Determine whether or not the vector field is conservative. If it is, find a function f such that F=▽f.F(x,y,z)=3y2z3i+6xyz3j+9xy2z2kf(x,y,z)=+K

Question

asked Nov 12, 2020 219k views

1 Answer

Juan Lanus · Answer 1 · 2020-11-17T02:32:22+0000

If
$\vec F$ is conservative, then there is a scalar function
such that
$\\abla f=\vec F$ . This means

f_x=3y^2z^3

f_y=6xyz^3

f_z=9xy^2z^2

Integrate both sides of the first PDE wrt
:

f(x,y,z)=3xy^2z^3+g(y,z)

Differentiate wrt
:

$f_y=6xyz^3=6xyz^3+g_y\implies g_y=0\implies g(y,z)=h(z)$

Differentiate wrt
:

$f_z=9xy^2z^2=9xy^2z^2+g_z=9xy^2z^2+h_z\implies h_z=0\implies h(z)=C$

Then

f(x,y,z)=3xy^2z^3+C

so
$\vec F$ is conservative.