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Question
Verify `(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)` for u = x3 + 3x2 y2 + y3.
Solution
Given u = x3 + 3x2 y2 + y3
Differentiating partially with respect to 'y' we get,
`(del "u")/(del "y") = 0 + 3x^2 (2y) + 3y^2 = 6x^2y + 3y^2`
Differentiating again partially with respect to 'x' we get,
`(del^2 "u")/(del x del "y")` = 6y(2x) + 0 = 12xy ...(1)
Differentiating 'u' partially with respect to 'x' we get,
`(del "u")/(del "x") = 3x2 + 3y2 (2x) + 0 = 3x2 + 6xy2`
Differentiating again partially with respect to 'y' we get,
`(del^2 "u")/(del "y" del x)` = 0 + 6x(2y) = 12xy ....(2)
From (1) and (2)
`(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)`
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