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Sum

Verify `(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)` for u = x^{3} + 3x^{2} y^{2} + y^{3}.

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#### Solution

Given u = x^{3} + 3x^{2} y^{2} + y^{3}

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") = 3x^{2} + 3y^{2} (2x) + 0 = 3x^{2} + 6xy^{2}`

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)`

Concept: Applications of Partial Derivatives

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