Vectors and ODEs

(a)      Let F be the vector field defined as

Find div( F).

Show that curl( F)=0.

Find a scalar function such that

Can you find such a function which in addition satisfies

(9 marks)

(b)       Use the gradient operator to find the direction of the normal to the surface

at the point

Hence find the tangent plane and normal line to the surface at this point.

(8 marks)

(c)      Consider the following system

where b ≥ 0 and k ≠ 0 are constants.

(i)        For what value of b does this system have periodic solutions?

(2 marks)

(ii)       Use the result of (c) (i) to find these periodic solutions of this system.

(4 marks)

(d)     Consider the following ordinary differential equation:

where c a positive constant.

• For what values of c are there two distinct equilibrium solutions?

(2 marks)

• Find the equilibrium solutions corresponding to the values of c which you have found above. Describe the behavior of the solutions to this system as.                                                            (4 marks)