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Question
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `tan^-1 (x/y)`
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Solution
`(del"f")/(delx) = 1/(1 + x^2/y^2) (1/y) = y/(x^2 + y^2)`
`(del"f")/(dely) = 1/(1 + x^2/y^2) ((-x)/y^2) = (-x)/(x^2 + y^2)`
`(del^2"f")/(delxdely) = del/(delx)[(del"f")/(dely)]`
= `del/(delx) [(-x)/(x^2 + y^2)]`
= `((x^2 + y^2)[- 1] - (- x)[2x])/(x^2 + y^2)^2`
= `(x^2 - y^2)/(x^2 + y^2)^2` ........(1)
`(del^2"f")/(delydelx) = del/(dely) [(del"f")/(delx)]`
= `del/(dely)[y/(x^2 + y^2)]`
= `((x^2 + y^2)[1] - y[2y])/(x^2 + y^2)^2`
= `(x^2 - y^2)/(x^2 + y^2)^2` ..........(2)
From (1) and (2)
⇒ `(del^2"f")/(delxdely) = (del^2"f")/(delydelx)`
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