Question

Determine if the field Bold Upper F equals 10 yz Bold i plus 10 xz Bold j plus 10 xy Bold k is conservative or not conservative.

Answer 1

F is conservative if we can find a scalar funciton f such that grad(f) = F.

This would entail

$\dfrac{\partial f}{\partial x}=10yz$

$\dfrac{\partial f}{\partial y}=10xz$

$\dfrac{\partial f}{\partial z}=10xy$

Integrate both sides of the first equation with respect to x :

$f(x,y,z)=10xyz+g(y,z)$

Differentiate both sides with respect to y :

$\dfrac{\partial f}{\partial y}=10xz=10xz+\dfrac{\partial g}{\partial y}$

$\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

Differentiate both sides with respect to z :

$\dfrac{\partial f}{\partial z}=10xy=10xy+\dfrac{\mathrm dh}{\mathrm dz}$

$\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

So we have

$f(x,y,z)=10xyz+C$

that satisfies

$\nabla f(x,y,z)=\mathbf F(x,y,z)$

and so F is indeed conservative.