# If X = U V , Y = U V . Prove that J J , = 1 - Applied Mathematics 1

If x=uv, y=u/v."prove that"  jj,=1

#### Solution

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`

Concept: Partial Derivatives of First and Higher Order
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