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Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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we will talk about special functions of vector fields and about operators which act on vector fields. We begin with the concept of a line integral. We are familiar with normal integrals, which can be regarded as a sum. Let us consider a force acting on a particle in one dimension. The work done in moving the particle from a to b is given by the integral  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The particle in this case can move parallel to the x-axis. How do we generalize this to higher dimension? Suppose, the force F acts on a particle taking it along a curve in two or three dimensions.

Work done is given by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where the integral is taken along the path in which the particle moves, the path C can be either an open path or a closed path which terminates at its starting point.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Unlike the case of normal integrals that we have learnt in school, the integral here does not depend only on the end-points but they may depend on the details of the path traversed. There are of course exceptions to this. For instance, the work done by gravitational force or electrostatic force depends only on the end points. Such forces are called “conservative forces”.

Along the curve one can always define a single parameter. For instance, taking time as the parameter, we can express the work done along the curve as  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is the velocity of the particle at time t.

Example 1 :

We will illustrate calculation of the work by computing the work done by a force given by a two dimensional force Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) along three different paths connecting the origin O(0,0) to the point P(2,1).

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

(i) The path 1 is a straight line connecting O with P. The equation to the path is y=x/2

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Substituting y=x/2 in the first integral and x=2y in the second, we get, 

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

(ii) The second path is a parabola given by the equation  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The calculation proceeds exactly as in case (i) and we have,

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

(iii) The third path is in two segments, OQ which connects (0,0) to (0,1) and then along QP connecting (0,1) to (2,1). Note that in the first case, x=0 so that  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the integral reduces to Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) However, along this path dx=0. Thus the work done along this path is zero. From Q to P, dy is zero, and y=1, thus for this path  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We can see from the above example that the integral is path dependent.

Example 2 :

As a second example, consider a force  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) along the first quadrant of a circle  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) taken in anti-clockwise fashion.
Look at the sketch of the path. In this quadrant x and y are both positive. You could sketch the vector field and see that vectors at each point is “somewhat” directed upward, implying that the direction of the tangent to the path and the force makes an acute angle. Thus we expect Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)to be positive.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the particle moves on a circular path of unit radius , we can parameterize the position by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

In general, the line integral depends on the path of integration and not just on its end points. In cases where the value of the integral depends on the path of integration, the force is called “nonconservative”. If  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) depends only on its end points, the force is said to be “conservative” In this case, it can be further seen that Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where the circle symbol over the integration sign means an integral over a closed contour. This becomes obvious if we look at the following. Suppose, we are computing the line integral over a closed path which goes from A to B along a path 1 but return from B to the initial point A through path 2.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the integral depends only on the end points we can write for the path  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the value of the indefinite integral. In the returning path 2 : we have Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus over the closed contour, we have Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We will spend some time on conservative field. This is because, we will be dealing with electrostatic field which is a conservative field.

Suppose, we have a force field which can be expressed as a gradient of some scalar function, i.e.  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Note that not all forces can be written this way. But in those cases, where it can be so written, we have,

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

So if a force field is expressible as a gradient of a scalar function, then the line integral would depend only on end points and not depend on path. Such a field is therefore conservative. We make a statement here that a conservative field can be expressed as a gradient of a scalar field. Generally, in Physics, we define the force as negative gradient of a scalar function φ , i.e.,   Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  We call such a function as “potential”. (A mathematician would call φ to be a potential, whether we have the negative sign or not; in Physics, the negative gradient of potential is the force).

Surface Integral :

We define surface integral as an integral of a vector function over a surface. It is given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the “outward” normal to the surface over which the integral is taken. We will shortly define the meaning of the phrase “outward”. The quantity Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is also known as the flux of the vector field through the surface S.

Let us consider an open surface. An open surface is bounded by a curve, for instance, a cup or a butterfly net. In both cases, there is a rim which is the bounding curve of the surface. The direction of the outward normal is defined by the “right handed rule”—if the bounding curve of a surface is traversed in the direction of rotation of a right handed screw, the direction in which the head of the screw moves is the direction of the outward normal. For being able to define a surface integral, we need what are known as “two sided surfaces”. Most of the common surfaces that we meet are two sided, they have an inside surface and an outside surface, separated by an edge or a rim. Take for example, the butterfly net shown here. The circular rim separates the inside surface from the outside surface. One cannot go from a point on the outside surface to the inside surface or vice versa without crossing the edge.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

There are one-sided surfaces too for which one cannot define a surface integral. One such example is a Mobius Strip. A Mobius strip is easily constructed by taking a strip of paper and sticking the short edges after giving a half-turn son that the edge on one side is glued to the other side. In such a case one can seamlessly traverse the entire surface without having to cross an edge. Note that in this case the normal at a point cannot be uniquely defined.

Divergence of a Vector : 

We define divergence of a vector field at a point as the limit of surface integral to the volume enclosed by such a surface  as the volume enclosed goes to zero. Thus

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is a point relationship, i.e. a relationship defined at every point in the region where a vector field exists. We imagine an infinitely small volume around the point and compute the surface integral of the vector field over the surface defining this volume.

