Question

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `tan^-1 (x/y)`

Answer 1

`(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)`