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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

User Parth Modi
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If
\vec F is conservative, then there is a scalar function
f 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
x:


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

Differentiate wrt
y:


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

Differentiate wrt
z:


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.

User Juan Lanus
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