Mathematics: 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

Determine whether or not the vector field is conservative.

Ответ:

Braxtonw875

13.03.2022

$\dfrac{\partial f}{\partial x}=3y^2z^3\implies f(x,y,z)=3xy^2z^3+g(y,z)$

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

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

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

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

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Ответ:

animaemaster

14.12.2021

Mathematics

360 divided by 5 equal pieces equals to 72 a whole circle is 360 degrees

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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

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