verify whether the vector field F

verify whether the vector field
F= <-sin(x-y), sin(x-y)-z*e^(-y*z)+1, -y*e^(-y*z)-cos(z)+3*z^2
has a potential function. If a potential function exists, find it.

If F= to have a potential we need to have
dM/dy=dN/dx, dM/dz=dP/dx,dN/dz=dP/dy

dM/dy=cos(x-y)=dN/dx

dM/dz=0=dP/dx

dN/dz=-yze^-yz =dP/dy

Suppose we have a potential g

dg/dx=-sin(x-y)

g=cos(x-y)+C(y,z)

dg/dy=sin(x-y)-z*e^(-y*z)+1. We integrate

g=cos(x-y)+e^-yz+y+C(x,z)

dg/dz=-y*e^(-y*z)-cos(z)+3*z^2. We integrate

g=e^-yz+sin(z)+z³+C(x,y)

We compae them all and we obtain the potential

g=cos(x-y)+e^-yz+y+sin(z)+z³

verify whether the vector field F= <-sin(x-y), sin(x-y)-z*e^(-y*z)+1, -y*e^(-y*z)-cos(z)+3*z^2 has a potential function. If a potential function exists, find

verify whether the vector field F

Solution

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