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taurus [48]
3 years ago
11

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) = 3y2z3 i + 6xyz3 j + 9xy2z2 k
Mathematics
1 answer:
amm18123 years ago
7 0
\dfrac{\partial f}{\partial x}=3y^2z^3\implies f(x,y,z)=3xy^2z^3+g(y,z)

\dfrac{\partial f}{\partial y}=6xyz^3+\dfrac{\partial g}{\partial y}=6xyz^3
\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)

\dfrac{\partial f}{\partial z}=9xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}=9xy^2z^2
\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C

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

The potential function exists, so \mathbf f is indeed conservative.
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