Question

If `x=uv, y=u/v."prove that"  jj,=1`

Answer 1

x=uv                               .............(1)

∴` x_u=(delx)/(delu)=v`        and   ` x_v=(delx)/(delv)=u`  ..........(2)

And,` y=u/v`         ...............(3)

∴` yu=(dely)/(delu)=1/v`            and ` yv=
(dely)/(delv)=u -1/v^2`      ..............(4)

∴ `J = (del(x,y))/(del(u,v))=|[x_u,x_v],[y_u,y_v]|`

=`x_u  y_v x_v  y_u` 

=`v u (-1)/v^2 - u 1/v`                    …(From 2 & 4) 

=` (-u)/v-u/v`

=`(-2u)/v`

∴ `J=-2y 

From (3), u = vy 

Substituting ‘u’ in (1) we get, x= (vy)v 

`x/y=v^2` 

∴ `v=sqrtx/sqrty=x^(1/2)y^(1/2)`

∴` v_x=y^(1/2).x^(-1/2)`    and  `v_y=x^(1/2).-1/2 y^(-3/2)`

From (6) and (7),                      `u= (x^(1/2)  y^(-1/2))y` 

∴ `u=x^(1/2) y^ (1/2)`

∴ `u_x=y^(1/2)`.     and    `u_y=x^ (1/2)1/2  y^(-1/2)`  ......(9)

J'= `(del(u,v))/(del(x,y))=|[u_x,u_y],[v_x,v_y]|`

= `u_xv_y-u_y v_x` 

=`(y^(1/2). 1/2x^(-1/2))(x^(1/2).-1/2 y^(-3/2))-(x^(1/2).1/2 y^(-1/2)) (y^-(1/2).1/2 x^((-1)/2))`      …(From 8 & 9)

= `-1/4 x^(-1/2+1/2).y^(1/2-3/2)  -1/4 x^(1/2-1/2).y^(-1/2-1/2)` 

`(-1)/4.y^-1-1/4.y^-1`

=`-2/4.y^-1` 

∴` J' =-1/(2y)`                     .................(10)

From (5) and (10),                      ` J.J'=-2y. -1/(2y)`

∴  `J.J'=1`