(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
(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.
(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?
(ii) Use the result of (c) (i) to find these periodic solutions of this system.
(d) Consider the following ordinary differential equation:
where c a positive constant.
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