Determine whether or not the vector field is conservative. If it is, find a function f such that F = ∇f. If the vector field is not conservative, enter NONE. F(x, y, z) = 5y2z3 …

Question

asked Mar 14, 2020 115k views

1 Answer

CharlieP · Answer 1 · 2020-03-18T17:29:07+0000

$\vec F$ is conservative if we can find a scalar function
such that
$\\abla f=\vec F$ . This would require

$(\partial f)/(\partial x)=5y^2z^3$

$(\partial f)/(\partial y)=10xyz^3$

$(\partial f)/(\partial z)=15xy^2z^2$

Integrate both sides of the first PDE with respect to
:

f(x,y,z)=5xy^2z^3+g(y,z) (*)

Differentiate both sides of (*) with respect to
:

$(\partial f)/(\partial y)=10xyz^3=10xyz^3+(\partial g)/(\partial y)\implies(\partial g)/(\partial y)=0\implies g(y,z)=h(z)$

Differentiate both sides of (*) with respect to
:

$(\partial f)/(\partial z)=15xy^2z^2=15xy^2z^2+(\mathrm dh)/(\mathrm dz)\implies(\mathrm dh)/(\mathrm dz)=0\implies h(z)=C$

So we have

f(x,y,z)=5xy^2z^3+C

and so
$\vec F$ is indeed conservative.