Consider the following.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

One the left is a rectangular parallelepiped whose outward normals from the top and the bottom faces are shown. If we imagine the parallelepiped to be sliced into two parts, as shown to the right, while there is no change in the outward normals of the top face of the upper half and the bottom face of the lower half. However at the interface where the section is made, the normals are oppositely directed and the contributions to the surface integral from the two surfaces cancel. The same would be true for the other three pairs of faces of the parallelepiped. Thus, if we have a macroscopic volume defined by a bounding surface, we could split the volume into a large number of small volumes and the surface integrals of all volumes which are inside cancel out leaving us with contribution from the outside surface only. Thus the flux over a closed surface can be written as a sum over the surfaces of elemental volumes which make up the total volume.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The quantity in the bracket was defined as the divergence of Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the location of the elemental volume. Thus the sum is nothing but an integral of the divergence over the volume of the system. This gives us the Divergence Theorem :

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Expression for Divergence in Cartesian Coordinates :

Since divergence is meaningful in the limit of the volume going to zero, we can calculate it by taking an infinitesimally small parallelepiped of dimensions Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which is oriented along the axes of the coordinate system.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Consider the flux from two opposite faces shown in the figure. Note that the left face is at some value of y and the normal is directed along  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) direction. Thus the flux from this face is  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The opposite face is at y+∆y and the normal to this face is along direction. The flux from the face is Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  The net outward flux from these two faces can be obtained by subtracting the former expression from the latter. Retaining only first order term in a Taylor series expansion of Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Thus the net flux from the two faces is Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

By symmetry, one can get expressions for the contributions from the other two pairs of faces. The net outward flux from the six faces is thus,

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Summing over, we have
Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  

However, the left hand side of this equation is identically equal to  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Since the volume we have taken in arbitrary, we have

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is the Cartesian expression for the divergence of a vector  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
We recall that the gradient operator  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Using this, one can write

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Physically, divergence, as the name suggests, shows how much does the field at a point diverges from its value at that point to the neighbouring region. In the figure below, the vector field  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which has a positive divergence has been plotted in the x-y plane in the range  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) can be seen that the fields spread outward from the origin. A field with negative divergence would instead converge.

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Tutorial :

1. Calculate the line integral of the vector field  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) along the following two paths joining the origin to the point P(1,1,1). (a) Along a straight line joining the origin to P, (ii) along a path parameterized by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
2. Calculate the line integral of  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) over a quarter circle in the upper half plane along the path connecting (3,0) to (0,3). What would be the result if the path was taken along the same circular path but in the reverse direction?
3. Calculate the line integral of the scalar function  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) over the right half of the semi-circle  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) along the counterclockwise direction from (0,-2) to (0,+2).
4. A two dimensional force field is given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Find a potential function for this force field.
5. Calculate the flux of a constant vector field  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) through the curved surface of a hemisphere of radius R whose base is in the x-y plane.
6. Calculate the divergence of the position vector  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
7. Calculate the divergence of  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
8. Calculate the divergence of  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Solution to Tutorial Problems :

1. (a) Along the straight line path, we have, x=y=z, so that  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) can be expressed as  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
(b) Substituting the parameterized form, we have, Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The line integral is  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
2. The path integral is most conveniently done by parameterizing  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) so that Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The integral is then given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) If the curve is traversed in the opposite direction, the line integral would become +18
3. Since the path is along a circle of radius 2, we can parameterize by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The line integral is 

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

4. Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
The potential function is therefore given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where C is an arbitrary constant.
5. In this case it is convenient to use the spherical polar coordinates, the surface element is  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and it is along the radial direction, which makes an angle θ with the z-axis. Thus Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

6. Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) (This is an important relation which we use frequently.)
7. Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
8.  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Self Assessment Quiz

1. Find the line integral of the vector field  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) from (0,0) to (1,1).
2. Calculate the line integral of the vector field Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) along the following two paths joining the origin to the point P(1,1,1). (a) Along a straight line joining the origin to P, (ii) along a path parameterized by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
3. From the result of Problem 2, can you conclude that the force is conservative? If so, determine a potential function for this vector field.
4. A potential function is given by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where a, b and c are constants. Find the force field. 
5. A conservative force field is given by Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Calculate the work done by the force in taking a particle from the origin to the point (1,1,2).
6. A force field is given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where r is the distance of the point from the origin. Calculate the divergence at a point other than the origin.

Solutions to Self Assessment Quiz

1. Substituting  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the line integral can be expressed as Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
2. (i) For the straight line path  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
(ii) For the second path  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
3. Just from the fact that line integrals along two different paths give the same result, one cannot conclude that the force is conservative. However, in this particular case, the vector field happens to be conservative. Let the potential function be  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Equating components of the force, we get,

Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Clearly, the function is given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where C is an arbitrary constant, which can be taken to be zero.The line integral can therefore be written as
Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

As expected.

4. The components of the force are obtained as follows Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

5. Since the force is conservative it can be expressed as a gradient of a scalar potential. Writing  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) We can show that the potential is given by  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The work done is Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 9 - 0 = 9 units.

6. Given Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Using chain rule (since the function depends only on the distance r,  Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The document Line & Surface Integrals - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electromagnetic Fields Theory (EMFT).
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