$Show that the line integral $ \oint_C y z d x + x z d y + x y d z$ is zero along any closed contour C .$

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I need to prove that line integral
$$ \oint_C y z d x + x z d y + x y d z$$ is zero along any closed contour C. I would like to see all the steps and a full explanation. Thanks.

Calculus Multivariable Calculus Integrals

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Answer 1

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Let $F=(yz,xz,xy)$. Then
\[\nabla F=(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})\]
\[=(x-x, y-y, z-z)=(0,0,0).\]
By Stokes' Theorem
\[\oint_Cyzdx+xady+xydz=\int_C F\cdot dr=\iint_D \nabla \times F \cdot dS=\iint 0 \cdot dS=0.\]

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$Show that the line integral $ \oint_C y z d x + x z d y + x y d z$ is zero along any closed contour C .$

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