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.



(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)


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