3 Vector integration

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 – Gauss’ theorem

5.2 Green’s theorem in a plane

5.3 Fundamental theorem for curl – Stokes’ theorem

5.4 Some other important theorems

 

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

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 ⃗ 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

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

2.1 Evaluation of line integral

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

Hence

without any reference to the path taken.

Theorem-2

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:

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

 

3.  Surface integrals

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.

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

4. Volume integrals

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

Example

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

 

5. Fundamental theorems of vector calculus

The fundamental theorem of calculus states that

The total change in function will be
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

  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.

 

Proof

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

So that

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

Example

5.2 Green’s theorem in a plane

Proof

On adding the two results we obtain the desired result:

 

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

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

We have to demonstrate that

Consider the first term

Hence equation (15) becomes

Using this expression in the surface integral, we have

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

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

Example

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.

SUMMARY

 

  • 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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