Mathematics: Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = â‡f. (if the vector field is not conservative, enter dne.) f(x, y, z) = eyzi + xzeyzj + xyeyzk f(x, y, z) = incorrect: your answer is incorrect.

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Determine whether or not the vector field is conservative.

milkshakegrande101

10.03.2022

We have to have

$\dfrac{\partial f}{\partial x}=e^{yz}$

$\dfrac{\partial f}{\partial y}=xze^{yz}$

$\dfrac{\partial f}{\partial z}=xye^{yz}$

Integrate the first PDE with respect to :

$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int e^{yz}\,\mathrm dx\implies f(x,y,z)=xe^{yz}+g(y,z)$

Differentiate with respect to :

$\dfrac{\partial f}{\partial y}=xze^{yz}=xze^{yz}+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

Now differentiate with respect to :

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

So we have

$f(x,y,z)=xe^{yz}+C$

which means is conservative.

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