpreted as follows: 𝑑𝑓 =^{𝑑𝑓(𝑥)}_𝑑𝑥 𝑑𝑥 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 𝑎⃗ to a point 𝑏⃗⃗ along a certain path, then the change in the function over an infinitesimal interval 𝑑𝑟⃗⃗⃗⃗⃗

will be

𝑑𝜑 =^𝜕𝜑

𝜕𝑥𝑑𝑥 +^𝜕𝜑

𝜕𝑦𝑑𝑦 +^𝜕𝜑

𝜕𝑧𝑑𝑧 = ∇⃗⃗⃗𝜑. 𝑑𝑟⃗⃗⃗⃗⃗

The total change in the function will be

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∫^𝑏⃗⃗ ∇⃗⃗⃗𝜑. 𝑑𝑟⃗⃗⃗⃗⃗

𝑎⃗⃗,𝒫 = 𝜑(𝑏⃗⃗) − 𝜑(𝑎⃗) (11)

This may be regarded as the fundamental theorem for gradients.

5.1 Fundamental theorem for divergence – Gauss’ theorem The fundamental theorem for divergence states that

∫ ∇_𝑉⃗⃗⃗. 𝐹⃗𝑑𝑉= ∯ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 (12)

This theorem is most often referred to as Gauss theorem and sometimes as Greens theorem. The “boundary” 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

𝑛̂

S 𝑆₁

𝑆₂ 𝑛̂

y

ℛ

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

𝑧 = 𝑓₁(𝑥, 𝑦); 𝑧 = 𝑓₂(𝑥, 𝑦)

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

∭_𝑉^𝜕𝐹_𝜕𝑧³𝑑𝑉= ∭ ^𝜕𝐹3

𝜕𝑧 𝑑𝑥𝑑𝑦𝑑𝑧

𝑉 = ∬ 𝑑𝑥𝑑𝑦[∫ ^𝜕𝐹3

𝜕𝑧 𝑓₁(𝑥,𝑦) 𝑓₂(𝑥,𝑦)

ℛ. 𝑑𝑧] = ∬ [𝐹_{ℛ 3}(𝑥, 𝑦, 𝑓₁) −𝐹₃(𝑥, 𝑦, 𝑓₂)]𝑑𝑥𝑑𝑦

For the upper portion of the surface, 𝑆₁, let 𝛼₁ be the angle between the normal and the z-axis. This angle being acute, we have

𝑑𝑥𝑑𝑦 = cos 𝛼₁𝑑𝑆₁= 𝑘̂. 𝑛̂𝑑𝑆₁

Similarly, the lower portion of the surface makes an obtuse angle 𝛼₂ with the z-axis and we have 𝑑𝑥𝑑𝑦 = cos 𝛼₂𝑑𝑆₂= −𝑘̂. 𝑛̂𝑑𝑆₂

As a result

∬ [𝐹_ℛ 3(𝑥, 𝑦, 𝑓₁)= ∬ 𝐹_{𝑆 3}𝑘̂. 𝑛̂𝑑𝑆₁

1 ; ∬ [𝐹_{ℛ 3}(𝑥, 𝑦, 𝑓₂)= − ∬ 𝐹_{𝑆 3}𝑘̂. 𝑛̂𝑑𝑆₁

2 ;

∬ [𝐹_ℛ 3(𝑥, 𝑦, 𝑓₁) −𝐹₃(𝑥, 𝑦, 𝑓₂)]𝑑𝑥𝑑𝑦 = ∬ 𝐹_𝑆 3𝑘̂. 𝑛̂𝑑𝑆₁

1 + ∬ 𝐹_{𝑆 3}𝑘̂. 𝑛̂𝑑𝑆₁

2 = ∬ 𝐹_{𝑆 3}𝑘̂. 𝑛̂𝑑𝑆 So that

∭_𝑉^𝜕𝐹_𝜕𝑧³𝑑𝑉= ∬ 𝐹_{𝑆 3}𝑘̂. 𝑛̂𝑑𝑆

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

∭ (^𝜕𝐹_𝜕𝑥¹+^𝜕𝐹2

𝜕𝑦 +^𝜕𝐹3

𝜕𝑧)𝑑𝑉

𝑉 = ∬ (𝐹_{𝑆 1}𝑖̂ + 𝐹₂𝑗̂ + 𝐹₃𝑘̂). 𝑛̂𝑑𝑆 Or

∫ ∇_𝑉⃗⃗⃗. 𝐹⃗𝑑𝑉= ∯ 𝐹⃗. 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 Q.E.D.

