Determine if F is conservative F 2xy1ysec2xy x2xy2sec2xy3y2

Determine if F is conservative F = (2xy+1/ysec2x/y, x2-x/y2sec2x/y-3y2)

Solution

F(x) = (2xy + (1/y)(sec^2 (x/y)) F(y) = x^2 - ((x/y^2) sec^2 (x/y) - 3y^2)) This problem is complicated take the F(x) derivative with respect to x first then with respect to y take the F(y) derivative with respect to y first then with respect to x F\'(x) = (2 y+(2 sec^2(x/y) tan(x/y))/y^2) F\'(x,y) = (2 (y^4-x sec^4(x/y)-2 sec^2(x/y) tan(x/y) (y+x tan(x/y))))/y^4) F\'(y) = (2 (3 y^5+x sec^2(x/y) (y+x tan(x/y))))/y^4) F\'(y,x) = (sec^4(x/y) (4 x^2+y^2+(-2 x^2+y^2) cos((2 x)/y)+4 x y sin((2 x)/y)))/y^5) since F\'(x,y) = F\'(y,x) F is Conservative