1. Find the line integral of the vector field   along a path  from (0,0) to (1,1).

Substituting , the line integral can be expressed as

2. Calculate the line integral of the vector field  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  

(i) For the straight line path x=y=z,  
(ii) For the second path , so that

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.

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 . Equating components of the force, we get,

Clearly, the function is given by , where C is an arbitrary constant, which can be taken to be zero.The line integral can therefore be written as

As expected.

4. A potential function is given by , where a, b and c are constants. Find the force field.

The components of the force are obtained as follows : .

 

5. A conservative force field is given by  . Calculate the work done by the force in taking a particle from the origin to the point (1,1,2).

Since the force is conservative it can be expressed as a gradient of a scalar potential. Writing

We can show that the potential is given by . The work done is  units.

 

6. A force field is given by  , where r is the distance of the point from the origin. Calculate the divergence at a point other than the origin.

Given , where Using chain rule (since the function depends only on the distance r,