Another vector task to solve!

[sky_tm]

vector 3

i) If \[F = y^2 i - 3x^2 j + yzk\], find \[\nabla XF\] and \[\nabla \bullet F\]

ii) Show that \[G = 2xy^3 i + (1 + 3x^2 y^2 )j\] is conservative vector field on the entire plane.

iii) Find a potential function \[\Phi \] so that \[\nabla \Phi = G\].

 

[steve10]

Re: vector 3

i) If \[F = y^2 i - 3x^2 j + yzk\], find \[\nabla XF\] and \[\nabla XF\]

ii) Show that \[G = 2xy^3 i + (1 + 3x^2 y^2 )j\] is conservative vector field on the entire plane.

iii) Find a potential function \[\Phi \] so that \[\nabla \Phi = G\].

i) you mean
...find \[\nabla XF\] and \[\nabla F\]?

ii) If G=f1(x,y) +f2(x,y)j. Then that G is conservative is equivalent to
df2/dx=df1/dy
G indeed satisfies this requirement by a trivial checking;

iii) \[ \Phi =y + x^2 y^3\]