Physically the meaning of the theorem is clear. Let 𝐹⃗ refer to the velocity of a fluid (it may even refer to an electric field). As we have already seen ∇⃗⃗⃗. 𝐹⃗𝑑𝑉 is the volume of the fluid coming out of the volume element dV per second, and ∭ ∇⃗⃗⃗. 𝐹⃗𝑑𝑉 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 𝐹⃗. 𝑛̂𝑑𝑆 so that the total fluid crossing the closed surface enveloping the volume is ∬ 𝐹⃗. 𝑛̂𝑑𝑆. 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 ∬ 𝐹⃗. 𝑛̂𝑑𝑆 over the surface of the cube described by 0 ≤ 𝑥, 𝑦, 𝑧 ≤ 2 for the function 𝐹⃗ = 4𝑥𝑦𝑖̂ + 𝑦𝑧𝑗̂ + 𝑥𝑦𝑧𝑘̂.

Using Gauss theorem, the required integral is

∭ ∇⃗⃗⃗. ( 4𝑥𝑦𝑖̂ + 𝑦𝑧𝑗̂ + 𝑥𝑦𝑧𝑘̂)𝑑𝑉 = ∭( 4𝑦 + 𝑧 + 𝑥𝑦)𝑑𝑉 = ∭( 4𝑦 + 𝑧 + 𝑥𝑦)𝑑𝑥𝑑𝑦𝑑𝑧

= ∫ 𝑑𝑥

2 0

∫ 𝑑𝑦 ∫ 𝑑𝑧(4𝑦 + 𝑧 + 𝑥𝑦) = 48

2 0 2 0

5.2 Green’s theorem in a plane

Green’s theorem in the plane can be regarded as the special case of the more general Stokes’ 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, Green’s theorem in a plane states that

∮ (𝑀𝑑𝑥 + 𝑁𝑑𝑦)_𝐶 = ∬ (^𝜕𝑁

𝜕𝑥−^𝜕𝑀

𝜕𝑦) 𝑑𝑥𝑑𝑦

𝑅

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 𝑦 = 𝐹₁(𝑥) and that of ADB be 𝑦 = 𝐹₂(𝑥). Then

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∬_𝑅^𝜕𝑀_𝜕𝑦𝑑𝑥𝑑𝑦= ∫ [∫ ^𝜕𝑀

𝜕𝑦𝑑𝑦

𝐹₂(𝑥)

𝐹₁(𝑥) ] 𝑑𝑥

𝑏

𝑎 = ∫ [𝑀(𝑥, 𝐹_𝑎^𝑏 ₂) − 𝑀(𝑥, 𝐹₁)]𝑑𝑥

= − ∫ 𝑀(𝑥, 𝐹₁)𝑑𝑥

𝑏 𝑎

− ∫ 𝑀(𝑥, 𝐹₂)𝑑𝑥

𝑎 𝑏

= − ∮ 𝑀𝑑𝑥

𝐶

Or

∮ 𝑀𝑑𝑥_𝐶 = − ∬ ^𝜕𝑀

𝜕𝑦𝑑𝑥𝑑𝑦

𝑅

Similarly, let the equation of the parts EAD and EBD of the curve be 𝑥 = 𝐺₁(𝑦) and 𝑥 = 𝐺₂(𝑦), respectively.

