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enyata [817]
2 years ago
12

Consider the vector field f(x,y,z)=(2z+3y)i+(3z+3x)j+(3y+2x)kf(x,y,z)=(2z+3y)i+(3z+3x)j+(3y+2x)k.

Mathematics
1 answer:
natima [27]2 years ago
6 0
\dfrac{\partial f}{\partial x}=2z+3y\implies f(x,y,z)=2xz+3xy+g(y,z)

\dfrac{\partial f}{\partial y}=3x+\dfrac{\partial g}{\partial y}=3z+3x
\dfrac{\partial g}{\partial y}=3z\implies g(y,z)=3yz+h(z)
\implies f(x,y,z)=3xz+3xy+3yz+h(z)


\dfrac{\partial f}{\partial z}=3x+3y+\dfrac{\mathrm dh}{\mathrm dz}=3y+2x
\dfrac{\mathrm dh}{\mathrm dz}=-x

But we assume h is a function of z alone, so there is no solution for h and hence no scalar function f such that \nabla f=\mathbf f.
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