Question

u=x/y-z,v=y/z-x,w=z/x-y. (u,v,w)/(x,y,z) Jacobian method​

Answer 1

Answer:

If u= x/(y-z), v = y/(z-x), w=z/(x-y)

then uv + vw + uw = ?

uv + vw + uw

= (x/(y - z))(y/(z-x)) + (y/(z-x))(z/(x-y)) + (x/(y-z))(z/(x-y))

= xy/(y - z)(z-x) + yz/(z-x)(x-y) + xz/(y-z)(x-y)

= (xy(x - y) + yz(y-z) + xz(z - x)) / ((x-y)(y-z)(z-x) )

Let solve numerator only

xy(x - y) + yz(y-z) + xz(z - x)

= xy(x - y) + y²z - yz² + xz² - x²z

= xy(x - y) + z²(x - y) -z(x² - y²)

= xy(x - y) + z²(x - y) -z(x + y)(x - y)

= (x - y) ( xy + z² - z(x + y))

= (x - y) ( xy - zx + z² - zy)

=(x - y) (x(y - z) - z(y - z) )

= (x - y) (x - z)(y - z)

= (x - y )(y - z)( -(z - x))

= - (x-y)(y-z)(z-x)

=> uv + vw + uw = - (x-y)(y-z)(z-x) / ((x-y)(y-z)(z-x) )

=> uv + vw + uw = -1