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

∮ 𝑁𝑑𝑦_𝐶 = ∬ ^𝜕𝑁

𝜕𝑥𝑑𝑥𝑑𝑦

𝑅

On adding the two results we obtain the desired result:

∮ (𝑀𝑑𝑥 + 𝑁𝑑𝑦)_𝐶 = ∬ (^𝜕𝑁

𝜕𝑥−^𝜕𝑀

𝜕𝑦) 𝑑𝑥𝑑𝑦

𝑅

5.3 Fundamental theorem for curl - Stokes’ theorem

We now take up the Stokes’ 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

∬ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆= ∬ (∇⃗⃗⃗𝘹𝐹⃗). 𝑑𝑆⃗⃗⃗⃗⃗

𝑆 = ∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝐶 (13)

Here S is an open two-sided surface, bounded by a closed non-intersecting curve 𝐶. Since the surface is an open surface, direction of the normal 𝑛̂ 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 Stokes’ 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. Stokes’ theorem states that the surface integral of (∇⃗⃗⃗𝘹𝐹⃗). 𝑛̂ 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 ∬ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆 depends only on the boundary of the surface.

 The integral ∯ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆 over a closed surface vanishes, since the boundary line shrinks to a point.

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Proof of Stokes’ theorem

We now come to the proof of Stokes’ theorem, which as we have mentioned can be regarded as the generalization of the above proved Green’s 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

𝑥 = 𝑓(𝑦, 𝑧); 𝑦 = 𝑔(𝑥, 𝑧); 𝑧 = ℎ(𝑥, 𝑦)

We have to demonstrate that

∬ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆= ∬ [∇_𝑆 ⃗⃗⃗𝘹(𝐹_𝑥𝑖̂ + 𝐹_𝑦𝑗̂ + 𝐹_𝑧𝑘̂)]. 𝑛̂𝑑𝑆= ∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝐶

z

dS

𝑛̂

dSd

S dS

SS

C

y

dxdy

R

C’

x

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Consider the first term

∬ [∇_𝑆 ⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆

Now

[∇⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆 = (^𝜕𝐹𝑥

𝜕𝑧 𝑛̂. 𝑗̂ −^𝜕𝐹𝑥

𝜕𝑦𝑛̂. 𝑘̂)𝑑𝑆 (14)

If we take the equation of the surface as 𝑧 = ℎ(𝑥, 𝑦), the position vector of any point in S is 𝑟⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂ = 𝑥𝑖̂ + 𝑦𝑗̂ + ℎ(𝑥, 𝑦)𝑘̂

Regarding this as the parametric form of the equation of a surface, ^𝜕𝑟⃗

𝜕𝑥 and ^𝜕𝑟⃗

𝜕𝑦 are vectors tangent to the surface.

Hence 𝑛̂.^𝜕𝑟⃗

𝜕𝑦= 𝑛̂. 𝑗̂ +^𝜕𝑧

𝜕𝑦𝑛̂. 𝑘̂ = 0 ⟹ 𝑛̂. 𝑗̂ = −^𝜕𝑧

𝜕𝑦𝑛̂. 𝑘̂

Substituting in (14) we have (^𝜕𝐹𝑥

𝜕𝑧 𝑛̂. 𝑗̂ −^𝜕𝐹𝑥

𝜕𝑦 𝑛̂. 𝑘̂) = −^𝜕𝐹𝑥

𝜕𝑧

𝜕𝑧

𝜕𝑦𝑛̂. 𝑘̂ −^𝜕𝐹𝑥

𝜕𝑦𝑛̂. 𝑘̂ = −(^𝜕𝐹𝑥

𝜕𝑧

𝜕𝑧

𝜕𝑦+^𝜕𝐹𝑥

𝜕𝑦)𝑛̂. 𝑘̂

Or

[∇⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆 = −(^𝜕𝐹𝑥

𝜕𝑧

𝜕𝑧

𝜕𝑦+^𝜕𝐹𝑥

𝜕𝑦)𝑛̂. 𝑘̂𝑑𝑆 (15)

Now on the given surface S, 𝐹_𝑥(𝑥, 𝑦, 𝑧) = 𝐹_𝑥(𝑥, 𝑦, ℎ(𝑥, 𝑦)) = 𝐺(𝑥, 𝑦) ⟹ (^𝜕𝐹𝑥

𝜕𝑧

𝜕𝑧

𝜕𝑦+^𝜕𝐹𝑥

𝜕𝑦) =^𝜕𝐺

𝜕𝑦

Hence equation (15) becomes [∇⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆 = −^𝜕𝐺

