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
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"
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