Question

For the following functions find the fx, and fyand show that fxy= fyx

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)

$(del^2"f")/(delxdely) = (del^2"f")/(delydelx)$