𝜕𝑦𝑛̂. 𝑘̂𝑑𝑆 = −^𝜕𝐺

𝜕𝑦𝑑𝑥𝑑𝑦 Using this expression in the surface integral, we have

∬ [∇_𝑆 ⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆= − ∬ ^𝜕𝐺

𝑅𝜕𝑦𝑑𝑥𝑑𝑦

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

∬ [∇_𝑆 ⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆= − ∬ ^𝜕𝐺

𝑅𝜕𝑦𝑑𝑥𝑑𝑦 = ∮ 𝐺𝑑𝑥_𝐶′

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where 𝐶^′ is the boundary of the planar region R. Since at each point (x,y) of 𝐶^′ the value of G is the same as that of 𝐹_𝑥 at each point (x,y,z) of C, and since dx is the same for both the curves, it follows that

∮ 𝐺𝑑𝑥_𝐶′ = ∮ 𝐴_{𝐶 𝑥}𝑑𝑥⟹ ∬ [∇_𝑆 ⃗⃗⃗𝘹(𝐹_𝑥𝑖̂)]. 𝑛̂𝑑𝑆= ∮ 𝐴_{𝐶 𝑥}𝑑𝑥

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

∬ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆= ∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝐶 Q.E.D.

Example

Verify Stoke’s theorem for the vector field 𝐹⃗ = (4𝑥 − 𝑦)𝑖̂ + 𝑦²𝑧³𝑗̂ + 𝑦³𝑧²𝑘̂ for the upper surface of the sphere 𝑥²+ 𝑦²+ 𝑧²= 1.

The surface in this case is the upper half sphere and the boundary is the circle 𝑥²+ 𝑦²= 1. If the path is traversed along the counter clockwise direction, the normal to the surface will be “upwards”. Write the equation of the circle in the parametric form

𝑥 = sin 𝑡 ; 𝑦 = − cos 𝑡 ; 0 ≤ 𝑡 ≤ 2𝜋

Then

∮ 𝐹⃗. 𝑑𝑟⃗⃗⃗⃗⃗

𝐶 = ∫ (4 sin 𝑡 + cos 𝑡) cos 𝑡 𝑑𝑡₀^2𝜋 = 𝜋 On the other hand

∇⃗⃗⃗𝘹𝐹⃗ = 𝑘̂

Hence

∬ (∇_𝑆 ⃗⃗⃗𝘹𝐹⃗). 𝑛̂𝑑𝑆= ∬ 𝑘̂. 𝑛̂𝑑𝑆_𝑆 = ∬ 𝑑𝑥𝑑𝑦_𝑅 = 𝜋

Here R is the projection of the hemisphere on the x-x plane, i.e., 𝑥²+ 𝑦²= 1. Hence the rightmost integral is nothing but the area of a unit circle which equals π.

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.

 Green’s first identity

Given scalars Φ and Ψ, Green’s first identity states that ∭ [Φ∇_𝑉 ²Ψ + (∇⃗⃗⃗Ψ). (∇⃗⃗⃗Φ)]dV= ∬ (Φ∇⃗⃗⃗

𝑆 Ψ). 𝑑𝑆⃗⃗⃗⃗⃗

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 Green’s second identity

Given scalars Φ and Ψ, Green’s first identity states that

∭ [Φ∇_𝑉 ²Ψ − Ψ∇²Φ)]dV= ∬ (Φ∇⃗⃗⃗

𝑆 Ψ − Ψ∇⃗⃗⃗Φ). 𝑑𝑆⃗⃗⃗⃗⃗

 Divergence like theorem for curl ∫ ∇_𝑉⃗⃗⃗𝘹𝐹⃗𝑑𝑉= ∯ 𝑑𝑆_𝑆⃗⃗⃗⃗⃗𝘹𝐹⃗

 Stoke’s like theorem for scalars

∮ Φdr⃗⃗⃗⃗⃗

𝐶 = ∬ 𝑑𝑆_𝑆⃗⃗⃗⃗⃗𝘹∇⃗⃗⃗Φ

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

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

 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.

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

 Then we explain evaluation of volume integral by an example.

 We next state and prove the very important fundamental theorems regarding the line, volume and surface integrals; viz., the Gauss theorem, Green’s theorem in a plane and the Stoke’s theorem.

 Finally we state some other useful theorems without proof.