Question

State a necessary condition for the functional dependence of two differentiable 

functions f and g on an open subset D of .

R² Verify this theorem for the 

functions f and g, defined by f(x,y)=(y-x)/(y+x) and g(x,y) =x/y

Answer 1

Let F be a differentiable transformation. If $f(x,y),g(x,y)$ are functionally dependent, then the Jacobian of F is identically zero.

For the given functions:

$f_x=\frac{-(x+y)-(y-x)}{(x+y)^2}=-\frac{2y}{(x+y)^2}$

$f_y=\frac{(x+y)-(y-x)}{(x+y)^2}=\frac{2x}{(x+y)^2}$

$g_x=\frac{1}{y},\ g_y=-\frac{x}{y^2}$

$\frac{\partial (f,g)}{\partial (x,y)}=f_xg_y-f_yg_x=\frac{2y}{(x+y)^2}\cdot \frac{x}{y^2}-\frac{2x}{(x+y)^2}\cdot \frac{1}{y}=0$