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Question
If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`
Solution
U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`
`(del"U")/(delx) = ((xy)(2x) - (x^2 + y^2)(y))/(xy)^2` + 0
= `(2x^2y - x^2y - y^3)/(xy)^2`
= `(x^2y - y^3)/(xy)^2`
= `(y(x^2 - y^2))/(x^2y^2)`
= `(x^2 - y^2)/(x^2y)`
`(del"U")/(dely) = ((xy)(2y) - (x^2 + y^2)(x))/(xy)^2 + 3z^2`
= `(2xy^2 - x^3 - y^2x)/(xy)^2 + 3z^2`
= `(xy^2 - x^3)/(xy)^2 + 3z^2`
= `(x(y^2 - x^2))/(x^2y^2)`
= `(y^2 - x^2)/(y^2x) + 3z^2`
`(del"U")/(delz)` = 0 + 6zy = 6zy
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