(a) (i) Determine whether the following vector field is conservative, (i.e. curl
F=0
).
F(x,y,z)=(8xy^(3)z)i+(12x^(2)y^(2)z-2yz^(3))j+(4x^(2)y^(3)-3y^(2)z^(2))k
(ii) Find the directional derivative of
\phi (x,y,z)=9x^(4)y^(3)-8xz^(3)+5x^(2)y-6yz^(5)
at the point
(1,1,1)
in the direction of the vector
a=2i-j+3k
(15 marks) (b) Show that the following partial differential equation is exact and find its general solution
(12x^(3)y^(3)-12ycos4x)dx+(9x^(4)y^(2)-3sin4x-42e^(-7y))dy=0
(c) Show that the function
f(x,y)=30x^(2)y-45x^(2)+4y^(3)-30y^(2)+7
has four critical points and determine their nature. (40 marks)