Question

Does there exist a function F (x, y) such that Fx(x, y) = xy, Fy(x,y) = y^2?

Answer 1

Do you need help starting this question?

Answer 2

yes some guildelines will be very helpful

Answer 3

We need to integrate both Fx and Fy and see if we can match up the equations, if we can't the function F(x,y) does not exist.

Answer 4

For Fx
\[F(x,y)=\frac{1}{2}x^2y+F(y)+c\]

For Fy
\[F(x,y)=\frac{1}{3}y^3+F(x)+c\]

Now the function F(y) could be y^3/3 however F(x) cannot be x^2y/2 because if you derive that with respect to y you are left with x^2/2.

So the function F(x,y) does not exist.

Answer 5

We expect the result\[F_{xy}(x,y) = F_{yx}(x,y),\]but this is impossible because \[F_x(x,y) = xy\] and \[F_y(x,y) = y^2\] yields\[(F_x)_y(x,y) =x \ne (F_y)_x(x,y)=0,\]

Answer 6

Good explanation Broken Fixer, very easy to understand and follow :D

Answer 7

Great! thank you very much! Both of the explanations were very helpful!