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( u=f(e^{y-z},e^{z-x},e^{x-y}) ext{find} dfrac{partial u}{partial x}+dfrac{partial u}{partial y}+dfrac{partial u}{partial z} )
written 3.9 years ago by |
Answer: Let u1=ey−z,u2=ez−x,u3=ex−y. Then by chain rule,
∂u∂x=ux=∂u∂u1∂u1∂x+∂u∂u2∂u2∂x+∂u∂u3∂u3∂x
=∂u∂u1.0−∂u∂u2.ez−x+∂u∂u3ex−y.
∴∂u∂x=−u2∂u∂u2+u3∂u∂u3...................................(1)
Similarly,
∂u∂y=uy=∂u∂u1∂u1∂y+∂u∂u2∂u2∂y+∂u∂u3∂u3∂y.
=∂u∂u1ey−z+∂u∂u2.0−∂u∂u3ex−y
=u1∂u∂u1−u3∂u∂u3.................................(2)
∂u∂z=uz=∂u∂u1∂u1∂z+∂u∂u2∂u2∂z+∂u∂u3∂u3∂z
=−∂u∂u1ey−z+∂u∂u2ez−x+∂u∂u3.0
=−u1∂u∂u1+u2∂u∂u2...............................(3)
Adding (1), (2) and (3), we get ux+uy+uz=0. Answer.