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Question
If w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`, find `x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)`
Solution
w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`
Convert into exponential function
ew = `((5x^3y^4 + 7y^2xz^4 75y^3z^4)/(x^2 + y^2))` = f(x, y)
`"f"(lambdax, lambday) = (5lambda^3x^3lambda^4y^4 + 7lambda^2y^lambdaxlambda^4z^4 - 75lambda^3y^3lambda^4z^4)/(lambda^2x^2 + lambda^2y^2)`
= `(lambda^7(5x^3y^4 + 7y^2xz^4 - 75y^3y^4))/(lambda^2(x^2 + y^2))`
f is a homogeneous function of degree 5
By Euler's Theorem,
`x (del"f")/(delx) + y (del"f")/(dely) + z (del"f")/(delz)` = 5f
`x del/(delx) + "e"^"w" + y del/(dely) "e"^"w" + z del/(delz) "e"^"w"` = 5ew
`"e"^"w" x (del"w")/(delx) + "e"^"w" y (del"w")/(dely) + "e"^"w" z (del"w")/(delz)` = 5ew
Divided by ew
`x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz) = (5"e"^"w")/"e"^"w"`
`x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)` = 5
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